Maximum Likelihood Estimation 

of the Parameters of a Bivariate 

Gaussian–Weibull Distribution 

from Machine Stress-Rated Data

Steve P. Verrill

David E. Kretschmann

James W. Evans

Research Paper

FPL–RP–685

United States Department of Agriculture

Forest 

Service

Forest Products

Laboratory

July

2016

0 1000 2000 3000 4000 5000 6000

0.80 0.85 0.90 0.95

full data CI

.4,.8 data CI

Sample size

Coverage
Abstract

Two important wood properties are stiffness (modulus of 

elasticity, MOE) and bending strength (modulus of rupture, MOR). In the past, MOE has often been modeled as 

a Gaussian and MOR as a lognormal or a two- or threeparameter Weibull. It is well known that MOE and MOR are 

positively correlated. To model the simultaneous behavior of 

MOE and MOR for the purposes of wood system reliability 

calculations, a 2012 paper by Verrill, Evans, Kretschmann, 

and Hatfield introduced a bivariate Gaussian–Weibull 

distribution and the associated pseudo-truncated Weibull. 

In that paper, they obtained an asymptotically efficient 

estimator of the parameter vector of the bivariate Gaussian–

Weibull. This estimator requires data from the full bivariate 

MOE,MOR distribution.

In practice, such data are often not available. Instead, in 

some cases “Machine Stress-Rated” (MSR) data are available. An MSR data set consists of MOE,MOR pairs, where 

a pair is accepted into the data set (a piece of lumber is 

accepted) if and only if the MOE value lies between predetermined lower and upper bounds. For such a data set, the 

asymptotically efficient methods appropriate for a full data 

set cannot be used. In this paper we present an approach that 

is effective for MSR data.

July 2016

Verrill, Steve P.; Kretschmann, David E.; Evans, James W. 2016. Maximum 

likelihood estimation of the parameters of a bivariate Gaussian–Weibull 

distribution from machine stress-rated data. Research Paper FPL-RP-685. 

Madison, WI: U.S. Department of Agriculture, Forest Service, Forest Products Laboratory. 72 p.

A limited number of free copies of this publication are available to the 

public from the Forest Products Laboratory, One Gifford Pinchot Drive, 

Madison, WI 53726–2398. This publication is also available online at 

www.fpl.fs.fed.us. Laboratory publications are sent to hundreds of libraries 

in the United States and elsewhere.

The Forest Products Laboratory is maintained in cooperation with the 

University of Wisconsin. 

The use of trade or firm names in this publication is for reader information 

and does not imply endorsement by the United States Department of 

Agriculture (USDA) of any product or service.

In accordance with Federal civil rights law and U.S. Department of Agriculture (USDA) civil rights regulations and policies, the USDA, its Agencies, offices, 

and employees, and institutions participating in or administering USDA programs are prohibited from discriminating based on race, color, national origin, 

religion, sex, gender identity (including gender expression), sexual orientation, disability, age, marital status, family/parental status, income derived from a 

public assistance program, political beliefs, or reprisal or retaliation for prior civil rights activity, in any program or activity conducted or funded by USDA 

(not all bases apply to all programs). Remedies and complaint filing deadlines vary by program or incident. 

Persons with disabilities who require alternative means of communication for program information (e.g., Braille, large print, audiotape, American Sign Language, etc.) should contact the responsible Agency or USDA’s TARGET Center at (202) 720–2600 (voice and TTY) or contact USDA through the Federal 

Relay Service at (800) 877–8339. Additionally, program information may be made available in languages other than English. 

To file a program discrimination complaint, complete the USDA Program Discrimination Complaint Form, AD-3027, found online at http://www.ascr.usda.

gov/complaint_filing_cust.html and at any USDA office or write a letter addressed to USDA and provide in the letter all of the information requested in 

the form. To request a copy of the complaint form, call (866) 632–9992. Submit your completed form or letter to USDA by: (1) mail: U.S. Department of 

Agriculture, Office of the Assistant Secretary for Civil Rights, 1400 Independence Avenue, SW, Washington, D.C. 20250–9410; (2) fax: (202) 690–7442; or 

(3) email: program.intake@usda.gov.

USDA is an equal opportunity provider, employer, and lender.

Keywords: reliability, modulus of rupture, modulus of elasticity, normal distribution, Weibull distribution, bivariate 

Gaussian–Weibull distribution, bivariate normal–Weibull 

distribution, pseudo-truncated Weibull distribution, likelihood methods, machine stress-rated data, MSR data

Contents

1 Introduction.................................................................... 1

2 Simulations of Gaussian–Weibull Fits........................... 3

3 Web Program to Estimate the Parameters of a 

Bivariate Gaussian–Weibull from MSR Data.................... 10

4 Summary........................................................................ 12

5 References...................................................................... 12

6 Appendix A—Partial Derivatives of log( ftr(x, w))........ 14

7 Appendix B—Obtaining an Estimate of the 

Fifth Percentile of a Pseudo-Truncated Weibull and 

an Approximate One-sided, Lower 75% Confidence 

Bound on the Fifth Percentile ............................................ 16

8 Appendix C—Algorithmic Details of the Web 

Program that Estimates the Parameters of a Bivariate 

Gaussian–Weibull from MSR Data.................................... 18

9 Appendix D—“Kludges” Used in the Optimization 

Programs............................................................................ 19

Tables and Figures............................................................. 22
Maximum Likelihood Estimation of the Parameters of a

Bivariate Gaussian–Weibull Distribution from Machine

Stress-Rated Data

Steve P. Verrill, Mathematical Statistician

David E. Kretschmann, Research General Engineer

James W. Evans, Mathematical Statistician

Forest Products Laboratory, Madison, Wisconsin

1 Introduction

Two important wood properties are stiffness (modulus of elasticity or MOE) and bending strength

(modulus of rupture or MOR). In the past, MOE has often been modeled as a Gaussian and MOR

as a lognormal or a two- or three-parameter Weibull (for example, ASTM 2010a, Evans and Green

1988, Green and Evans 1988).

Design engineers must ensure that the loads to which wood systems are subjected rarely exceed

the systems’ strengths. To this end, ASTM D 2915 (ASTM 2010a), and ASTM D 245 or ASTM

D 1990 (ASTM 2010b,c) describe the manner in which “allowable properties” are assigned to

populations of visually graded structural lumber. In essence, an allowable strength property is

calculated by estimating a fifth percentile of a population (actually a 95% content, lower, 75%

tolerance bound) and then dividing that value by “duration of load” (aging) and safety factors.

The intent is that the population can be used only in applications in which the load does not

exceed the allowable property. Of course there are stochastic issues associated with variable loads,

uncertainty in estimation, and the division of a percentile with no consideration of population

variability. Thus, this is not an ideal approach to ensuring reliability of wood systems. However,

it is the currently codified approach.

To apply this approach, one must obtain estimates of the fifth percentiles of MOR distributions.

Currently, one method for obtaining estimates involves fitting a two-parameter Weibull distribution

to a sample of MORs. To obtain this fit, either a maximum likelihood approach or a linear regression

approach based on order statistics is permitted under ASTM D 5457 (ASTM 2010d).

Unfortunately, these methods are often applied to populations that are not really distributed

as two-parameter Weibulls. For example, in the United States, construction grade 2 by 4’s are

often classified into visual categories—select structural, number 1, number 2—or into machine

stress-rated (MSR) grades. In the case of MSR grades, MOE boundaries are selected, MOE is

measured nondestructively, and boards are placed into categories based upon the MOE bins into

which the boards fall. Because MOE and MOR are correlated, bins with higher MOE boundaries

also tend to contain board populations with higher MOR values. The fifth percentiles of these

MOR populations are sometimes estimated by fitting Weibull distributions to these populations.

Upon reflection, statisticians and reliability engineers recognize that this poses a problem. Even

if the full population of lumber strengths were distributed as a Weibull, we would not expect

that subpopulations formed by visual grades or MOE binning would continue to be distributed as

Weibulls.

In fact, such a subpopulation is not distributed as a Weibull. Instead, if the full joint MOE–

MOR population were distributed as a bivariate Gaussian–Weibull, the subpopulation would be

1
distributed as a “pseudo-truncated Weibull” (PTW). Verrill et al. (2012a) obtained the distribution

of a PTW and showed how to obtain estimates of its parameters by using asymptotically efficient

methods to fit a bivariate Gaussian–Weibull to the full MOE–MOR distribution.

However, currently, full MOE–MOR data sets are generally not available. Instead, one might

have, for example, a sample of number 2 data or a sample of MSR data.

In this paper, we use computer simulations to investigate the properties of maximum likelihood

estimators of the parameters of a bivariate Gaussian–Weibull population (and the corresponding

PTW) that are based on MSR-type data. We note that we do not provide a rigorous mathematical

proof that our estimators are asymptotically efficient. However, our simulation results strongly

suggest that our estimators do indeed have the desired optimality properties.

Our simulation results are described in Section 2. In Section 3 we describe a Web-based computer program that calculates estimates of the parameters of a bivariate Gaussian–Weibull distribution (and thus of a PTW distribution) from MSR-type data.

In the course of performing the computer simulations, we also found that if a full bivariate MOE–

MOR population is truly bivariate Gaussian–Weibull, we can better characterize the properties of

an associated MSR population and the corresponding PTW distribution (and thus probabilities

of failure) by working with a sample of n observations from the full bivariate Gaussian–Weibull

population than by working with n observations from the MSR population. This result might be

counterintuitive to some readers. We discuss this result and try to make it more intuitive in Section

2.1.

As an aside, we remark that the bivariate Gaussian–Weibull distribution has uses other than as

a generator of PTWs. For example, engineers who are interested in simulating the performance of

wood systems must begin with a model for the joint stiffness/strength distribution of the members

of the system. Provided that we are considering the full population, a Gaussian–Weibull is one

possible model for this joint distribution.

Bivariate Gaussian–Weibull distributions first appeared in the literature in Verrill et al. (2012a).

Gumbel (1960), Freund (1961), Marshall and Olkin (1967), Block and Basu (1974), Clayton (1978),

Lee (1979), Hougaard (1986), Sarkar (1987), Lu and Bhattacharyya (1990), Patra and Dey (1999),

Johnson et al. (1999), and others had previously investigated bivariate Weibulls.

We note that the bivariate Gaussian–Weibull distribution discussed in Verrill et al. (2012a,b;

2013; 2014) and the current paper is not the only possible bivariate distribution with Gaussian and

Weibull marginals. In essence we begin with a “Gaussian copula”—a bivariate uniform distribution

generated by starting with a bivariate normal distribution and then applying normal cumulative

distribution functions to its marginals. However, there is a large literature on alternative copulas

(multivariate distributions with uniform marginals). See, for example, Nelsen (1999) and Jaworski

(2010). (Also see Wang et al. (2008) for an application of copulas to joint models of tree heights and

diameters.) These alternatives would lead to alternative bivariate Gaussian–Weibulls. Ultimately,

the test of the usefulness of our proposed version of a Gaussian–Weibull for a particular application

will depend on the match between the theoretical distribution and data. Still, we believe that the

ability to fit the version discussed in the current paper represents a useful step in the construction

and evaluation of bivariate Gaussian–Weibull distributions.

2
2 Simulations of Gaussian–Weibull Fits

In Verrill et al. (2012a) we found that the joint probability density function of the proposed

Gaussian–Weibull was

f(x, w; μ, σ, ρ, γ, β) ≡ γββwβ−1 exp 

−(γw)

β



(1)

×

1

√2π

1

σ

1 − ρ2 exp 

−

x − μ

σ − ρy2

/(2(1 − ρ2))

where x ∈ (−∞, ∞) is the Gaussian value, w > 0 is the Weibull value, μ and σ are the mean and

the standard deviation of the marginal Gaussian, ρ is the generating correlation, γ and β are the

inverse scale and the shape of the marginal two-parameter Weibull (we assumed that β > 1 in our

development),

y = Φ−1



1 − exp −(γ × w)

β

 (2)

and Φ denotes the N(0,1) cumulative distribution function. (In figures 1 – 9 of Verrill et al. (2012a),

we provide contour plots of our version of a bivariate Gaussian–Weibull distribution for coefficients

of variation equal to 0.35, 0.25, and 0.15, and generating correlations equal to 0.5, 0.7, and 0.9.)

In section 4 of the 2012a paper, we noted that the appropriate joint probablity density function

of the corresponding truncated (on the Gaussian) bivariate Gaussian–Weibull distribution is given

by

ftr(x, w; μ, σ, ρ, γ, β) = f(x, w; μ, σ, ρ, γ, β)/ (Φ((cu − μ)/σ) − Φ((cl − μ)/σ)) (3)

for x between cl and cu, and 0 elsewhere. In the MSR case, cl and cu are the (known) limits on the

permitted MOE values.

In section 5 of the 2012a paper, we demonstrated that the univariate probablity density function

of the PTW associated with this truncated (on the Gaussian) bivariate Gaussian–Weibull is

fPTW(w) = d

dwFPTW(w) (4)

= γββwβ−1 exp 

−(γw)

β



×

	

Φ



(cu − μ)/



σ

1 − ρ2



− ρy/1 − ρ2



− Φ



(cl − μ)/



σ

1 − ρ2



− ρy/1 − ρ2


 ÷ [Φ((cu − μ)/σ) − Φ((cl − μ)/σ)]

where

y = Φ−1



1 − exp −(γw)

β



Note that we can characterize a univariate PTW (obtain estimates of its parameters and thus,

for example, its fifth percentile) by obtaining estimates of the (same) parameters of either the

corresponding truncated (on the Gaussian) bivariate Gaussian–Weibull (density given by Equation

3) or the corresponding full bivariate Gaussian–Weibull (density given by Equation 1).

In the 2012a paper we also proved that

√n

⎛

⎜⎜⎜⎜⎝

⎛

⎜⎜⎜⎜⎝

μˆ

σˆ

ρˆ

γˆ

βˆ

⎞

⎟⎟⎟⎟⎠ −

⎛

⎜⎜⎜⎜⎝

μ

σ

ρ

γ

β

⎞

⎟⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎠

D

→ N(0, I(θ)

−1) (5)

3
where n is the sample size, θ ≡ (μ, σ, ρ, γ, β)T , ˆμ and ˆσ are one-step Newton estimators based on the

bivariate Gaussian–Weibull theory (that is, the gradient and Hessian used to calculate the Newton

step correspond to the first and second partials of the full Gaussian–Weibull likelihood) that start at

the standard univariate normal maximum likelihood estimators of the mean and standard deviation

of a Gaussian, ˆγ and βˆ are one-step Newton estimators based on the bivariate Gaussian–Weibull

theory that start at the standard univariate maximum likelihood estimators of 1/scale and shape

for a Weibull, ˆρ is a one-step Newton estimator based on the bivariate Gaussian–Weibull theory

that starts at the √n-consistent estimator of ρ introduced in appendix B of Verrill et al. (2012a),

and the elements of I(θ) are listed in appendix A of Verrill et al. (2012b). The D

→ notation here

means “converges in distribution to.”

Ideally we would like to be able to prove a result similar to (5) when we are working with MSR

data rather than with full bivariate Gaussian–Weibull data. However, to prove (5) we showed (for

example) that our initial parameter estimates were √n-consistent, we calculated the elements of

the information matrix I(θ) exactly, we proved that I(θ) is positive definite, and we were able

to establish various regularity conditions that were needed to invoke theorem 4.2 in chapter 6 of

Lehmann (1983). The proof becomes much more difficult when we are working with truncated

Gaussian and PTW data. Thus, in the current paper we numerically optimize the truncated (on

the Gaussian) bivariate Gaussian–Weibull likelihood function appropriate for MSR data (Equation

3) and establish via simulations that

√n

⎛

⎜⎜⎜⎜⎝

⎛

⎜⎜⎜⎜⎝

μˆ

σˆ

ρˆ

γˆ

βˆ

⎞

⎟⎟⎟⎟⎠ −

⎛

⎜⎜⎜⎜⎝

μ

σ

ρ

γ

β

⎞

⎟⎟⎟⎟⎠

⎞

⎟⎟⎟⎟⎠ ≈ N(0, I(θˆ)

−1) (6)

where θˆ ≡ (ˆμ, σ, ˆ ρ, ˆ γ, ˆ βˆ)T , ˆμ, ˆσ, ˆρ, ˆγ and βˆ are the results of the numerical optimization, and the

j,kth element of I(θˆ) is

−

n

i=1

∂2 log (ftr(xi, wi; θ))

∂θj∂θk

|

θˆ/n (7)

where ftr is the density given by Equation (3), xi, wi is the ith Gaussian–Weibull pair, and n is

the sample size. (That is, I(θˆ) is the approximate information matrix.) The values of the second

partials in Equation (7) are described in Appendix A.

We argue that the approximation in Equation (6) is justified by our simulations. In particular we

show that as sample sizes increase, the biases in our parameter estimates tend to zero. Further, our

simulations demonstrate that as sample sizes increase, the actual coverages of confidence intervals

based on Equation (6) approach the nominal coverages.

In Subsection 2.1 we discuss a non-intuitive fact illustrated by these simulations. In Subsection

2.2 we discuss the “mechanics” of the simulation. In Subsection 2.3 we discuss the results of the

simulation.

2.1 Estimating a pseudo-truncated population from non-truncated data — Intuition

An important conclusion of this paper is that if we want to estimate the parameters and the fifth

percentile of a population of MSR lumber, it is better to fit a sample of n observations from the

corresponding full bivariate population of MOE/MORs than a sample of n observations from the

4
population of MSR MOE/MOR pairs. (This assumes that the full population actually is a bivariate

Gaussian–Weibull.)

This might not be intuitively clear. We will demonstrate that the claim is correct via simulations,

but it is also useful to develop some intuition. This can be done by considering a simple regression

analogy.

Suppose that two variables, y and x, are related by the equation

yi = a + b × xi + 	i

where a and b are constants and the 	i are independent random errors centered at 0. We often

assume that

	 ∼ N(0, σ2)

That is, the random errors are normally distributed and have variance σ2.

Now suppose that, in practice, we are interested only in (x, y) pairs for x’s in the interval

[40, 80] (in analogy to MSR data with the MOE lying between two bounds, cl and cu). To model

the relationship between x and y, we could obtain n (x, y) pairs with the x values lying between 40

and 80, and do a regression fit to obtain estimates of a, b, and σ. Or we could recognize that even

though we will eventually be working just with a population for which x lies between 40 and 80, in

fact, the y ≈ a + b × x relationship holds for x’s beyond the [40,80] range and we know (if we have

taken the necessary statistics course) that we can obtain our best estimate of b by taking half of

our observations at the bottom of the range of x’s for which y ≈ a + b × x holds and half at the top

of that range. That is, even though we will eventually be exclusively working with a population in

which 40 ≤ x ≤ 80, we can obtain a better b estimate to predict things about the x, y relationship

for this population by taking a sample that has x values that lie outside the range appropriate for

the population.

Similarly, in practice we might be interested in an MSR bivariate population for which cl (the

lower MOE cutoff) and cu (the upper MOE cutoff) correspond to, for example, the 40th and 80th

percentiles of the MOE population. In this paper, we refer to this as a .4,.8 MSR population (or

a .4,.8 truncated [on the Gaussian] bivariate Gaussian–Weibull population). From Equation 4, we

can see that to estimate the fifth percentile of the corresponding MOR (a PTW), we need estimates

of the parameters μ, σ, ρ, γ, and β. (Of course, we also need the known cl and cu values.) We can

use maximum likelihood techniques and Equation (3) to obtain estimates of these parameters and

the covariance structure of the estimates (needed to obtain a lower confidence bound on the fifth

percentile of the PTW MOR population) from a sample of the .4,.8 MSR population. However,

alternatively we can use Equation (1) and maximum likelihood techniques to obtain estimates of μ,

σ, ρ, γ, and β, and the covariance structure of the estimates from a sample of the full (untruncated)

bivariate Gaussian–Weibull population.

Just as we can do better in the linear regression example in estimating the b needed to predict

the relationship between x and y for x constrained to lie between 40 and 80 by considering x values

that lie below 40 and above 80, we assert that we can do better in the MSR case by considering

MOE values that lie below cl and above cu. (Of course, this assumes that the full population is

distributed as a bivariate Gaussian–Weibull.) In particular, we do better by drawing a sample from

the whole bivariate Gaussian–Weibull MOE,MOR distribution than by taking a sample of only

MSR data.

We quantify the advantage of the full distribution sample in Subsection 2.3.

5
2.2 Simulation “mechanics”

Each of the 28 simulation runs (4 correlations, 7 sample sizes) involved 10,000 trials. In each of

these simulations the generating μ was set at 100, the generating σ was set at 20 (for coefficients

of variation equal to 0.20), the generating β was set at 5.7974 (for coefficients of variation equal to

0.20), and the generating γ was set at a value that would yield a Weibull median of 100 (given the

β value).

In the optimizations that were performed by the programs on full bivariate Gaussian–Weibull

data sets, no constraints were placed on the μ estimate, the ρ estimate was constrained to lie

within the interval [−1, 1], the σ and γ estimates were constrained to be non-negative, and the β

estimate was constrained to lie within the interval [1, 50.59] (that is, the coefficient of variation was

constrained to lie between 1 and 0.025).

In the optimizations that were performed by the programs on truncated (on one or both ends

of the Gaussian data) bivariate Gaussian–Weibull data sets, the μ estimate was constrained to

vary from its initial estimate by no more than a factor of 2 (it could be halved or doubled), the

σ estimate was constrained to vary from its initial estimate by no more than a factor of 8, the ρ

estimate was constrained to lie within the interval [−1, 1], and the γ and β values were constrained

to be non-negative.

In Appendix D these constraints are discussed in slightly greater detail.

Listings of the computer programs that were used to perform these simulations can be found

at http://www1.fpl.fs.fed.us/sim tr gauss weib.html and

http://www1.fpl.fs.fed.us/sim left tr gauss weib.html.

A single instance of a bivariate Gaussian–Weibull was generated as follows: Obtain independent

N(0,1)’s,



X1, X2, via a Gaussian random number generator. Set X = μ + σX1 and Y = ρX1 +

1 − ρ2X2. Then X is distributed as a N(μ, σ2), Y is distributed as a N(0,1), and their correlation

is ρ. Now let U = Φ(Y ). Then U is a Uniform(0,1) random variable that is correlated with X.

Finally, let W = (− log(1 − U))1/β/γ. Then W is distributed as a Weibull with shape parameter

β and scale parameter 1/γ, and together X and W have our joint bivariate Gaussian–Weibull

distribution.

To simulate MSR data we added an additional step. pl (lower truncation probability level)

and pu (upper truncation probability level) values were chosen to obtain full and MSR data. We

considered the five cases listed in Table 1:

Data type pl pu

full 0.0 1.0

0.1 0.9

MSR 0.2 0.8

0.4 0.8

0.4 1.0

Table 1: pl and pu values

The pl = .4, pu = .8 case was chosen to roughly represent Number 2 data. The pl = .4, pu = 1.0

case was chosen to roughly represent Number 2 and better data.

Given a particular pl, pu combination, we drew X,Y values until we had obtained the necessary

sample size. In the course of the draw, we rejected X values that lay below cl ≡ μ + σ × Φ−1(pl)

or above cu ≡ μ + σ × Φ−1(pu).

We considered two types of confidence intervals on the parameters.

6
1. Simulation-based (sim) confidence intervals:

ˆθ ± z1−α/2 × σsim (8)

2. Asymptotic theory-based confidence intervals:

ˆθ ± z1−α/2 × σth/

√n (9)

Here, θ denotes one of the five parameters (μ, σ, ρ, γ, β). z1−α/2 denotes the appropriate N(0,1)

quantile. (For example, for a 95% confidence interval, α = 0.05 and z1−α/2 = 1.96.) ˆθ denotes the

maximum likelihood estimate of the parameter obtained by maximizing Equation (1) in the full

case or Equation (3) in the truncated cases.

σth denotes the square root of the appropriate element of the estimated asymptotic covariance

matrix (obtained from the inverse of the estimated information matrix — see (7)). σsim is obtained

from the simulation. It is given by

σsim =



N

i=1



ˆθi −  ̄ˆθ

2

/(N − 1) (10)

where N is the number of trials, ˆθi is the estimate of the parameter in the ith simulation trial, and  ̄ˆθ is the average of the N ˆθi’s.

For a given correlation/sample size condition, the actual coverage associated with the sim (for

example) type of confidence interval was the fraction of trials in which ˆθ ± z1−α/2 × σsim included

the true θ value.

2.3 Simulation results

For a 0.20 coefficient of variation (for both the Gaussian and Weibull marginals), generating correlations of 0.5, 0.6, 0.7, and 0.8, (the “generating correlations” are approximately equal to the

observed correlations between the Gaussian random variable and the Weibull random variable—

see table 1 of Verrill et al. (2012b)), and sample sizes of 100, 200, 400, 800, 1600, 3200, and 6400,

we performed simulations that permitted us to produce the following tables:

1. Tables 2,μ – 2,β. These five tables contain simulation estimates of the percent biases and

percent root mean squared errors (rmse) of the parameter estimates. They demonstrate that

both the percent biases and the percent rmses of the parameter estimates decline to very

small values (at least in the full bivariate Gaussian–Weibull case) as sample sizes increase.

They also demonstrate that full data sets yield much better parameter estimates than do .4,.8

data sets.

2. Tables 3,μ – 3,β. These five tables present ratios of the rmses of parameter estimates obtained

from full bivariate–Gaussian data sets with the rmses of parameter estimates obtained from

.1,.9, .2,.8, .4,.8, and .4,1 data sets. The tables also present the inverses of these ratios

squared. These latter values are estimates of the relative sample sizes needed to obtain the

same quality of result. Thus, for example, for μ estimates, the rmses for full data sets of a

given size tend to be roughly 1/5 or less of the rmses of .4,.8 data sets of the same size, and

for a .4,.8 data set to yield estimates of the parameters as good as those obtained from a full

data set, it would generally have to contain 25 or more times as many observations (for μ

estimates).

7
3. Tables 4,75,μ – 4,99,β. These 20 tables provide the actual coverages (in 10,000) trials of

nominal 75%, 90%, 95%, or 99% confidence intervals on the parameters for sample sizes 100,

200, 400, 800, 1600, 3200, and 6400. The larger sample sizes (e.g., 800, 1600, 3200, 6400) were

included in the simulation because the more highly truncated data sets (e.g., a .4,.8 data set)

sometimes require sets this large for actual coverages to closely approach nominal coverages.

In designing an experiment in which one wants to obtain confidence intervals on bivariate

Gaussian–Weibull parameters, one should consult Table 4 to choose an adequate sample size.

However, our Web program provides some protection against inadequate sample sizes at the

analysis stage. It provides nominal 75%, 90%, 95%, and 99% simulation and theoretical confidence intervals on the parameters. However, it also provides on-the-fly simulation estimates

of the actual coverages of these confidence intervals. (Details on the manner in which on-thefly simulation estimates of coverages are calculated are provided in point 7 of Appendix C.)

If these simulation estimates of coverages significantly diverge from the nominal coverages,

then this fact should be reported and the simulation estimates of coverages should be used

rather than the nominal coverages.

We note that it is important to draw a distinction between two types of “needed sample sizes.”

We have been talking about the n’s needed to ensure that we can trust nominal confidence

levels. That is, we want the sample size that ensures that a confidence interval constructed to

cover the true value of a parameter at least 95% (for example) of the time really does cover

the parameter at least 95% of the time. This is distinct from a separate sample size issue.

The separate issue is whether the sample size is large enough to ensure that a confidence

interval is narrow enough or that our ability to detect a difference (statistical power) is large

enough. It is quite possible that the n needed to ensure that actual confidence levels are

close to nominal confidence levels could be as low as 15 (in the full case — see Verrill et al.

(2012b)), while the n needed to ensure that confidence interval widths are sufficiently small

could be much higher than 15. These are two separate issues. We do not consider the second

issue in this paper.

4. Table 5 presents results from a more careful analysis of a portion of the results presented in

Table 4.

For both full and .4,.8 data sets, a nominal 90% confidence level, and ρ values 0.5, 0.6, 0.7,

and 0.8, we used least squares to fit the interpolating model

actual coverage − nominal coverage = a1/

√n + a2/n + a3/n3/2 (11)

to the data. Here n denotes the sample size. Two examples of the data and the associated

fits to the data are given in Figures 1 and 2.

Given these fits, we could calculate the n needed to ensure that actual coverages lay in, for

example, [.88,.92] (the “Wide” case). In this case, if a curve approached the horizontal 0.92

line from above, then we would obtain the needed n by using a nonlinear equation solver to

solve

0.92 = 0.90 + a1/

√n + a2/n + a3/n3/2

for n where a1, a2, and a3 were obtained from the least squares fit of model (11). If a curve

approached the horizontal 0.88 line from below (as do the full and .4,.8 curves in both Figures

1 and 2), then we would obtain the needed n by using a nonlinear equation solver to solve

0.88 = 0.90 + a1/

√n + a2/n + a3/n3/2

8
for n.

It is clear from Tables 4 and 5 that we need much smaller sample sizes to yield good confidence

interval coverages of the parameters when we have a full bivariate Gaussian–Weibull data set

than when we have a .4,.8 truncated (on the Gaussian) bivariate Gaussian–Weibull data set

(that is, a MSR-type data set).

5. Table 6 presents the biases and rmses in estimates of the fifth percentile of a .4,.8 PTW

population from fits of corresponding full bivariate Gaussian–Weibull, and .1,.9, .2,.8, .4,.8,

and .4,1 truncated (on the Gaussian) bivariate Gaussian–Weibull data sets. The manner in

which we obtain estimates of the fifth percentile of a PTW from estimates of its parameters

is described in the first subsection of Appendix B. (Recall from the discussion surrounding

Equations (1), (3), and (4) that we can obtain estimates of the parameters of a univariate

PTW population from fits of data sets from the corresponding full bivariate Gaussian–Weibull

or truncated (on the Gaussian) bivariate Gaussian–Weibull populations.)

It is interesting to note that there is relatively little difference among the full, .1,.9, .2,.8,

.4,.8, and .4,.1 data sets in the qualities of their estimates of the fifth percentile of the .4,.8

population even though there are large differences in the qualities of their estimates of the

population parameters (as displayed in Tables 2 - 5). This illustrates the fact that maximum likelihood estimation focuses on matching predicted probabilities of observations with

observed densities of observations rather than on matching true and estimated parameters.

6. Table 7 presents the actual coverages of one-sided, lower nominal 75% confidence intervals on

the 5th percentile of the PTW associated with a .4,.8 MSR population from fits to full or .1,.9,

.2,.8, .4,.8, .4,1 samples from the corresponding full or truncated (on the Gaussian) Gaussian–

Weibull populations. (The manner in which confidence intervals on the 5th percentile of the

PTW are obtained is described in Appendix B.)

7. Table 8 presents results from a more careful analysis of a portion of the results presented in

Table 7.

For both full and .4,.8 data sets, and for ρ values .5, .6, .7, and .8, we again used least squares

to fit the interpolating model (11) to the data. Two examples of the data and the associated

fits to the data are given in Figures 3 and 4.

Given these fits, we could calculate the n needed to ensure that coverages lay in, for example,

[.73,.77] (the “Wide” case). In this case, if a curve approached the horizontal 0.77 line from

above (as does the .4,.8 curve in Figure 3), then we would obtain the needed n by using a

nonlinear equation solver to solve

0.77 = 0.75 + a1/

√n + a2/n + a3/n3/2

for n where a1, a2, and a3 were obtained from the least squares fit of model (11). That is,

we would find the n at which the upper curved line in Fig. 3 intersected the upper horizontal

line. If a curve approached the horizontal 0.73 line from below (as does the full data curve in

Fig. 4), then we would obtain the needed n by using a nonlinear equation solver to solve

0.73 = 0.75 + a1/

√n + a2/n + a3/n3/2

for n.

9
It is clear from Tables 7 and 8 that we need much smaller sample sizes to yield good confidence

interval coverages on the fifth percentile of the PTW when we have a full bivariate Gaussian–

Weibull data set than when we have a .4,.8 truncated (on the Gaussian) bivariate Gaussian–

Weibull data set.

Tables 7 and 8 also illustrate the fact that fairly large sample sizes are required to ensure that

the actual coverages of nominal 75% confidence intervals on the fifth percentile of the PTW

closely approach the nominal value.

8. Tables 9,.1,.9; 9,.2,.8; 9,.4,.8; 9,.4,1. These four tables are related to tables 6-9 of Verrill et al.

(2013). They assume that we want to estimate the probability, pBr,true that a piece of lumber

drawn from a .4,.8 MSR population (approximately a No. 2 population) will break when it

is subjected to a strain equal to its true fifth percentile divided by 2.1 (approximately, the

population “allowable property”). In the simulations, for each of 10,000 samples we calculated

the ratio of this true probability of breakage to a probability of breakage calculated from

the parameters estimated from the sample. In the four versions of Table 9, we report the

fraction of the time that these ratios lay in the intervals [0,.01], (.01,.02], (.02,.1], (.1,.2],

(.2,.5], (.5,1), (1,2), [2,5), [5,10), [10,50), [50,100), [100,∞). Ideally, these ratios would be

close to 1. Values below 1 indicate overly conservative estimates. Values above 1 indicate

nonconservative estimates.

These tables illustrate the fact that, in general, a full bivariate Gaussian–Weibull data set

yields better estimates of the probability of breakage at the allowable property than do truncated (on the Gaussian) bivariate Gaussian–Weibull data sets. However, the left-truncated

.4,1 data sets yielded estimates of the probability of breakage that were very similar to the

estimates obtained from the full bivariate Gaussian–Weibull data sets. This suggests that

truncated data sets that are truncated only on one side might be an acceptable alternative

to full data sets. We plan to perform an additional set of simulations to investigate this

possibility.

3 Web Program to Estimate the Parameters of a Bivariate Gaussian–

Weibull from MSR Data

We developed a computer program that obtains maximum likelihood estimates of the parameters

of a bivariate Gaussian–Weibull population (and the associated PTW) from a sample of truncated

(on the Gaussian) bivariate Gaussian–Weibull data. That is, it fits MSR-type data. The program

also returns nominal 75%, 90%, 95%, and 99% sim and th (Eq. (8) and (9)) confidence intervals

on the parameters. Finally, it performs simulations to obtain estimates of the actual coverages of

these intervals. Algorithmic details of the program are provided in Appendices C and D. The Web

program can be run at http://www1.fpl.fs.fed.us/fit tr gauss weib.html. The code for a

standalone FORTRAN program that performs these same functions can be found at

http://www1.fpl.fs.fed.us/fit tr gauss weib code.html.

In Figures 5–9 we provide screen shots of the Web program.

A user can choose to fit one of four types of data: full bivariate Gaussian–Weibull data, left and

right truncated (on the Gaussian) bivariate Gaussian–Weibull data, left truncated (on the Gaussian)

bivariate Gaussian–Weibull data, or right truncated (on the Gaussian) bivariate Gaussian–Weibull

data.

Six fields on the web page (five fields in the untruncated case) need to be filled:

10
1. The user must specify a data file name. The data file must be a txt file that contains two

columns. (By default, Notepad and Emacs create txt files. Wordpad and Word can be

directed to create txt files.) The first column must contain the values from the Gaussian

distribution. The second column must contain the corresponding values from the Weibull

distribution. The user must have previously sent the data file by FTP to our Web server.

Directions for doing this are provided at the “provide the data file” link near the top of the

Web page.

2. The user can specify a results file name. Directions for retrieving the results file appear above

the results field on the Web page. In fact, however, the user need not specify a results file

as the results are displayed in tabular form after the execute button is clicked. These results

can be printed or saved to the user’s machine.

3. The user must specify the sample size (n in this paper). Currently, the program cannot

handle sample sizes greater than 6400 observations. If this presents a problem for you, please

contact Steve Verrill at sverrill@fs.fed.us.

4. If the data set is doubly truncated, the user must provide the bounds cl and cu on the Gaussian

(MOE in our applications). If the data set is left truncated, the user must provide the bound

cl. If the data set is right truncated, the user must provide the bound cu.

5. The user must specify the number of trials in the simulation. Currently, this cannot exceed

10000. If this presents a problem for you, please contact Steve Verrill at sverrill@fs.fed.us.

6. The user must provide an integer starting value for the random number generator. This

starting value cannot exceed 231 − 1 = 2147483647.

After filling the six (or five) fields, the user clicks the execute button and the program runs.

Results are then displayed in tabular form. The program produces results for four confidence

levels—75%, 90%, 95%, and 99%. For each confidence level, it produces two tables. The first table

(see, for example, Fig. 7) contains the maximum likelihood estimates of the five parameters together

with simulation-based and asymptotic theory-based confidence intervals on those parameters. The

second table (see, for example, Figs. 7 and 8) contains simulation estimates of the actual coverages

(rather than the nominal 75%, 90%, 95%, or 99%) of the two types of confidence intervals. It

also contains confidence intervals (based on the arcsin square root transformation) on these actual

coverages. If simulation estimates of actual coverages differ significantly from nominal coverages,

then this fact should be reported and the simulation estimates of coverages should be used rather

than the nominal coverages.

Important The response will not be immediate. Because simulations are being run, there will

be a delay before the results appear. An approximate formula for the number of seconds needed

to perform the simulations is (n/2) × N/10000 where n is the sample size and N is the number of

trials. For example, the time needed to run 10000 trials of samples of size 200 is approximately

100 seconds. Given equal sample sizes, trials for less-truncated data sets (e.g., a .1,.9 data set) take

less time to complete than trials for more-truncated data sets (e.g., a .4,.8 data set).

If you encounter problems while running this program, please contact Steve Verrill at sverrill@fs.fed.us

or 608-231-9375. As of September 2014, the program is a beta program. That is, we have tried to

be very careful in its development. However, it might still contain bugs. If you believe that you

have encountered a bug, please contact us.

11
4 Summary

In the context of wood strength modeling, Verrill et al. (2012a) introduced a bivariate Gaussian–

Weibull distribution and the associated PTW distribution. In that paper, we also developed asymptotically efficient estimators of the parameters of the bivariate Gaussian–Weibull.

In this paper, we discuss a maximum likelihood method for fitting truncated (on the Gaussian)

bivariate Gaussian–Weibull data. This method can be used to fit machine stress-rated MOE-MOR

data sets. We describe a Web-based computer program that implements the maximum likelihood

estimation technique. We also discuss computer simulations that investigate the small sample

properties of this technique.

In the course of conducting these computer simulations we found that full bivariate Gaussian–

Weibull data sets of size n yield much better estimates of parameters and probabilities of breakage

at allowable properties than do truncated (on the Gaussian) bivariate Gaussian–Weibull data sets

of the same size. That is, provided that the full bivariate population really is a bivariate Gaussian–

Weibull, it is much more efficient to work with data from the full population than with data from

the truncated (MSR) population even if we are interested in the allowable property of the MOR

from the MSR population.

This implies that it would be very worthwhile to investigate whether full bivariate MOE-MOR

populations (as opposed to visual grade populations or MSR populations) really are bivariate

Gaussian–Weibull. If they are, we should be assigning allowable properties of MSR grades (and,

perhaps, visual grades) based on estimates of the 5th percentiles of (for example) .4,.8 populations

calculated from parameter estimates obtained from fits of full population data sets.

5 References

ASTM. 2010a. Standard practice for evaluating allowable properties for grades of structural

lumber. D 2915-10. West Conshohocken, PA: ASTM International.

ASTM. 2010b. Standard practice for establishing structural grades and related allowable properties for visually graded lumber. D 245-06. West Conshohocken, PA: ASTM International.

ASTM. 2010c. Standard practice for establishing allowable properties for visually-graded dimension lumber from in-grade tests of full-size specimens. D 1990-07. West Conshohocken, PA:

ASTM International.

ASTM. 2010d. Standard specification for computing reference resistence of wood-based materials and structural connections for load and resistance factor design. D 5457-04a. West

Conshohocken, PA: ASTM International.

Block, H.W. and Basu, A.P. (1974). “A Continuous Bivariate Exponential Extension,” J. Amer.

Statist. Assoc., 69, pages 1031–1037.

Clayton, D.G. (1978). “A Model for Association in Bivariate Life Tables and its Application in

Epidemiological Studies of Familial Tendency in Chronic Disease Incidence,” Biometrika, 65,

pages 141–151.

Evans, J.W. and Green, D.W. (1988). Mechanical Properties of Visually Graded Dimension Lumber: Vol 2. Douglas Fir–Larch. Pub. PB-88-159-397. Springfield, VA: National Technical

Information Service. 476 p.

12
Freund, J.E. (1961). “A Bivariate Extension of the Exponential Distribution,” J. Amer. Statist.

Assoc., 56, pages 971–977.

Green, D.W. and Evans, J.W. (1988). Mechanical Properties of Visually Graded Dimension Lumber: Vol 4. Southern Pine. Pub. PB-88-159-413. Springfield, VA: National Technical Information Service. 412 p.

Gumbel, E.J. (1960). “Bivariate Exponential Distributions,” J. Amer. Statist. Assoc., 55, pages

698–707.

Hougaard, P. (1986). “A Class of Multivariate Failure Time Distributions,” Biometrika, 73, pages

671–678.

Jaworski, P. (2010). Copula Theory and Its Applications. Springer, Dordrecht.

Johnson, R.A., Evans, J.W., and Green, D.W. (1999). “Some Bivariate Distributions for Modeling the Strength Properties of Lumber.” Research Paper FPL-RP-575. Madison, WI: U.S.

Department of Agriculture, Forest Service, Forest Products Laboratory. 11 pages.

Lee, L. (1979). “Multivariate Distributions having Weibull Properties,” Journal of Multivariate

Analysis, 9, pages 267–277.

Lehmann, E.L. (1983), Theory of Point Estimation, New York: John Wiley & Sons.

Lu, J.C. and Bhattacharyya, G.K. (1990). “Some New Constructions of Bivariate Weibull Models,” Ann. Inst. Statist. Math., 42, Number 3, pages 543–559.

Marshall, A.W. and Olkin, I. (1967). “A Multivariate Exponential Distribution,” J. Amer. Statist.

Assoc., 62, pages 30–44.

Nelsen, R.B. (1999). An Introduction to Copulas. Springer, New York.

Patra, K. and Dey, D.K. (1999). “A Multivariate Mixture of Weibull Distributions in Reliability

Modeling,” Statistics & Probability Letters, 45, pages 225–235.

Sarkar, S.K. (1987). “A Continuous Bivariate Exponential Distribution,” J. Amer. Statist. Assoc.,

82, pages 667–675.

Verrill, S.P., Evans, J.W., Kretschman, D.E., and Hatfield, C.A. (2012a). “Asymptotically Efficient Estimation of a Bivariate Gaussian–Weibull Distribution and an Introduction to the

Associated Pseudo-truncated Weibull.” USDA Forest Products Laboratory Research Paper

666. 76 pages.

Verrill, S.P., Evans, J.W., Kretschman, D.E., and Hatfield, C.A. (2012b). “Small Sample Properties of Asymptotically Efficient Estimators of the Parameters of a Bivariate Gaussian–Weibull

Distribution.” USDA Forest Products Laboratory Research Paper 667. 46 pages.

Verrill, S.P., Evans, J.W., Kretschman, D.E., and Hatfield, C.A. (2013). “An Evaluation of

a Proposed Revision of the ASTM D 1990 Grouping Procedure.” USDA Forest Products

Laboratory Research Paper 671. 34 pages.

Verrill, S.P., Evans, J.W., Kretschman, D.E., and Hatfield, C.A. (2014). “Reliability Implications

in Wood Systems of a Bivariate Gaussian–Weibull Distribution and the Associated Univariate

Pseudo-truncated Weibull.” ASTM Journal of Testing and Evaluation, 42(2), pages 412-419.

Paper 671. 34 pages.

13
Wang, M., Rennolls, K., and Tang, S. (2008). “Bivariate Distribution Modeling of Tree Diameters

and Heights: Dependency Modeling Using Copulas,” Forest Science, 54(3), pages 284–293.

6 Appendix A—Partial Derivatives of log(ftr(x, w))

Given equality (3), we have

log(ftr(x, w)) = log(f(x, w)) − log (Φ((cu − μ)/σ) − Φ((cl − μ)/σ)) (12)

Thus the partials of the logs of the probability density functions in the full and truncated (on

the Gaussian) cases do not differ if they involve the parameters ρ, γ, or β. These partials are

provided in appendix C of Verrill et al. (2012a).

Partials involving μ and/or σ are equal to the corresponding values in appendix C of Verrill et

al. (2012a) plus the appropriate partials of

T ≡ − log (Φ((cu − μ)/σ) − Φ((cl − μ)/σ)) (13)

The first and second partials of T with respect to μ and σ are provided below.

From (13) we have

∂T

∂μ = φ ((cu − μ)/σ) − φ ((cl − μ)/σ)σ (Φ ((cu − μ)/σ) − Φ ((cl − μ)/σ)) (14)

∂T

∂σ = φ ((cu − μ)/σ) ((cu − μ)/σ) − φ ((cl − μ)/σ) ((cl − μ)/σ)σ (Φ ((cu − μ)/σ) − Φ ((cl − μ)/σ)) (15)

∂2T

∂μ2 = (φ ((cu − μ)/σ) − φ ((cl − μ)/σ))2

σ2 (Φ ((cu − μ)/σ) − Φ ((cl − μ)/σ))2

+ φ ((cu − μ)/σ) ((cu − μ)/σ) − φ ((cl − μ)/σ) ((cl − μ)/σ)

σ2 (Φ ((cu − μ)/σ) − Φ ((cl − μ)/σ)) (16)

∂2T

∂σ2 = −2 (φ ((cu − μ)/σ) ((cu − μ)/σ) − φ ((cl − μ)/σ) ((cl − μ)/σ))σ2 (Φ ((cu − μ)/σ) − Φ ((cl − μ)/σ))

+ (φ ((cu − μ)/σ) ((cu − μ)/σ) − φ ((cl − μ)/σ) ((cl − μ)/σ))2

σ2 (Φ ((cu − μ)/σ) − Φ ((cl − μ)/σ))2

+ φ ((cu − μ)/σ) ((cu − μ)/σ)

3 − φ ((cl − μ)/σ) ((cl − μ)/σ)3

σ2 (Φ ((cu − μ)/σ) − Φ ((cl − μ)/σ)) (17)

∂2T

∂μ∂σ = (φ ((cu − μ)/σ) − φ ((cl − μ)/σ)) (φ ((cu − μ)/σ) ((cu − μ)/σ) − φ ((cl − μ)/σ) ((cl − μ)/σ))

σ2 (Φ ((cu − μ)/σ) − Φ ((cl − μ)/σ))2

+ φ ((cu − μ)/σ) ((cu − μ)/σ)

2 − φ ((cl − μ)/σ) ((cl − μ)/σ)2

σ2 (Φ ((cu − μ)/σ) − Φ ((cl − μ)/σ))

− φ ((cu − μ)/σ) − φ ((cl − μ)/σ)

σ2 (Φ ((cu − μ)/σ) − Φ ((cl − μ)/σ)) (18)

The corresponding partials in the left truncated case (the case in which cu = ∞) are

14
∂T

∂μ = −φ ((cl − μ)/σ)σ (1 − Φ ((cl − μ)/σ)) (19)

∂T

∂σ = −φ ((cl − μ)/σ) ((cl − μ)/σ)σ (1 − Φ ((cl − μ)/σ)) (20)

∂2T

∂μ2 = φ2 ((cl − μ)/σ)

σ2 (1 − Φ ((cl − μ)/σ))2 − φ ((cl − μ)/σ) ((cl − μ)/σ)

σ2 (1 − Φ ((cl − μ)/σ)) (21)

∂2T

∂σ2 = 2φ ((cl − μ)/σ) ((cl − μ)/σ)σ2 (1 − Φ ((cl − μ)/σ)) + φ2 ((cl − μ)/σ) ((cl − μ)/σ)

2

σ2 (1 − Φ ((cl − μ)/σ))2

− φ ((cl − μ)/σ) ((cl − μ)/σ)

3

σ2 (1 − Φ ((cl − μ)/σ)) (22)

∂2T

∂μ∂σ = φ ((cl − μ)/σ)σ2 (1 − Φ ((cl − μ)/σ)) + φ2 ((cl − μ)/σ) ((cl − μ)/σ)

σ2 (1 − Φ ((cl − μ)/σ))2

− φ ((cl − μ)/σ) ((cl − μ)/σ)

2

σ2 (1 − Φ ((cl − μ)/σ)) (23)

The corresponding partials in the right truncated case (the case in which cl = −∞) are

∂T

∂μ = φ ((cu − μ)/σ)σΦ ((cu − μ)/σ) (24)

∂T

∂σ = φ ((cu − μ)/σ) ((cu − μ)/σ)σΦ ((cu − μ)/σ) (25)

∂2T

∂μ2 = φ2 ((cu − μ)/σ)σ2Φ2 ((cu − μ)/σ)

+ φ ((cu − μ)/σ) ((cu − μ)/σ)

σ2Φ ((cu − μ)/σ) (26)

∂2T

∂σ2 = −2φ ((cu − μ)/σ) ((cu − μ)/σ)σ2Φ ((cu − μ)/σ)

+ φ2 ((cu − μ)/σ) ((cu − μ)/σ)

2

σ2Φ2 ((cu − μ)/σ)

+ φ ((cu − μ)/σ) ((cu − μ)/σ)

3

σ2Φ ((cu − μ)/σ) (27)

∂2T

∂μ∂σ = φ2 ((cu − μ)/σ) ((cu − μ)/σ)σ2Φ2 ((cu − μ)/σ)

+ φ ((cu − μ)/σ) ((cu − μ)/σ)

2

σ2Φ ((cu − μ)/σ)

− φ ((cu − μ)/σ)

σ2Φ ((cu − μ)/σ) (28)

15
7 Appendix B — Obtaining an Estimate of the Fifth Percentile

of a Pseudo-Truncated Weibull and an Approximate One-sided,

Lower 75% Confidence Bound on the Fifth Percentile

7.1 Estimation of the 5th percentile

Our simulation program estimates the fifth percentile, ξ.05, by numerically solving the nonlinear

equation

 ξ.05

0

fPTW(w; ˆμ, σ, ˆ ρ, ˆ γ, ˆ β, c ˆ l, cu)dw = 0.05

for ξ.05 where cl and cu are the known truncation values, ˆμ, ˆσ, ˆρ, ˆγ, and βˆ are the maximum

likelihood parameter estimates, and fPTW(w; μ, σ, ρ, γ, β, cl, cu) is the probability density function

of a PTW:

γββwβ−1 exp 

−(γw)

β



(29)

×



Φ



(cu − μ)/



σ

1 − ρ2



− ρy/1 − ρ2



− Φ



(cl − μ)/



σ

1 − ρ2



− ρy/1 − ρ2



÷ (Φ((cu − μ)/σ) − Φ((cl − μ)/σ))

where

y = Φ−1



1 − exp −(γw)

β



(See section 5 of Verrill et al. (2012a) for the derivation of this density.)

7.2 Estimation of the one-sided, lower 75% confidence bound on the 5th percentile

In accordance with ASTM D1990, to calculate the allowable MOR property of a population, we

must obtain a one-sided, lower 75% confidence bound on the fifth percentile of that population.

This can be done via nonparametric methods. However, it is also currently permissible to use a

parametric Weibull assumption. We argued in Verrill et al. (2012a, 2013, 2014) that a parametric

PTW assumption (or an assumption of a mixture of PTWs) would be more appropriate.

To obtain an approximate parametric one-sided, lower 75% confidence bound on the fifth percentile of a PTW, we use the “delta method.” That is we start with the simple linear Taylor series

approximation:

ˆξ.05 ≈ ξ.05 +

∂ξ.05

∂μ

∂ξ.05

∂σ

∂ξ.05

∂ρ

∂ξ.05

∂γ

∂ξ.05

∂β 

|θ

(θˆ − θ)

where

θˆ =

⎛

⎜⎜⎜⎜⎝

μˆ

σˆ

ρˆ

γˆ

βˆ

⎞

⎟⎟⎟⎟⎠

and

θ =

⎛

⎜⎜⎜⎜⎝

μ

σ

ρ

γ

β

⎞

⎟⎟⎟⎟⎠

16
This immediately yields

Var(ˆξ.05) ≈

∂ξ.05

∂μ

∂ξ.05

∂σ

∂ξ.05

∂ρ

∂ξ.05

∂γ

∂ξ.05

∂β 

|θ

×



Cov(θˆ)

 ∂ξ.05

∂μ

∂ξ.05

∂σ

∂ξ.05

∂ρ

∂ξ.05

∂γ

∂ξ.05

∂β T

|θ (30)

In our program, in accordance with maximum likelihood theory, we approximate Cov(θˆ), the

covariance matrix of the parameter vector estimate θˆ, by the inverse of the approximate information

matrix, I(θˆ), where the j,kth element of I(θˆ) is given by (7) in the truncated case, and I(θ) is

given in appendix A of Verrill et al. (2012b) in the full case.

For j = 1,..., 5, our simulation program approximates

∂ξ.05

∂θj

|θ

in Equation (30) by

∂ξ.05

∂θj

|

θˆ (31)

Our simulation program approximates, for example,

∂ξ.05

∂μ |

θˆ

by

(P2 − P1)/h

where

 P1

0

fPTW(w; θˆ, cl, cu)dw = 0.05

(that is, P1 is the fifth percentile when the true parameter vector is θˆ), and

 P2

0

fPTW(w; θˆ + hu1, cl, cu)dw = 0.05

where u1 =  10000 T (that is P2 is the fifth percentile when the true parameter vector

is θˆ + hu1).

Given the fact that (6) is an asymptotic (large sample) result, the fact that we are using an

approximate information matrix, the fact that we are using a two-term Taylor series approximation

to obtain (30), and the fact that we are using a finite difference (of values obtained from numerically

solving equations involving numerically approximated integrals) to estimate (31), it is somewhat

remarkable that we obtain reasonable fifth percentile coverages for (somewhat) reasonable sample

sizes — see Tables 7 and 8.

If there is sufficient interest in one-sided, lower 75% confidence intervals on the fifth percentiles

of PTWs, it might be worthwhile to attempt to reduce these sample sizes further by using a threeterm Taylor series, and by being more careful with our numerical integrations, numerical equation

solving, and numerical estimates of derivatives.

17
8 Appendix C—Algorithmic Details of the Web Program that

Estimates the Parameters of a Bivariate Gaussian–Weibull from

MSR Data

The algorithm for full data sets is described in appendix B of Verrill et al. (2012b).

The program to fit truncated (on the Gaussian) bivariate Gaussian–Weibull data is straightforward. It performs the following tasks:

1. It obtains initital estimates of μ and σ. These are simply the standard univariate estimates—

x ̄ and n

i=1(xi − x ̄)2/n. Note that these are not optimal for univariate truncated data.

However, in practice, we found that we had occasional convergence problems in the bivariate

optimization when we began with univariate, truncated data maximum likelihood estimates of

μ and σ. Thus, we adopted this rough approach to obtaining starting values, and it appeared

to eliminate the convergence problems.

2. Similarly, our initial estimates of γ and β are linear regression estimates that are appropriate

for a true univariate Weibull distribution. These estimates are described in detail in appendix

B of Verrill et al. (2012b) (and in appendix X.2 of ASTM D 5457 (ASTM 2010d)). Again,

these estimates are not optimal for a PTW, but we found that they led to good convergence

behavior.

3. Our initial estimate of ρ is simply the sample correlation coefficient between the corresponding

Gaussian and Weibull values.

4. Given these initial estimates of μ, σ, ρ, γ, and β, we employ the public domain UNCMIN

optimization routine. It makes use of the Newton method modified by a backtracking line

search technique to find the ˆμ, ˆσ, ˆρ, ˆγ, βˆ that yield a maximum of the of the log likelihood

function (based on Eq. 1 or Eq. 3 depending upon whether we have a full or a truncated [on

the Gaussian] bivariate Gaussian–Weibull data set).

5. We calculate the approximate information matrix via expression (7) and the results in Appendix A.

6. We invert the information matrix using the LINPACK routines dpofa and dpodi.

7. We perform a simulation that has two purposes. First, it permits us to calculate the σsim used

to calculate the simulation-based confidence intervals (8). Second, it permits us to estimate

the coverage of both the simulation-based and theory-based (9) confidence intervals.

Let n denote the size of a sample provided by a user of the Web program. From the sample,

the program first obtains maximum likelihood estimates ˆμ, ˆσ, ˆρ, ˆγ, and βˆ as noted in point 4

above. As noted in points 5 and 6 above, it also obtains an approximation to the inverse of

the information matrix. This yields ˆσth for all five parameters. Then it generates N samples

of size n from a truncated (on the Gaussian) bivariate Gaussian–Weibull distribution with

parameters ˆμ, ˆσ, ˆρ, ˆγ, and βˆ. (Here, N is the number of trials specified by the user.) For

the ith generated sample, it obtains estimates (as above) of the five parameters, ˆμˆi, ˆ

σˆi, ˆρˆi,

ˆ

γˆi, and ˆ

βˆi. For each of the five parameters, the program calculates

σˆsim =



N

i=1



ˆˆθi −  ̄ˆˆθ

2

/(N − 1)

18
For all five parameters, the program reports the theory-based confidence intervals

ˆθ ± z1−α/2 × σˆth/

√n

and the simulation-based confidence intervals

ˆθ ± z1−α/2 × σˆsim

It obtains estimates of the coverages of these intervals by going back through the N groups

of ˆμˆi, ˆ

σˆi, ˆρˆi, ˆγˆi, and ˆ

βˆi and calculating the fraction of the time in which (for the theory-based

intervals)

ˆθ ∈ ˆˆθi ± z1−α/2 × σˆth,i/

√n (32)

or (for the simulation-based confidence intervals)

ˆθ ∈ ˆˆθi ± z1−α/2 × σˆsim

Here, a new ˆσth,i is calculated for each of the N samples of size n (see points 5 and 6 above).

9 Appendix D — “Kludges” Used in the Optimization Programs

To ensure that the maximum likelihood programs converged, we placed constraints on the fitted

bivariate Gaussian–Weibull parameters and on the fitted truncated (on the Gaussian) bivariate

Gaussian–Weibull parameters in our computer programs. We do not claim that these constraints

are necessary or optimal. However, in practice, they work. We describe them here. They are

appropriate for the parameter ranges that we have fitted. However, the programs might have to be

modified for other parameter ranges.

The fitbiv.f routine (see http://www1.fpl.fs.fed.us/sim tr gauss weib.html) is used to fit a full

bivariate Gaussian–Weibull distribution. It finds the parameters p1, p2, p3, p4, and p5 that maximize

the likelihood where

μ = p1

σ = p2

2

ρ = sin(p3) (33)

γ = p2

4

β = sin(p5) × 24.795 + 25.795

Thus, σ and γ are constrained to be non-negative, ρ is constrained to lie between -1 and 1, and

β is constrained to lie between 1 and 50.59. That is, the coefficient of variation of the Weibull is

constrained to lie between 0.025 and 1 if we define it as a fraction (2.5% and 100% if we define it

as a percent).

The fitbivtr.f, routine (see http://www1.fpl.fs.fed.us/sim tr gauss weib.html) is used to fit a

truncated (on the Gaussian) bivariate Gaussian–Weibull distribution. It finds the parameters p1,

p2, p3, p4, and p5 that maximize the likelihood where

μ = exp(log(mubase) + sin(p1) × log(2))

σ = exp(log(sigbase) + sin(p2) × log(8))

ρ = sin(p3) (34)

γ = p2

4

β = p2

5

19
Thus, γ and β are constrained to be non-negative, ρ is constrained to lie between -1 and 1,

μ is constrained to lie between 1/2 and 2 times “mubase,” and σ is constrained to lie between

1/8 and 8 times “sigbase.” In the current implementation, “mubase” is just the sample average of

the truncated x values (MOE in the MOE,MOR case) and “sigbase” is just the sample standard

deviation of the truncated x values.

These parameterizations lead to new partials of the log(f(x, w)) values. Let θ1, θ2, θ3, θ4, θ5

denote μ, σ, ρ, γ, β.

We have

∂ log(f(x, w; θ))

∂pi

= ∂ log(f)

∂θi

×

∂θi

∂pi

(35)

∂2 log(f(x, w; θ))

∂p2

i

= ∂2 log(f)

∂θ2

i

×

∂θi

∂pi

2

+

∂ log(f)

∂θi

×

∂2θi

∂p2

i

(36)

and

∂2 log(f(x, w; θ))

∂pi∂pj

= ∂ log(f)

∂θiθj

×

∂θi

∂pi

×

∂θj

∂pj

(37)

The ∂ log(f)

∂θi and ∂2 log(f)∂θi∂θj values can be found in (or calculated from) appendix C of Verrill et al.

(2012a) and Appendix A.

For fitbiv.f the ∂θi

∂pi values are

∂μ

∂p1

= 1

∂σ

∂p2

= 2 × p2

∂ρ

∂p3

= cos(p3) (38)

∂γ

∂p4

= 2 × p4

∂β

∂p5

= cos(p5) × 24.795

For fitbiv.f the ∂2θi

∂p2

i

values are

∂2μ

∂p2

1

= 0

∂2σ

∂p2

2

= 2

∂2ρ

∂p2

3

= − sin(p3) (39)

∂2γ

∂p2

4

= 2

∂2β

∂p2

5

= − sin(p5) × 24.795

20
For fitbivtr.f, the ∂θi

∂pi values are

∂μ

∂p1

= exp(log(mubase) + sin(p1) × log(2)) × cos(p1) × log(2)

∂σ

∂p2

= exp(log(sigbase) + sin(p2) × log(8)) × cos(p2) × log(8)

∂ρ

∂p3

= cos(p3) (40)

∂γ

∂p4

= 2 × p4

∂β

∂p5

= 2 × p5

For fitbivtr.f, the ∂2θi

∂p2

i

values are

∂2μ

∂p2

1

= exp(log(mubase) + sin(p1) × log(2)) × (cos(p1) × log(2))2

− exp(log(mubase) + sin(p1) × log(2)) × sin(p1) × log(2)

∂2σ

∂p2

2

= exp(log(sigbase) + sin(p2) × log(8)) × (cos(p2) × log(8))2

− exp(log(sigbase) + sin(p2) × log(8)) × sin(p2) × log(8)

∂2ρ

∂p2

3

= − sin(p3) (41)

∂2γ

∂p2

4

= 2

∂2β

∂p2

5

= 2

(42)

21
Estimated percent bias of ˆμ Estimated percent rmse of ˆμ

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 0.0 0.2 0.8 -3.0 -1.1 2.0 7.0 14.7 16.7 8.7

200 0.0 0.0 0.3 -2.7 -0.5 1.4 2.6 7.3 12.6 5.6

400 0.0 0.0 0.1 -2.1 -0.2 1.0 1.6 3.2 9.0 3.7

.5 800 0.0 0.0 0.0 -1.3 -0.1 0.7 1.1 1.8 5.7 2.6

1600 0.0 0.0 0.0 -0.6 0.0 0.5 0.8 1.1 3.4 1.8

3200 0.0 0.0 0.0 -0.3 0.0 0.4 0.5 0.8 2.2 1.3

6400 0.0 0.0 0.0 -0.1 0.0 0.2 0.4 0.6 1.5 0.9

100 0.0 0.2 0.8 -2.7 -1.1 2.0 6.2 12.9 16.5 8.5

200 0.0 0.0 0.3 -2.5 -0.5 1.4 2.6 6.3 11.7 5.4

400 0.0 0.0 0.0 -1.5 -0.2 1.0 1.6 3.0 7.6 3.7

.6 800 0.0 0.0 0.0 -0.8 -0.1 0.7 1.1 1.7 4.5 2.6

1600 0.0 0.0 0.0 -0.4 -0.1 0.5 0.8 1.1 2.9 1.8

3200 0.0 0.0 0.0 -0.2 0.0 0.4 0.5 0.8 1.9 1.3

6400 0.0 0.0 0.0 -0.1 0.0 0.2 0.4 0.5 1.3 0.9

100 0.0 0.2 0.5 -2.7 -1.1 2.0 5.4 11.8 15.8 8.4

200 0.0 0.0 0.2 -1.9 -0.5 1.4 2.4 5.5 10.3 5.3

400 0.0 0.0 0.0 -1.2 -0.2 1.0 1.6 2.7 6.6 3.6

.7 800 0.0 0.0 0.0 -0.5 -0.1 0.7 1.1 1.7 3.9 2.5

1600 0.0 0.0 0.0 -0.2 0.0 0.5 0.8 1.1 2.5 1.7

3200 0.0 0.0 0.0 -0.1 0.0 0.4 0.5 0.8 1.7 1.2

6400 0.0 0.0 0.0 -0.1 0.0 0.2 0.4 0.5 1.2 0.9

100 0.0 0.2 0.4 -2.5 -0.8 2.0 4.8 11.1 15.3 7.8

200 0.0 0.0 0.1 -1.8 -0.5 1.4 2.3 5.1 9.9 5.2

400 0.0 0.0 0.1 -1.0 -0.2 1.0 1.6 2.6 6.1 3.5

.8 800 0.0 0.0 0.0 -0.4 -0.1 0.7 1.1 1.6 3.5 2.4

1600 0.0 0.0 0.0 -0.2 0.0 0.5 0.8 1.1 2.3 1.7

3200 0.0 0.0 0.0 -0.1 0.0 0.4 0.5 0.8 1.6 1.2

6400 0.0 0.0 0.0 -0.1 0.0 0.2 0.4 0.5 1.1 0.8

Table 2,μ: Estimated percent biases and rmse’s of ˆμ (from 10,000 trials)
Estimated percent bias of ˆσ Estimated percent rmse of ˆσ

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 -0.8 9.9 25.8 14.2 0.3 7.1 51.2 82.9 68.5 18.1

200 -0.4 3.2 15.7 13.0 0.0 5.0 22.9 57.5 59.5 12.4

400 -0.2 1.1 7.3 9.8 0.0 3.5 12.3 33.7 46.4 8.7

.5 800 -0.1 0.5 3.3 6.6 0.0 2.5 8.0 19.9 33.5 6.1

1600 0.0 0.4 1.4 2.9 -0.1 1.8 5.6 12.4 22.2 4.3

3200 0.0 0.1 0.7 1.7 0.0 1.2 3.9 8.4 15.6 3.1

6400 0.0 0.1 0.5 0.8 0.1 0.9 2.7 5.9 10.6 2.2

100 -0.8 9.0 22.0 14.4 0.2 7.1 46.2 74.7 66.4 18.1

200 -0.4 2.7 11.2 12.3 0.0 4.9 19.8 48.2 55.1 12.3

400 -0.2 1.2 5.5 7.2 0.1 3.5 12.0 29.0 39.8 8.6

.6 800 -0.1 0.6 2.4 3.9 0.0 2.5 8.0 17.2 27.3 6.1

1600 -0.1 0.2 1.2 2.1 0.0 1.8 5.5 11.4 18.8 4.2

3200 0.0 0.1 0.6 0.8 0.0 1.2 3.9 7.8 12.8 3.0

6400 0.0 0.1 0.3 0.5 0.0 0.9 2.7 5.5 8.9 2.1

100 -0.7 6.6 18.2 14.5 0.3 7.0 40.3 67.8 63.9 17.7

200 -0.4 2.6 9.2 9.1 0.1 5.0 18.9 42.2 48.8 12.1

400 -0.2 0.9 4.2 6.3 0.0 3.5 11.6 24.6 35.3 8.5

.7 800 -0.1 0.4 1.9 2.8 0.0 2.5 7.7 15.6 23.6 6.0

1600 -0.1 0.2 0.9 1.2 0.0 1.8 5.4 10.3 15.6 4.1

3200 0.0 0.0 0.7 0.7 0.0 1.2 3.7 7.2 10.9 2.9

6400 0.0 0.0 0.3 0.4 0.0 0.9 2.6 5.0 7.7 2.1

100 -0.9 5.5 16.8 14.4 -0.2 7.0 35.8 64.7 62.2 16.8

200 -0.4 1.7 7.6 9.8 0.0 4.9 16.9 37.2 46.6 11.7

400 -0.2 0.9 3.4 5.3 0.0 3.4 10.9 22.0 31.5 8.1

.8 800 -0.1 0.4 1.4 2.3 -0.1 2.4 7.3 14.0 20.4 5.7

1600 0.0 0.2 0.8 1.3 0.0 1.7 5.2 9.5 13.9 4.1

3200 0.0 0.1 0.4 0.5 0.0 1.2 3.6 6.6 9.7 2.9

6400 0.0 0.1 0.1 0.3 0.0 0.9 2.5 4.6 6.7 2.0

Table 2,σ: Estimated percent biases and rmse’s of ˆσ (from 10,000 trials)
Estimated percent bias of ˆρ Estimated percent rmse of ˆρ

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 -0.4 1.6 2.5 -7.7 -1.7 15.0 27.7 39.6 51.4 23.6

200 -0.1 0.8 2.7 -3.1 -1.0 10.7 18.6 30.1 38.4 16.7

400 -0.2 0.3 1.6 -0.6 -0.4 7.5 12.8 21.6 28.2 11.8

.5 800 -0.1 0.0 0.6 0.3 -0.2 5.2 9.1 15.1 20.8 8.3

1600 0.0 0.1 0.2 0.1 -0.2 3.7 6.4 10.5 15.3 5.8

3200 0.0 -0.1 0.1 0.0 -0.1 2.6 4.5 7.3 11.0 4.2

6400 0.0 0.1 0.1 0.0 0.0 1.9 3.2 5.3 7.7 2.9

100 -0.3 1.2 0.8 -6.0 -1.4 10.7 20.0 28.7 37.6 16.8

200 -0.2 0.2 0.6 -2.7 -0.9 7.6 13.7 21.2 27.7 12.1

400 -0.1 0.1 0.6 -1.3 -0.4 5.3 9.5 15.7 20.6 8.3

.6 800 0.0 0.1 0.3 -0.6 -0.2 3.8 6.6 10.9 14.9 5.9

1600 0.0 0.0 0.1 -0.3 -0.1 2.6 4.7 7.7 10.9 4.1

3200 0.0 0.0 0.2 -0.2 -0.1 1.9 3.3 5.5 7.7 2.9

6400 0.0 0.0 0.1 -0.1 -0.1 1.3 2.3 3.9 5.4 2.1

100 -0.3 0.0 -0.4 -5.0 -1.1 7.3 13.8 20.3 27.2 11.7

200 -0.1 0.2 -0.1 -2.9 -0.5 5.2 9.6 15.1 20.0 8.2

400 -0.1 -0.1 0.1 -1.1 -0.3 3.6 6.8 10.8 14.5 5.7

.7 800 -0.1 0.0 0.0 -0.7 -0.2 2.6 4.7 7.6 10.5 4.1

1600 0.0 0.0 0.0 -0.4 -0.1 1.8 3.3 5.4 7.3 2.8

3200 0.0 0.0 0.1 -0.1 0.0 1.3 2.3 3.8 5.2 2.0

6400 0.0 0.0 0.0 -0.1 0.0 0.9 1.7 2.7 3.7 1.4

100 -0.2 -0.2 -0.9 -4.1 -0.8 4.5 8.7 13.3 18.8 7.3

200 -0.1 -0.2 -0.3 -1.8 -0.4 3.1 6.1 9.5 13.4 5.0

400 -0.1 -0.1 -0.2 -0.9 -0.2 2.2 4.2 6.8 9.5 3.5

.8 800 0.0 0.0 -0.2 -0.4 -0.1 1.6 3.0 4.8 6.6 2.5

1600 0.0 0.0 0.0 -0.2 -0.1 1.1 2.1 3.4 4.6 1.7

3200 0.0 0.0 0.0 -0.1 0.0 0.8 1.5 2.4 3.3 1.2

6400 0.0 0.0 0.0 0.0 0.0 0.6 1.0 1.7 2.3 0.9

Table 2,ρ: Estimated percent biases and rmse’s of ˆρ (from 10,000 trials)
Estimated percent bias of ˆγ Estimated percent rmse of ˆγ

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 0.1 -0.1 -0.3 1.1 0.6 1.8 3.2 5.5 7.7 4.6

200 0.0 0.0 -0.2 0.9 0.2 1.3 1.6 3.1 5.2 2.8

400 0.0 0.0 -0.1 0.6 0.1 0.9 1.1 1.6 3.3 1.9

.5 800 0.0 0.0 0.0 0.4 0.1 0.6 0.8 1.0 2.1 1.3

1600 0.0 0.0 0.0 0.2 0.0 0.5 0.5 0.7 1.3 0.9

3200 0.0 0.0 0.0 0.1 0.0 0.3 0.4 0.5 0.9 0.6

6400 0.0 0.0 0.0 0.0 0.0 0.2 0.3 0.3 0.6 0.5

100 0.1 -0.1 -0.4 1.5 0.7 1.8 3.3 6.0 8.9 5.0

200 0.0 0.0 -0.2 1.0 0.3 1.3 1.7 3.3 5.6 3.1

400 0.0 0.0 -0.1 0.6 0.2 0.9 1.1 1.8 3.4 2.1

.6 800 0.0 0.0 0.0 0.3 0.1 0.6 0.8 1.1 2.1 1.4

1600 0.0 0.0 0.0 0.2 0.0 0.5 0.6 0.7 1.3 1.0

3200 0.0 0.0 0.0 0.1 0.0 0.3 0.4 0.5 0.9 0.7

6400 0.0 0.0 0.0 0.0 0.0 0.2 0.3 0.4 0.6 0.5

100 0.1 -0.1 -0.3 1.7 0.9 1.8 3.5 6.6 9.8 5.6

200 0.0 0.0 -0.2 1.0 0.4 1.3 1.8 3.4 5.9 3.4

400 0.0 0.0 0.0 0.5 0.2 0.9 1.2 1.9 3.5 2.3

.7 800 0.0 0.0 -0.1 0.2 0.1 0.6 0.8 1.2 2.1 1.5

1600 0.0 0.0 0.0 0.1 0.0 0.5 0.6 0.8 1.3 1.1

3200 0.0 0.0 0.0 0.1 0.0 0.3 0.4 0.6 0.9 0.7

6400 0.0 0.0 0.0 0.0 0.0 0.2 0.3 0.4 0.6 0.5

100 0.1 -0.2 -0.4 1.9 0.8 1.8 3.5 7.3 11.1 5.6

200 0.0 0.0 -0.2 1.1 0.4 1.3 1.9 3.7 6.8 3.6

400 0.0 0.0 -0.1 0.6 0.2 0.9 1.3 2.0 4.0 2.3

.8 800 0.0 0.0 -0.1 0.2 0.1 0.6 0.9 1.3 2.2 1.6

1600 0.0 0.0 0.0 0.1 0.0 0.5 0.6 0.9 1.4 1.1

3200 0.0 0.0 0.0 0.1 0.0 0.3 0.4 0.6 1.0 0.8

6400 0.0 0.0 0.0 0.0 0.0 0.2 0.3 0.4 0.7 0.6

Table 2,γ: Estimated percent biases and rmse’s of ˆγ (from 10,000 trials)
Estimated percent bias of βˆ Estimated percent rmse of βˆ

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 1.4 -0.7 -2.7 -0.8 1.2 8.2 12.9 17.5 21.6 13.8

200 0.6 -0.1 -2.2 -1.5 0.7 5.6 8.1 12.8 17.3 9.6

400 0.4 0.0 -1.2 -1.4 0.4 3.9 5.4 8.7 12.9 6.8

.5 800 0.2 0.0 -0.5 -1.2 0.1 2.8 3.7 5.8 9.6 4.8

1600 0.1 0.0 -0.2 -0.5 0.1 2.0 2.6 3.9 6.9 3.4

3200 0.0 0.0 -0.1 -0.3 0.0 1.4 1.9 2.7 4.9 2.4

6400 0.0 0.0 -0.1 -0.1 0.0 1.0 1.3 1.9 3.4 1.7

100 1.4 -0.8 -2.4 -0.1 1.5 8.0 14.1 19.6 24.8 15.1

200 0.7 -0.1 -1.5 -0.8 0.9 5.6 9.0 14.0 19.4 10.6

400 0.3 0.0 -1.0 -0.7 0.4 3.9 6.0 9.9 14.6 7.5

.6 800 0.2 0.0 -0.5 -0.5 0.2 2.8 4.2 6.6 10.6 5.2

1600 0.1 0.0 -0.2 -0.3 0.1 2.0 3.0 4.6 7.7 3.7

3200 0.1 0.0 -0.2 -0.1 0.1 1.4 2.1 3.3 5.4 2.6

6400 0.0 0.0 -0.1 0.0 0.0 1.0 1.5 2.3 3.8 1.9

100 1.4 -0.3 -1.8 1.2 1.9 8.1 15.3 22.3 28.9 16.8

200 0.7 -0.2 -1.2 0.2 1.0 5.6 10.1 16.0 22.1 11.7

400 0.4 0.1 -0.8 -0.3 0.5 3.9 6.9 11.2 16.5 8.2

.7 800 0.2 0.0 -0.3 -0.1 0.2 2.8 4.8 7.7 12.0 5.8

1600 0.1 0.0 -0.1 0.0 0.2 1.9 3.4 5.3 8.4 4.1

3200 0.0 0.0 -0.2 -0.1 0.1 1.4 2.4 3.8 6.0 2.9

6400 0.0 0.0 -0.1 0.0 0.0 0.9 1.7 2.6 4.3 2.1

100 1.4 0.1 -1.1 3.0 2.6 8.1 16.7 25.5 34.2 18.0

200 0.8 0.2 -0.8 0.6 1.1 5.5 11.2 18.0 26.3 12.5

400 0.3 0.1 -0.3 0.1 0.6 3.8 7.7 12.6 19.2 8.7

.8 800 0.2 0.0 -0.1 0.1 0.3 2.7 5.4 8.8 13.6 6.3

1600 0.1 0.0 -0.1 -0.1 0.2 1.9 3.8 6.2 9.6 4.5

3200 0.0 0.0 -0.1 0.0 0.0 1.4 2.7 4.4 6.8 3.1

6400 0.0 0.0 0.0 0.0 0.0 1.0 1.9 3.0 4.7 2.2

Table 2,β: Estimated percent biases and rmse’s of βˆ (from 10,000 trials)
(full rmse)/(tr rmse) (tr n needed)/(full n needed)

ρ n .1,.9 .2,.8 .4,.8 .4,1 .1,.9 .2,.8 .4,.8 .4,1

100 .29 .14 .12 .23 12.0 53.8 69.2 18.2

200 .54 .19 .11 .25 3.4 26.5 79.0 15.7

400 .62 .31 .11 .27 2.6 10.4 79.4 13.9

.5 800 .64 .40 .12 .27 2.4 6.4 66.0 13.4

1600 .65 .44 .15 .28 2.4 5.1 45.8 12.9

3200 .67 .45 .16 .27 2.2 5.0 39.4 13.3

6400 .65 .46 .17 .28 2.4 4.8 34.6 13.2

100 .32 .16 .12 .23 10.0 41.0 68.8 18.4

200 .55 .22 .12 .26 3.4 20.0 66.9 14.6

400 .63 .34 .13 .27 2.6 8.6 56.8 13.9

.6 800 .64 .41 .16 .28 2.4 5.9 40.7 13.1

1600 .66 .44 .17 .28 2.3 5.2 33.9 12.7

3200 .67 .46 .19 .28 2.2 4.8 28.6 12.5

6400 .66 .46 .19 .29 2.3 4.8 26.5 12.2

100 .38 .17 .13 .24 7.1 35.0 62.3 17.7

200 .58 .25 .14 .27 3.0 15.5 53.7 14.2

400 .63 .37 .15 .28 2.5 7.4 43.1 13.2

.7 800 .64 .42 .18 .28 2.4 5.7 31.1 12.9

1600 .66 .44 .20 .28 2.3 5.1 24.3 12.4

3200 .66 .46 .21 .29 2.3 4.8 23.0 11.7

6400 .66 .46 .21 .29 2.3 4.7 21.9 11.8

100 .42 .18 .13 .26 5.8 30.4 58.7 15.4

200 .60 .28 .14 .27 2.7 12.9 49.4 13.2

400 .64 .38 .16 .28 2.4 6.8 37.2 12.5

.8 800 .65 .43 .20 .30 2.4 5.4 25.1 11.4

1600 .65 .45 .21 .29 2.4 5.0 21.7 12.0

3200 .65 .46 .23 .29 2.4 4.8 19.4 11.6

6400 .65 .46 .23 .30 2.3 4.8 18.9 11.2

Table 3,μ: (rmse in the full case) divided by (rmse in the truncated case), and (sample size needed to

obtain a given rmse in the truncated case) divided by (sample size needed to obtain the same rmse in the

full case) (from 10,000 trials)
(full rmse)/(tr rmse) (tr n needed)/(full n needed)

ρ n .1,.9 .2,.8 .4,.8 .4,1 .1,.9 .2,.8 .4,.8 .4,1

100 .14 .09 .10 .39 52.3 135.6 93.3 6.5

200 .22 .09 .08 .40 21.0 131.3 139.2 6.1

400 .28 .11 .08 .41 12.3 90.1 173.7 6.0

.5 800 .31 .13 .07 .40 10.4 62.1 181.7 6.1

1600 .31 .14 .08 .41 10.2 49.3 159.3 6.1

3200 .32 .15 .08 .41 10.0 44.9 155.6 5.9

6400 .32 .15 .08 .40 9.7 45.1 143.1 6.2

100 .15 .09 .11 .39 42.1 111.2 88.4 6.6

200 .25 .10 .09 .41 16.1 92.6 122.9 6.0

400 .29 .12 .09 .42 11.9 67.7 125.8 5.8

.6 800 .31 .14 .09 .41 10.4 47.7 121.7 5.8

1600 .32 .15 .09 .41 9.6 42.4 113.0 5.9

3200 .31 .16 .10 .40 10.1 39.8 104.6 6.1

6400 .33 .16 .10 .42 9.4 39.3 105.2 5.8

100 .17 .10 .11 .40 33.1 94.0 83.5 6.4

200 .26 .12 .10 .41 14.5 72.0 97.8 5.9

400 .30 .14 .10 .42 10.8 50.6 102.6 5.8

.7 800 .32 .16 .10 .41 9.5 39.9 92.0 5.9

1600 .33 .17 .11 .42 9.4 35.0 80.5 5.6

3200 .33 .17 .11 .42 9.1 33.8 79.8 5.7

6400 .33 .17 .11 .42 9.0 33.4 76.0 5.7

100 .19 .11 .11 .41 26.5 86.1 79.5 5.9

200 .29 .13 .11 .41 12.0 56.6 88.6 5.9

400 .32 .16 .11 .43 10.0 38.9 82.9 5.5

.8 800 .33 .18 .12 .43 8.9 32.0 67.5 5.5

1600 .33 .18 .12 .43 9.0 30.3 65.5 5.5

3200 .34 .19 .13 .43 8.8 28.4 62.5 5.4

6400 .34 .19 .13 .43 8.5 27.7 59.9 5.4

Table 3,σ: (rmse in the full case) divided by (rmse in the truncated case), and (sample size needed to

obtain a given rmse in the truncated case) divided by (sample size needed to obtain the same rmse in the

full case) (from 10,000 trials)
(full rmse)/(tr rmse) (tr n needed)/(full n needed)

ρ n .1,.9 .2,.8 .4,.8 .4,1 .1,.9 .2,.8 .4,.8 .4,1

100 .54 .38 .30 .65 3.4 6.8 11.4 2.4

200 .57 .36 .28 .64 3.1 7.9 12.9 2.4

400 .58 .35 .27 .64 2.9 8.4 14.1 2.5

.5 800 .58 .35 .25 .64 3.0 8.2 15.5 2.5

1600 .58 .36 .25 .64 3.0 7.9 16.5 2.4

3200 .59 .36 .24 .64 2.9 7.8 17.4 2.5

6400 .59 .35 .24 .63 2.9 8.0 17.1 2.6

100 .53 .37 .29 .64 3.5 7.1 12.2 2.4

200 .55 .36 .27 .63 3.3 7.7 13.4 2.6

400 .56 .34 .26 .64 3.2 8.8 14.9 2.4

.6 800 .57 .34 .25 .64 3.1 8.4 15.7 2.5

1600 .56 .35 .24 .64 3.2 8.4 17.1 2.4

3200 .57 .34 .24 .63 3.1 8.5 17.0 2.5

6400 .57 .34 .25 .64 3.1 8.6 16.5 2.5

100 .53 .36 .27 .64 3.5 7.6 13.8 2.5

200 .54 .34 .26 .63 3.4 8.8 14.8 2.5

400 .54 .34 .25 .64 3.5 8.8 16.1 2.5

.7 800 .55 .34 .24 .63 3.3 8.8 16.8 2.5

1600 .54 .33 .24 .64 3.5 8.9 16.7 2.5

3200 .54 .34 .24 .63 3.4 8.7 16.8 2.5

6400 .54 .33 .24 .63 3.5 8.9 17.0 2.6

100 .52 .34 .24 .63 3.7 8.6 17.1 2.5

200 .52 .33 .24 .63 3.8 9.1 18.0 2.5

400 .53 .33 .24 .64 3.6 9.3 18.0 2.4

.8 800 .53 .32 .24 .62 3.6 9.5 17.5 2.6

1600 .52 .32 .24 .64 3.7 9.6 17.3 2.5

3200 .53 .32 .24 .64 3.6 9.5 17.8 2.5

6400 .53 .33 .24 .64 3.6 9.2 17.6 2.5

Table 3,ρ: (rmse in the full case) divided by (rmse in the truncated case), and (sample size needed to

obtain a given rmse in the truncated case) divided by (sample size needed to obtain the same rmse in the

full case) (from 10,000 trials)
(full rmse)/(tr rmse) (tr n needed)/(full n needed)

ρ n .1,.9 .2,.8 .4,.8 .4,1 .1,.9 .2,.8 .4,.8 .4,1

100 .58 .33 .24 .39 2.9 9.3 17.8 6.4

200 .78 .42 .24 .45 1.6 5.6 17.0 4.9

400 .84 .57 .28 .48 1.4 3.1 12.9 4.3

.5 800 .86 .65 .31 .49 1.4 2.4 10.5 4.1

1600 .86 .69 .35 .50 1.3 2.1 8.3 4.0

3200 .88 .70 .37 .50 1.3 2.1 7.3 4.0

6400 .86 .69 .39 .50 1.4 2.1 6.7 4.0

100 .55 .31 .20 .37 3.3 10.5 24.2 7.3

200 .74 .40 .23 .43 1.8 6.4 18.8 5.5

400 .80 .52 .26 .43 1.6 3.8 14.5 5.4

.6 800 .79 .60 .31 .45 1.6 2.8 10.3 4.9

1600 .82 .61 .34 .47 1.5 2.7 8.6 4.6

3200 .82 .64 .36 .46 1.5 2.5 7.7 4.7

6400 .82 .63 .38 .47 1.5 2.5 7.1 4.6

100 .52 .28 .19 .32 3.7 13.0 29.0 9.7

200 .71 .37 .21 .38 2.0 7.2 21.8 7.1

400 .75 .49 .26 .40 1.8 4.2 15.0 6.2

.7 800 .76 .53 .30 .41 1.8 3.6 11.0 5.9

1600 .78 .56 .34 .42 1.7 3.2 8.5 5.6

3200 .77 .56 .35 .43 1.7 3.2 8.2 5.3

6400 .76 .58 .36 .43 1.7 3.0 7.8 5.4

100 .52 .25 .16 .33 3.7 15.7 37.1 9.5

200 .67 .35 .19 .37 2.2 8.3 28.7 7.4

400 .71 .45 .23 .38 2.0 5.0 19.1 6.8

.8 800 .71 .48 .29 .40 2.0 4.3 11.9 6.4

1600 .72 .51 .32 .39 1.9 3.9 9.8 6.5

3200 .71 .51 .34 .40 2.0 3.8 8.8 6.2

6400 .72 .52 .35 .41 1.9 3.6 8.4 6.1

Table 3,γ: (rmse in the full case) divided by (rmse in the truncated case), and (sample size needed to

obtain a given rmse in the truncated case) divided by (sample size needed to obtain the same rmse in the

full case) (from 10,000 trials)
(full rmse)/(tr rmse) (tr n needed)/(full n needed)

ρ n .1,.9 .2,.8 .4,.8 .4,1 .1,.9 .2,.8 .4,.8 .4,1

100 .64 .46 .38 .59 2.5 4.7 6.9 2.9

200 .70 .44 .33 .58 2.1 5.1 9.5 3.0

400 .72 .45 .30 .58 1.9 4.9 11.2 3.0

.5 800 .74 .47 .29 .58 1.8 4.4 11.8 3.0

1600 .74 .51 .28 .57 1.8 3.9 12.7 3.0

3200 .74 .50 .28 .58 1.8 3.9 12.8 3.0

6400 .73 .50 .29 .56 1.9 4.0 12.0 3.1

100 .57 .41 .33 .54 3.0 5.9 9.4 3.4

200 .62 .40 .29 .53 2.6 6.3 12.0 3.6

400 .66 .39 .27 .53 2.3 6.5 13.6 3.6

.6 800 .65 .42 .26 .52 2.3 5.8 15.3 3.6

1600 .66 .42 .25 .53 2.3 5.7 15.6 3.6

3200 .65 .42 .25 .52 2.4 5.7 15.6 3.7

6400 .66 .42 .25 .52 2.3 5.6 16.0 3.7

100 .53 .36 .28 .49 3.6 7.5 13.1 4.2

200 .55 .35 .25 .48 3.3 8.2 15.6 4.4

400 .57 .35 .24 .47 3.1 8.2 18.0 4.5

.7 800 .58 .36 .23 .47 3.0 7.7 19.2 4.4

1600 .57 .36 .23 .48 3.1 7.7 19.0 4.4

3200 .57 .37 .23 .48 3.0 7.5 19.0 4.4

6400 .56 .37 .23 .47 3.2 7.5 19.5 4.5

100 .48 .32 .24 .44 4.3 10.0 17.8 5.1

200 .49 .31 .21 .44 4.1 10.6 22.5 5.1

400 .50 .31 .20 .44 4.0 10.3 24.2 5.1

.8 800 .50 .31 .20 .43 3.9 10.3 25.0 5.3

1600 .50 .31 .20 .43 4.1 10.4 25.0 5.4

3200 .51 .31 .20 .43 3.9 10.3 25.9 5.4

6400 .51 .32 .20 .44 3.9 9.9 24.8 5.2

Table 3,β: (rmse in the full case) divided by (rmse in the truncated case), and (sample size needed to

obtain a given rmse in the truncated case) divided by (sample size needed to obtain the same rmse in the

full case) (from 10,000 trials)
Theory-based confidence intervals Simulation-based confidence intervals

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 0.746 0.810 0.904 0.719 0.778 0.751 0.946 0.898 0.836 0.862

200 0.749 0.780 0.868 0.721 0.770 0.750 0.819 0.907 0.860 0.810

400 0.746 0.755 0.827 0.753 0.761 0.750 0.766 0.857 0.872 0.783

.5 800 0.743 0.749 0.781 0.769 0.747 0.744 0.756 0.799 0.870 0.769

1600 0.753 0.748 0.766 0.760 0.750 0.747 0.749 0.767 0.850 0.755

3200 0.747 0.755 0.757 0.746 0.746 0.749 0.752 0.758 0.808 0.750

6400 0.751 0.745 0.750 0.750 0.746 0.750 0.753 0.754 0.777 0.750

100 0.755 0.797 0.901 0.730 0.774 0.750 0.931 0.902 0.835 0.858

200 0.754 0.773 0.856 0.747 0.764 0.749 0.805 0.902 0.859 0.811

400 0.754 0.758 0.808 0.753 0.760 0.749 0.767 0.838 0.872 0.777

.6 800 0.753 0.754 0.778 0.760 0.754 0.749 0.755 0.786 0.848 0.766

1600 0.744 0.751 0.762 0.757 0.756 0.747 0.754 0.765 0.818 0.757

3200 0.744 0.755 0.758 0.751 0.752 0.751 0.753 0.757 0.785 0.754

6400 0.751 0.759 0.759 0.750 0.746 0.751 0.755 0.753 0.762 0.750

100 0.737 0.800 0.897 0.756 0.775 0.750 0.912 0.905 0.839 0.856

200 0.740 0.772 0.849 0.761 0.767 0.746 0.787 0.885 0.864 0.810

400 0.748 0.764 0.793 0.769 0.751 0.747 0.764 0.814 0.870 0.786

.7 800 0.752 0.750 0.770 0.760 0.750 0.749 0.757 0.781 0.840 0.765

1600 0.746 0.750 0.749 0.762 0.747 0.752 0.750 0.764 0.793 0.756

3200 0.752 0.753 0.753 0.754 0.749 0.752 0.754 0.756 0.778 0.750

6400 0.749 0.751 0.747 0.754 0.745 0.747 0.750 0.748 0.765 0.750

100 0.745 0.793 0.887 0.767 0.773 0.750 0.889 0.905 0.845 0.848

200 0.744 0.772 0.834 0.782 0.766 0.748 0.783 0.876 0.867 0.806

400 0.745 0.759 0.792 0.778 0.763 0.750 0.760 0.808 0.865 0.777

.8 800 0.749 0.757 0.771 0.764 0.754 0.746 0.756 0.774 0.821 0.760

1600 0.756 0.753 0.762 0.757 0.739 0.751 0.755 0.761 0.785 0.757

3200 0.756 0.746 0.746 0.750 0.751 0.751 0.750 0.749 0.768 0.757

6400 0.753 0.749 0.752 0.755 0.745 0.751 0.754 0.753 0.755 0.750

Table 4,75,μ: Actual coverages (in 10,000 trials) of nominal 75% confidence intervals on μ
Theory-based confidence intervals Simulation-based confidence intervals

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 0.743 0.783 0.748 0.690 0.764 0.744 0.932 0.851 0.802 0.793

200 0.747 0.793 0.752 0.709 0.755 0.745 0.897 0.872 0.819 0.770

400 0.749 0.768 0.755 0.722 0.758 0.745 0.801 0.875 0.839 0.760

.5 800 0.748 0.760 0.759 0.724 0.749 0.749 0.770 0.838 0.829 0.756

1600 0.750 0.753 0.760 0.733 0.748 0.749 0.761 0.791 0.797 0.751

3200 0.752 0.754 0.756 0.729 0.744 0.747 0.755 0.767 0.777 0.749

6400 0.758 0.759 0.749 0.742 0.745 0.755 0.754 0.758 0.765 0.753

100 0.741 0.793 0.758 0.709 0.751 0.749 0.927 0.860 0.801 0.781

200 0.750 0.782 0.749 0.726 0.750 0.749 0.858 0.880 0.825 0.764

400 0.750 0.764 0.746 0.724 0.756 0.748 0.795 0.866 0.832 0.756

.6 800 0.755 0.760 0.755 0.720 0.745 0.754 0.771 0.810 0.800 0.750

1600 0.748 0.760 0.752 0.735 0.755 0.752 0.762 0.780 0.781 0.752

3200 0.759 0.750 0.751 0.741 0.752 0.755 0.754 0.768 0.760 0.754

6400 0.745 0.755 0.747 0.756 0.750 0.749 0.757 0.755 0.760 0.746

100 0.743 0.779 0.763 0.728 0.754 0.743 0.920 0.867 0.808 0.789

200 0.740 0.773 0.763 0.740 0.757 0.742 0.850 0.874 0.837 0.769

400 0.744 0.752 0.751 0.739 0.745 0.749 0.791 0.839 0.820 0.762

.7 800 0.745 0.755 0.754 0.735 0.739 0.751 0.767 0.797 0.791 0.755

1600 0.744 0.746 0.754 0.744 0.753 0.748 0.756 0.776 0.764 0.750

3200 0.752 0.752 0.756 0.749 0.749 0.750 0.750 0.763 0.760 0.748

6400 0.752 0.752 0.755 0.749 0.745 0.754 0.750 0.757 0.753 0.746

100 0.740 0.783 0.773 0.738 0.755 0.746 0.918 0.874 0.812 0.773

200 0.747 0.760 0.780 0.767 0.754 0.750 0.830 0.872 0.835 0.767

400 0.749 0.756 0.759 0.752 0.754 0.748 0.789 0.826 0.819 0.759

.8 800 0.751 0.760 0.751 0.749 0.750 0.748 0.764 0.788 0.782 0.756

1600 0.746 0.749 0.752 0.755 0.748 0.747 0.757 0.771 0.768 0.755

3200 0.757 0.753 0.747 0.747 0.749 0.753 0.751 0.758 0.755 0.752

6400 0.751 0.747 0.753 0.754 0.752 0.752 0.747 0.752 0.753 0.749

Table 4,75,σ: Actual coverages (in 10,000 trials) of nominal 75% confidence intervals on σ
Theory-based confidence intervals Simulation-based confidence intervals

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 0.744 0.736 0.683 0.639 0.739 0.755 0.770 0.736 0.753 0.755

200 0.740 0.752 0.699 0.643 0.748 0.751 0.760 0.741 0.731 0.759

400 0.747 0.756 0.724 0.684 0.745 0.751 0.756 0.754 0.734 0.753

.5 800 0.756 0.751 0.741 0.708 0.744 0.750 0.752 0.760 0.742 0.749

1600 0.753 0.753 0.750 0.718 0.751 0.753 0.752 0.753 0.742 0.749

3200 0.750 0.752 0.751 0.735 0.746 0.749 0.750 0.753 0.750 0.753

6400 0.753 0.757 0.747 0.748 0.747 0.750 0.751 0.748 0.751 0.751

100 0.744 0.726 0.677 0.640 0.743 0.756 0.759 0.730 0.751 0.758

200 0.743 0.744 0.709 0.671 0.739 0.754 0.757 0.745 0.734 0.752

400 0.747 0.749 0.715 0.696 0.743 0.753 0.753 0.749 0.740 0.742

.6 800 0.748 0.754 0.742 0.713 0.749 0.752 0.752 0.752 0.740 0.751

1600 0.757 0.750 0.742 0.726 0.755 0.755 0.751 0.746 0.747 0.751

3200 0.748 0.749 0.742 0.741 0.745 0.751 0.750 0.749 0.752 0.746

6400 0.751 0.752 0.741 0.749 0.753 0.754 0.750 0.748 0.748 0.753

100 0.741 0.731 0.685 0.648 0.740 0.757 0.759 0.735 0.766 0.764

200 0.744 0.744 0.703 0.688 0.743 0.752 0.755 0.742 0.748 0.755

400 0.749 0.740 0.718 0.714 0.750 0.754 0.751 0.743 0.751 0.754

.7 800 0.739 0.752 0.740 0.726 0.748 0.750 0.752 0.752 0.751 0.756

1600 0.748 0.748 0.747 0.740 0.749 0.749 0.751 0.749 0.750 0.750

3200 0.753 0.752 0.750 0.745 0.751 0.754 0.752 0.752 0.749 0.750

6400 0.751 0.746 0.753 0.747 0.747 0.751 0.751 0.751 0.747 0.751

100 0.742 0.731 0.689 0.663 0.743 0.760 0.759 0.743 0.802 0.771

200 0.747 0.740 0.718 0.694 0.748 0.748 0.758 0.747 0.766 0.761

400 0.741 0.750 0.729 0.725 0.754 0.745 0.755 0.747 0.764 0.757

.8 800 0.748 0.748 0.736 0.734 0.742 0.753 0.749 0.746 0.754 0.751

1600 0.752 0.747 0.741 0.748 0.751 0.751 0.750 0.751 0.755 0.750

3200 0.745 0.750 0.743 0.741 0.752 0.745 0.749 0.749 0.752 0.750

6400 0.752 0.745 0.751 0.752 0.749 0.751 0.747 0.748 0.750 0.747

Table 4,75,ρ: Actual coverages (in 10,000 trials) of nominal 75% confidence intervals on ρ
Theory-based confidence intervals Simulation-based confidence intervals

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 0.740 0.781 0.852 0.828 0.793 0.750 0.883 0.881 0.869 0.861

200 0.754 0.764 0.824 0.799 0.769 0.756 0.784 0.874 0.872 0.813

400 0.745 0.753 0.800 0.795 0.760 0.746 0.755 0.814 0.866 0.780

.5 800 0.743 0.746 0.768 0.789 0.760 0.746 0.750 0.775 0.849 0.768

1600 0.752 0.754 0.758 0.767 0.751 0.748 0.748 0.758 0.820 0.751

3200 0.745 0.748 0.752 0.754 0.752 0.752 0.747 0.755 0.792 0.755

6400 0.750 0.743 0.740 0.759 0.751 0.750 0.748 0.748 0.772 0.752

100 0.748 0.785 0.854 0.824 0.793 0.754 0.871 0.884 0.862 0.864

200 0.748 0.761 0.821 0.805 0.776 0.748 0.780 0.871 0.869 0.810

400 0.745 0.753 0.786 0.781 0.761 0.747 0.754 0.813 0.859 0.782

.6 800 0.760 0.749 0.776 0.773 0.754 0.754 0.749 0.775 0.830 0.767

1600 0.748 0.751 0.758 0.765 0.758 0.750 0.748 0.756 0.803 0.758

3200 0.748 0.754 0.752 0.757 0.750 0.750 0.754 0.752 0.776 0.756

6400 0.755 0.747 0.753 0.755 0.755 0.753 0.744 0.754 0.760 0.756

100 0.737 0.779 0.854 0.827 0.794 0.752 0.873 0.886 0.854 0.872

200 0.739 0.765 0.818 0.813 0.775 0.746 0.771 0.860 0.869 0.827

400 0.750 0.760 0.782 0.801 0.764 0.750 0.756 0.799 0.861 0.791

.7 800 0.749 0.753 0.768 0.773 0.755 0.748 0.756 0.773 0.821 0.770

1600 0.745 0.755 0.753 0.765 0.748 0.750 0.754 0.756 0.785 0.759

3200 0.751 0.746 0.752 0.760 0.749 0.751 0.748 0.755 0.774 0.752

6400 0.758 0.751 0.744 0.753 0.746 0.748 0.749 0.749 0.759 0.749

100 0.740 0.787 0.847 0.838 0.787 0.752 0.858 0.889 0.861 0.864

200 0.743 0.769 0.812 0.825 0.778 0.750 0.778 0.862 0.878 0.818

400 0.741 0.756 0.788 0.807 0.764 0.744 0.759 0.794 0.870 0.783

.8 800 0.751 0.753 0.761 0.778 0.757 0.750 0.752 0.774 0.814 0.766

1600 0.753 0.752 0.755 0.759 0.748 0.750 0.752 0.760 0.784 0.761

3200 0.753 0.742 0.751 0.753 0.753 0.750 0.747 0.755 0.765 0.754

6400 0.751 0.749 0.754 0.755 0.744 0.752 0.751 0.750 0.757 0.747

Table 4,75,γ: Actual coverages (in 10,000 trials) of nominal 75% confidence intervals on γ
Theory-based confidence intervals Simulation-based confidence intervals

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 0.739 0.766 0.754 0.676 0.750 0.751 0.805 0.793 0.757 0.752

200 0.742 0.768 0.750 0.666 0.750 0.751 0.779 0.800 0.761 0.748

400 0.757 0.756 0.750 0.689 0.752 0.760 0.759 0.784 0.760 0.750

.5 800 0.748 0.754 0.754 0.718 0.754 0.750 0.754 0.779 0.766 0.755

1600 0.746 0.746 0.757 0.722 0.749 0.747 0.747 0.759 0.757 0.750

3200 0.743 0.756 0.751 0.731 0.745 0.746 0.755 0.753 0.755 0.745

6400 0.753 0.754 0.748 0.746 0.746 0.749 0.752 0.755 0.753 0.748

100 0.757 0.764 0.735 0.659 0.754 0.759 0.796 0.780 0.740 0.755

200 0.747 0.761 0.740 0.677 0.744 0.749 0.770 0.790 0.742 0.749

400 0.744 0.762 0.736 0.695 0.748 0.750 0.758 0.782 0.748 0.747

.6 800 0.750 0.760 0.751 0.714 0.750 0.753 0.757 0.765 0.746 0.749

1600 0.741 0.759 0.749 0.733 0.758 0.746 0.756 0.756 0.753 0.751

3200 0.750 0.747 0.749 0.740 0.749 0.750 0.748 0.757 0.751 0.751

6400 0.746 0.757 0.741 0.751 0.752 0.747 0.754 0.746 0.749 0.751

100 0.750 0.752 0.715 0.664 0.745 0.758 0.787 0.770 0.718 0.753

200 0.748 0.757 0.729 0.692 0.744 0.752 0.763 0.775 0.731 0.751

400 0.748 0.754 0.733 0.714 0.752 0.753 0.755 0.764 0.743 0.757

.7 800 0.748 0.757 0.752 0.727 0.744 0.752 0.756 0.765 0.748 0.750

1600 0.750 0.747 0.749 0.738 0.743 0.748 0.748 0.751 0.748 0.745

3200 0.750 0.757 0.756 0.743 0.753 0.745 0.753 0.754 0.747 0.752

6400 0.758 0.753 0.754 0.749 0.744 0.748 0.753 0.754 0.749 0.749

100 0.745 0.746 0.706 0.683 0.742 0.755 0.775 0.754 0.710 0.745

200 0.753 0.746 0.732 0.699 0.750 0.755 0.761 0.763 0.728 0.749

400 0.753 0.754 0.737 0.726 0.755 0.752 0.758 0.757 0.749 0.751

.8 800 0.754 0.755 0.743 0.729 0.746 0.751 0.756 0.752 0.747 0.749

1600 0.754 0.739 0.746 0.745 0.745 0.751 0.749 0.749 0.750 0.750

3200 0.750 0.752 0.747 0.744 0.750 0.751 0.754 0.751 0.750 0.748

6400 0.749 0.750 0.753 0.752 0.749 0.750 0.748 0.752 0.751 0.747

Table 4,75,β: Actual coverages (in 10,000 trials) of nominal 75% confidence intervals on β
Theory-based confidence intervals Simulation-based confidence intervals

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 0.891 0.951 0.986 0.796 0.865 0.902 0.972 0.931 0.882 0.929

200 0.897 0.930 0.979 0.801 0.886 0.901 0.930 0.945 0.902 0.924

400 0.896 0.908 0.966 0.832 0.900 0.899 0.903 0.934 0.921 0.917

.5 800 0.898 0.905 0.942 0.861 0.901 0.901 0.901 0.910 0.920 0.907

1600 0.905 0.903 0.921 0.876 0.895 0.902 0.898 0.907 0.924 0.902

3200 0.900 0.905 0.911 0.884 0.897 0.902 0.898 0.902 0.919 0.901

6400 0.903 0.896 0.901 0.889 0.897 0.902 0.900 0.900 0.915 0.902

100 0.897 0.947 0.986 0.809 0.862 0.900 0.967 0.935 0.884 0.927

200 0.900 0.927 0.977 0.827 0.889 0.897 0.923 0.947 0.909 0.920

400 0.903 0.914 0.960 0.836 0.896 0.902 0.905 0.924 0.922 0.916

.6 800 0.905 0.904 0.941 0.866 0.894 0.901 0.900 0.913 0.922 0.908

1600 0.895 0.904 0.917 0.884 0.902 0.899 0.898 0.902 0.923 0.906

3200 0.897 0.904 0.912 0.890 0.900 0.905 0.902 0.901 0.914 0.905

6400 0.900 0.899 0.908 0.901 0.898 0.900 0.899 0.904 0.908 0.899

100 0.889 0.947 0.984 0.828 0.867 0.900 0.962 0.938 0.887 0.925

200 0.896 0.930 0.975 0.839 0.888 0.899 0.918 0.941 0.916 0.925

400 0.902 0.913 0.953 0.864 0.899 0.900 0.901 0.915 0.926 0.915

.7 800 0.901 0.902 0.931 0.881 0.896 0.900 0.901 0.907 0.923 0.909

1600 0.896 0.904 0.910 0.894 0.899 0.902 0.902 0.900 0.919 0.903

3200 0.900 0.903 0.907 0.896 0.901 0.900 0.900 0.901 0.913 0.901

6400 0.900 0.901 0.902 0.896 0.899 0.899 0.899 0.902 0.906 0.905

100 0.894 0.944 0.982 0.843 0.868 0.900 0.959 0.941 0.892 0.931

200 0.898 0.926 0.974 0.859 0.894 0.901 0.913 0.944 0.919 0.921

400 0.894 0.910 0.950 0.879 0.900 0.898 0.902 0.913 0.928 0.911

.8 800 0.901 0.912 0.923 0.888 0.898 0.900 0.906 0.903 0.923 0.907

1600 0.900 0.903 0.911 0.896 0.896 0.897 0.898 0.900 0.914 0.908

3200 0.901 0.896 0.905 0.899 0.898 0.900 0.899 0.902 0.911 0.902

6400 0.901 0.897 0.905 0.904 0.897 0.900 0.899 0.904 0.908 0.900

Table 4,90,μ: Actual coverages (in 10,000 trials) of nominal 90% confidence intervals on μ
Theory-based confidence intervals Simulation-based confidence intervals

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 0.894 0.873 0.824 0.758 0.883 0.902 0.958 0.893 0.863 0.922

200 0.896 0.894 0.853 0.800 0.894 0.900 0.951 0.916 0.878 0.909

400 0.902 0.903 0.875 0.836 0.901 0.899 0.924 0.925 0.898 0.907

.5 800 0.900 0.907 0.889 0.860 0.900 0.903 0.908 0.925 0.911 0.904

1600 0.899 0.904 0.897 0.874 0.896 0.899 0.906 0.918 0.917 0.900

3200 0.902 0.903 0.901 0.876 0.902 0.900 0.902 0.908 0.914 0.901

6400 0.904 0.909 0.899 0.888 0.897 0.901 0.905 0.906 0.907 0.901

100 0.886 0.879 0.832 0.776 0.875 0.899 0.957 0.901 0.865 0.921

200 0.899 0.890 0.854 0.818 0.890 0.899 0.936 0.923 0.882 0.910

400 0.901 0.900 0.879 0.846 0.897 0.899 0.920 0.926 0.910 0.905

.6 800 0.898 0.903 0.889 0.863 0.897 0.897 0.913 0.922 0.919 0.904

1600 0.896 0.900 0.898 0.881 0.901 0.899 0.905 0.912 0.913 0.901

3200 0.899 0.896 0.900 0.883 0.898 0.896 0.902 0.906 0.905 0.900

6400 0.897 0.901 0.897 0.895 0.903 0.900 0.900 0.904 0.903 0.902

100 0.890 0.872 0.837 0.792 0.882 0.901 0.953 0.910 0.868 0.912

200 0.895 0.893 0.864 0.832 0.893 0.900 0.936 0.926 0.902 0.908

400 0.894 0.898 0.884 0.869 0.898 0.898 0.920 0.925 0.916 0.908

.7 800 0.895 0.899 0.893 0.880 0.894 0.900 0.911 0.917 0.917 0.902

1600 0.899 0.896 0.901 0.893 0.899 0.903 0.905 0.909 0.909 0.900

3200 0.903 0.900 0.902 0.902 0.901 0.901 0.902 0.904 0.905 0.903

6400 0.899 0.901 0.902 0.902 0.902 0.901 0.903 0.904 0.904 0.903

100 0.889 0.871 0.842 0.809 0.880 0.895 0.953 0.915 0.872 0.918

200 0.897 0.890 0.875 0.851 0.893 0.900 0.929 0.926 0.903 0.911

400 0.903 0.902 0.893 0.884 0.897 0.899 0.917 0.923 0.915 0.902

.8 800 0.900 0.903 0.894 0.891 0.896 0.899 0.907 0.915 0.914 0.902

1600 0.900 0.897 0.901 0.898 0.897 0.902 0.904 0.908 0.908 0.901

3200 0.906 0.903 0.899 0.896 0.901 0.902 0.904 0.904 0.905 0.898

6400 0.900 0.903 0.902 0.902 0.899 0.900 0.901 0.899 0.902 0.899

Table 4,90,σ: Actual coverages (in 10,000 trials) of nominal 90% confidence intervals on σ
Theory-based confidence intervals Simulation-based confidence intervals

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 0.893 0.878 0.820 0.785 0.891 0.904 0.900 0.890 0.929 0.904

200 0.892 0.901 0.839 0.793 0.896 0.901 0.909 0.892 0.912 0.903

400 0.897 0.899 0.864 0.825 0.897 0.900 0.900 0.900 0.900 0.903

.5 800 0.902 0.900 0.887 0.852 0.899 0.900 0.902 0.902 0.897 0.903

1600 0.903 0.898 0.893 0.864 0.899 0.902 0.899 0.898 0.899 0.899

3200 0.900 0.904 0.901 0.876 0.894 0.898 0.901 0.901 0.899 0.896

6400 0.900 0.904 0.898 0.893 0.898 0.898 0.900 0.901 0.898 0.901

100 0.890 0.875 0.818 0.782 0.890 0.900 0.901 0.894 0.920 0.902

200 0.889 0.889 0.850 0.813 0.889 0.900 0.902 0.896 0.908 0.904

400 0.894 0.897 0.862 0.835 0.901 0.901 0.900 0.900 0.904 0.905

.6 800 0.896 0.901 0.886 0.857 0.902 0.902 0.903 0.900 0.902 0.901

1600 0.900 0.902 0.895 0.874 0.900 0.899 0.901 0.899 0.900 0.900

3200 0.897 0.897 0.895 0.886 0.902 0.901 0.899 0.900 0.902 0.901

6400 0.896 0.903 0.892 0.896 0.904 0.897 0.901 0.898 0.902 0.903

100 0.893 0.878 0.827 0.789 0.891 0.907 0.898 0.904 0.908 0.903

200 0.894 0.889 0.848 0.825 0.895 0.902 0.900 0.900 0.910 0.904

400 0.900 0.893 0.870 0.854 0.902 0.903 0.901 0.902 0.905 0.904

.7 800 0.893 0.896 0.888 0.869 0.900 0.902 0.899 0.903 0.898 0.903

1600 0.901 0.895 0.896 0.888 0.898 0.899 0.898 0.899 0.901 0.899

3200 0.898 0.899 0.899 0.895 0.904 0.900 0.898 0.900 0.898 0.901

6400 0.900 0.896 0.898 0.894 0.895 0.899 0.898 0.896 0.899 0.898

100 0.891 0.877 0.826 0.803 0.894 0.904 0.901 0.915 0.902 0.908

200 0.898 0.886 0.864 0.834 0.899 0.903 0.900 0.903 0.915 0.905

400 0.899 0.896 0.874 0.863 0.903 0.904 0.902 0.899 0.904 0.900

.8 800 0.901 0.899 0.888 0.882 0.893 0.904 0.901 0.898 0.904 0.903

1600 0.901 0.894 0.896 0.894 0.898 0.900 0.898 0.900 0.901 0.900

3200 0.899 0.902 0.894 0.892 0.902 0.900 0.901 0.900 0.903 0.903

6400 0.901 0.902 0.901 0.901 0.902 0.900 0.901 0.901 0.902 0.900

Table 4,90,ρ: Actual coverages (in 10,000 trials) of nominal 90% confidence intervals on ρ
Theory-based confidence intervals Simulation-based confidence intervals

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 0.890 0.931 0.954 0.913 0.890 0.897 0.958 0.926 0.913 0.930

200 0.901 0.916 0.949 0.896 0.900 0.900 0.914 0.932 0.919 0.924

400 0.895 0.907 0.938 0.888 0.901 0.900 0.903 0.924 0.922 0.912

.5 800 0.897 0.901 0.922 0.899 0.901 0.899 0.901 0.908 0.922 0.910

1600 0.901 0.904 0.911 0.898 0.895 0.898 0.899 0.901 0.919 0.901

3200 0.896 0.903 0.903 0.899 0.899 0.901 0.902 0.903 0.914 0.900

6400 0.896 0.896 0.895 0.900 0.900 0.897 0.898 0.899 0.908 0.901

100 0.899 0.931 0.954 0.910 0.885 0.903 0.952 0.925 0.909 0.933

200 0.897 0.910 0.949 0.898 0.897 0.901 0.908 0.938 0.920 0.921

400 0.898 0.908 0.935 0.888 0.903 0.901 0.900 0.919 0.921 0.915

.6 800 0.905 0.901 0.926 0.898 0.902 0.900 0.901 0.908 0.920 0.907

1600 0.898 0.902 0.908 0.899 0.907 0.900 0.898 0.904 0.916 0.905

3200 0.895 0.903 0.908 0.901 0.903 0.899 0.901 0.904 0.911 0.904

6400 0.901 0.900 0.902 0.900 0.897 0.899 0.898 0.902 0.904 0.899

100 0.891 0.927 0.960 0.917 0.880 0.902 0.954 0.929 0.907 0.930

200 0.896 0.918 0.951 0.907 0.898 0.898 0.906 0.933 0.923 0.929

400 0.902 0.908 0.931 0.905 0.904 0.900 0.899 0.914 0.928 0.914

.7 800 0.902 0.902 0.919 0.905 0.897 0.899 0.903 0.904 0.923 0.910

1600 0.899 0.903 0.909 0.909 0.896 0.902 0.899 0.899 0.913 0.906

3200 0.899 0.896 0.902 0.899 0.900 0.898 0.900 0.902 0.909 0.904

6400 0.907 0.904 0.899 0.900 0.899 0.901 0.905 0.901 0.904 0.905

100 0.892 0.933 0.953 0.925 0.871 0.900 0.948 0.934 0.909 0.928

200 0.896 0.918 0.942 0.920 0.900 0.899 0.910 0.938 0.929 0.924

400 0.898 0.912 0.932 0.915 0.906 0.901 0.902 0.914 0.932 0.915

.8 800 0.901 0.907 0.917 0.913 0.899 0.899 0.903 0.905 0.922 0.910

1600 0.902 0.905 0.905 0.907 0.896 0.900 0.903 0.904 0.910 0.907

3200 0.898 0.897 0.901 0.909 0.899 0.896 0.901 0.900 0.905 0.901

6400 0.903 0.898 0.908 0.909 0.896 0.904 0.900 0.906 0.906 0.903

Table 4,90,γ: Actual coverages (in 10,000 trials) of nominal 90% confidence intervals on γ
Theory-based confidence intervals Simulation-based confidence intervals

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 0.900 0.912 0.899 0.826 0.901 0.901 0.919 0.899 0.901 0.900

200 0.897 0.915 0.896 0.815 0.899 0.899 0.912 0.906 0.904 0.899

400 0.901 0.906 0.895 0.835 0.903 0.899 0.899 0.906 0.907 0.902

.5 800 0.897 0.903 0.902 0.854 0.901 0.898 0.899 0.903 0.907 0.900

1600 0.902 0.901 0.905 0.869 0.898 0.901 0.901 0.903 0.902 0.899

3200 0.900 0.901 0.903 0.877 0.900 0.901 0.899 0.905 0.902 0.900

6400 0.904 0.899 0.898 0.895 0.896 0.901 0.899 0.901 0.900 0.898

100 0.903 0.910 0.879 0.801 0.899 0.898 0.915 0.897 0.904 0.899

200 0.901 0.911 0.883 0.820 0.898 0.903 0.909 0.909 0.907 0.896

400 0.899 0.913 0.884 0.840 0.899 0.903 0.903 0.912 0.905 0.900

.6 800 0.898 0.901 0.895 0.862 0.901 0.897 0.899 0.903 0.905 0.899

1600 0.896 0.899 0.896 0.874 0.900 0.898 0.898 0.900 0.900 0.896

3200 0.901 0.899 0.897 0.884 0.900 0.901 0.903 0.901 0.901 0.900

6400 0.899 0.903 0.895 0.896 0.894 0.899 0.899 0.899 0.900 0.896

100 0.903 0.904 0.862 0.804 0.886 0.904 0.914 0.901 0.905 0.897

200 0.899 0.904 0.868 0.831 0.897 0.898 0.904 0.905 0.899 0.902

400 0.898 0.903 0.884 0.854 0.898 0.900 0.901 0.905 0.900 0.900

.7 800 0.898 0.901 0.893 0.868 0.898 0.899 0.899 0.905 0.901 0.902

1600 0.899 0.897 0.898 0.887 0.903 0.899 0.899 0.903 0.900 0.903

3200 0.903 0.902 0.900 0.896 0.901 0.901 0.901 0.901 0.904 0.900

6400 0.908 0.898 0.901 0.898 0.900 0.901 0.898 0.900 0.899 0.903

100 0.897 0.896 0.853 0.815 0.895 0.896 0.909 0.898 0.903 0.895

200 0.900 0.900 0.875 0.842 0.900 0.898 0.901 0.905 0.894 0.900

400 0.902 0.898 0.888 0.866 0.902 0.897 0.900 0.906 0.896 0.897

.8 800 0.902 0.900 0.890 0.883 0.898 0.898 0.899 0.902 0.901 0.901

1600 0.902 0.900 0.894 0.892 0.895 0.900 0.905 0.899 0.897 0.898

3200 0.899 0.903 0.898 0.896 0.903 0.900 0.903 0.899 0.903 0.903

6400 0.897 0.901 0.900 0.902 0.903 0.898 0.900 0.898 0.901 0.899

Table 4,90,β: Actual coverages (in 10,000 trials) of nominal 90% confidence intervals on β
Theory-based confidence intervals Simulation-based confidence intervals

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 0.945 0.983 0.996 0.836 0.904 0.949 0.978 0.944 0.903 0.947

200 0.946 0.974 0.995 0.839 0.918 0.949 0.964 0.957 0.923 0.951

400 0.947 0.959 0.992 0.864 0.938 0.950 0.950 0.956 0.938 0.954

.5 800 0.950 0.954 0.983 0.894 0.945 0.951 0.951 0.946 0.941 0.952

1600 0.952 0.955 0.971 0.910 0.943 0.951 0.951 0.947 0.945 0.951

3200 0.949 0.952 0.959 0.923 0.948 0.950 0.948 0.950 0.946 0.953

6400 0.951 0.948 0.949 0.936 0.950 0.950 0.950 0.948 0.951 0.950

100 0.949 0.980 0.998 0.847 0.898 0.950 0.975 0.947 0.906 0.947

200 0.950 0.971 0.993 0.861 0.924 0.950 0.954 0.961 0.929 0.950

400 0.954 0.963 0.991 0.871 0.934 0.952 0.947 0.950 0.940 0.953

.6 800 0.951 0.956 0.982 0.903 0.940 0.950 0.950 0.948 0.945 0.952

1600 0.951 0.953 0.966 0.920 0.950 0.952 0.949 0.947 0.948 0.950

3200 0.948 0.955 0.960 0.931 0.951 0.951 0.950 0.947 0.950 0.949

6400 0.948 0.951 0.955 0.944 0.948 0.948 0.948 0.951 0.952 0.950

100 0.943 0.981 0.997 0.861 0.904 0.950 0.975 0.952 0.908 0.946

200 0.949 0.975 0.995 0.876 0.924 0.951 0.954 0.959 0.936 0.953

400 0.951 0.962 0.988 0.897 0.939 0.952 0.949 0.948 0.946 0.949

.7 800 0.952 0.956 0.974 0.917 0.943 0.950 0.950 0.944 0.948 0.951

1600 0.948 0.955 0.962 0.934 0.948 0.950 0.951 0.946 0.952 0.953

3200 0.947 0.952 0.956 0.939 0.948 0.949 0.949 0.951 0.952 0.952

6400 0.949 0.951 0.956 0.942 0.951 0.949 0.951 0.951 0.952 0.956

100 0.945 0.981 0.995 0.878 0.903 0.949 0.975 0.954 0.914 0.949

200 0.947 0.971 0.994 0.896 0.930 0.950 0.952 0.961 0.939 0.951

400 0.946 0.962 0.987 0.910 0.942 0.949 0.951 0.947 0.948 0.949

.8 800 0.953 0.957 0.972 0.923 0.945 0.952 0.949 0.947 0.950 0.951

1600 0.952 0.952 0.960 0.939 0.947 0.948 0.948 0.947 0.948 0.952

3200 0.950 0.950 0.957 0.945 0.947 0.948 0.952 0.951 0.952 0.948

6400 0.951 0.949 0.953 0.953 0.950 0.950 0.951 0.949 0.952 0.952

Table 4,95,μ: Actual coverages (in 10,000 trials) of nominal 95% confidence intervals on μ
Theory-based confidence intervals Simulation-based confidence intervals

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 0.941 0.904 0.860 0.788 0.918 0.948 0.966 0.913 0.891 0.952

200 0.948 0.926 0.887 0.832 0.933 0.951 0.968 0.934 0.904 0.952

400 0.951 0.935 0.909 0.871 0.944 0.951 0.952 0.944 0.924 0.953

.5 800 0.950 0.948 0.927 0.901 0.947 0.951 0.949 0.947 0.936 0.951

1600 0.947 0.951 0.937 0.919 0.947 0.948 0.952 0.951 0.947 0.950

3200 0.953 0.951 0.944 0.926 0.949 0.952 0.952 0.951 0.953 0.951

6400 0.950 0.956 0.945 0.939 0.948 0.948 0.951 0.950 0.951 0.952

100 0.936 0.911 0.866 0.807 0.911 0.949 0.966 0.918 0.898 0.954

200 0.947 0.924 0.889 0.851 0.934 0.952 0.956 0.940 0.913 0.952

400 0.948 0.937 0.914 0.882 0.940 0.949 0.952 0.946 0.934 0.952

.6 800 0.949 0.945 0.929 0.909 0.946 0.949 0.951 0.947 0.945 0.953

1600 0.948 0.949 0.942 0.927 0.947 0.950 0.948 0.950 0.951 0.949

3200 0.952 0.945 0.947 0.933 0.946 0.950 0.950 0.950 0.951 0.948

6400 0.950 0.950 0.945 0.943 0.950 0.951 0.950 0.951 0.949 0.950

100 0.939 0.907 0.868 0.822 0.919 0.949 0.964 0.929 0.898 0.950

200 0.947 0.930 0.896 0.863 0.933 0.952 0.956 0.942 0.926 0.952

400 0.946 0.937 0.923 0.900 0.946 0.950 0.954 0.948 0.943 0.951

.7 800 0.946 0.946 0.932 0.921 0.943 0.951 0.952 0.951 0.949 0.952

1600 0.950 0.948 0.943 0.937 0.949 0.953 0.951 0.949 0.953 0.951

3200 0.951 0.950 0.949 0.946 0.952 0.951 0.950 0.949 0.951 0.950

6400 0.948 0.949 0.951 0.948 0.951 0.949 0.951 0.950 0.952 0.952

100 0.939 0.907 0.871 0.837 0.916 0.948 0.965 0.933 0.899 0.956

200 0.947 0.926 0.907 0.881 0.936 0.951 0.954 0.949 0.927 0.953

400 0.951 0.942 0.924 0.914 0.944 0.952 0.954 0.948 0.944 0.948

.8 800 0.951 0.948 0.939 0.935 0.944 0.950 0.951 0.949 0.950 0.949

1600 0.950 0.945 0.948 0.943 0.949 0.951 0.949 0.950 0.949 0.949

3200 0.952 0.949 0.948 0.944 0.950 0.949 0.950 0.950 0.949 0.951

6400 0.951 0.952 0.949 0.951 0.950 0.951 0.951 0.951 0.951 0.950

Table 4,95,σ: Actual coverages (in 10,000 trials) of nominal 95% confidence intervals on σ
Theory-based confidence intervals Simulation-based confidence intervals

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 0.943 0.927 0.876 0.852 0.941 0.950 0.942 0.952 0.962 0.950

200 0.945 0.948 0.897 0.856 0.942 0.948 0.953 0.946 0.966 0.951

400 0.947 0.949 0.918 0.884 0.945 0.952 0.949 0.945 0.956 0.951

.5 800 0.951 0.948 0.935 0.900 0.946 0.951 0.949 0.947 0.954 0.949

1600 0.952 0.950 0.944 0.918 0.949 0.951 0.950 0.949 0.953 0.949

3200 0.950 0.953 0.949 0.928 0.949 0.951 0.951 0.951 0.950 0.951

6400 0.950 0.952 0.948 0.944 0.949 0.947 0.948 0.951 0.952 0.951

100 0.940 0.926 0.874 0.844 0.938 0.953 0.945 0.963 0.957 0.951

200 0.938 0.940 0.903 0.870 0.941 0.947 0.950 0.947 0.960 0.950

400 0.946 0.948 0.916 0.894 0.952 0.948 0.949 0.950 0.956 0.953

.6 800 0.948 0.950 0.937 0.917 0.949 0.952 0.948 0.948 0.956 0.949

1600 0.953 0.951 0.947 0.925 0.951 0.949 0.951 0.949 0.951 0.951

3200 0.947 0.948 0.943 0.939 0.951 0.950 0.949 0.951 0.951 0.950

6400 0.946 0.950 0.947 0.945 0.951 0.948 0.950 0.950 0.951 0.951

100 0.940 0.931 0.881 0.853 0.940 0.952 0.948 0.966 0.946 0.951

200 0.945 0.938 0.901 0.884 0.944 0.951 0.950 0.951 0.949 0.951

400 0.950 0.942 0.924 0.902 0.950 0.950 0.949 0.951 0.953 0.950

.7 800 0.945 0.947 0.938 0.919 0.946 0.949 0.948 0.951 0.949 0.951

1600 0.952 0.947 0.945 0.936 0.950 0.952 0.948 0.950 0.951 0.951

3200 0.947 0.951 0.946 0.944 0.950 0.950 0.950 0.950 0.953 0.951

6400 0.951 0.947 0.949 0.947 0.948 0.951 0.950 0.950 0.949 0.950

100 0.941 0.929 0.881 0.859 0.941 0.950 0.948 0.959 0.936 0.948

200 0.948 0.940 0.916 0.888 0.947 0.954 0.948 0.953 0.948 0.950

400 0.951 0.947 0.925 0.914 0.950 0.953 0.952 0.950 0.950 0.949

.8 800 0.948 0.949 0.938 0.933 0.945 0.951 0.952 0.949 0.954 0.949

1600 0.950 0.947 0.942 0.939 0.950 0.951 0.950 0.949 0.951 0.952

3200 0.951 0.950 0.949 0.943 0.951 0.952 0.949 0.950 0.949 0.950

6400 0.950 0.953 0.951 0.948 0.954 0.950 0.954 0.950 0.950 0.951

Table 4,95,ρ: Actual coverages (in 10,000 trials) of nominal 95% confidence intervals on ρ
Theory-based confidence intervals Simulation-based confidence intervals

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 0.942 0.967 0.979 0.946 0.923 0.949 0.971 0.942 0.932 0.950

200 0.949 0.961 0.980 0.934 0.932 0.947 0.958 0.950 0.939 0.954

400 0.951 0.954 0.975 0.926 0.942 0.951 0.949 0.952 0.941 0.953

.5 800 0.951 0.951 0.966 0.934 0.949 0.951 0.950 0.950 0.943 0.951

1600 0.954 0.956 0.957 0.936 0.945 0.952 0.950 0.948 0.947 0.950

3200 0.946 0.952 0.951 0.938 0.950 0.949 0.950 0.949 0.949 0.948

6400 0.947 0.946 0.946 0.942 0.950 0.948 0.949 0.949 0.951 0.948

100 0.947 0.969 0.982 0.944 0.919 0.952 0.969 0.943 0.931 0.951

200 0.948 0.961 0.977 0.933 0.932 0.950 0.951 0.957 0.940 0.949

400 0.950 0.956 0.976 0.926 0.942 0.949 0.949 0.949 0.943 0.952

.6 800 0.953 0.952 0.968 0.934 0.947 0.951 0.949 0.948 0.948 0.951

1600 0.948 0.952 0.958 0.940 0.951 0.950 0.951 0.948 0.951 0.951

3200 0.950 0.952 0.956 0.941 0.951 0.953 0.951 0.951 0.952 0.951

6400 0.950 0.953 0.951 0.950 0.948 0.949 0.951 0.950 0.951 0.950

100 0.944 0.969 0.984 0.949 0.912 0.948 0.973 0.947 0.929 0.948

200 0.951 0.961 0.981 0.944 0.928 0.953 0.948 0.955 0.942 0.951

400 0.950 0.956 0.969 0.942 0.942 0.951 0.948 0.947 0.948 0.949

.7 800 0.952 0.956 0.963 0.942 0.945 0.949 0.952 0.948 0.950 0.952

1600 0.949 0.952 0.956 0.947 0.948 0.950 0.950 0.948 0.948 0.952

3200 0.949 0.950 0.951 0.949 0.954 0.949 0.950 0.950 0.950 0.951

6400 0.954 0.952 0.950 0.950 0.948 0.950 0.951 0.949 0.949 0.953

100 0.946 0.974 0.978 0.952 0.907 0.950 0.968 0.949 0.929 0.949

200 0.947 0.964 0.977 0.951 0.928 0.951 0.951 0.960 0.945 0.950

400 0.949 0.959 0.967 0.948 0.943 0.951 0.949 0.949 0.953 0.949

.8 800 0.951 0.955 0.961 0.948 0.943 0.951 0.951 0.946 0.952 0.953

1600 0.949 0.953 0.954 0.951 0.947 0.949 0.950 0.949 0.950 0.952

3200 0.950 0.949 0.954 0.956 0.947 0.948 0.952 0.951 0.951 0.948

6400 0.948 0.950 0.954 0.956 0.950 0.948 0.951 0.952 0.952 0.952

Table 4,95,γ: Actual coverages (in 10,000 trials) of nominal 95% confidence intervals on γ
Theory-based confidence intervals Simulation-based confidence intervals

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 0.950 0.961 0.944 0.885 0.950 0.949 0.952 0.933 0.946 0.947

200 0.948 0.960 0.943 0.871 0.947 0.947 0.952 0.939 0.946 0.949

400 0.951 0.954 0.944 0.890 0.951 0.948 0.949 0.944 0.948 0.950

.5 800 0.952 0.954 0.949 0.908 0.951 0.952 0.947 0.944 0.946 0.952

1600 0.952 0.951 0.950 0.921 0.947 0.950 0.949 0.950 0.950 0.947

3200 0.950 0.949 0.951 0.930 0.952 0.950 0.948 0.951 0.952 0.953

6400 0.952 0.950 0.949 0.943 0.948 0.951 0.950 0.948 0.950 0.950

100 0.953 0.957 0.930 0.863 0.945 0.944 0.950 0.934 0.956 0.947

200 0.952 0.955 0.933 0.872 0.947 0.950 0.950 0.943 0.956 0.950

400 0.950 0.958 0.936 0.895 0.948 0.950 0.948 0.947 0.952 0.952

.6 800 0.948 0.954 0.944 0.916 0.952 0.948 0.951 0.947 0.955 0.948

1600 0.950 0.950 0.946 0.924 0.953 0.950 0.948 0.949 0.952 0.950

3200 0.948 0.949 0.948 0.933 0.951 0.948 0.950 0.948 0.952 0.951

6400 0.950 0.952 0.951 0.943 0.949 0.951 0.950 0.952 0.949 0.948

100 0.952 0.951 0.917 0.859 0.942 0.947 0.948 0.940 0.970 0.946

200 0.951 0.951 0.922 0.882 0.947 0.946 0.948 0.947 0.958 0.949

400 0.947 0.951 0.932 0.908 0.946 0.948 0.948 0.949 0.951 0.949

.7 800 0.948 0.949 0.943 0.922 0.945 0.948 0.949 0.949 0.951 0.949

1600 0.951 0.949 0.947 0.937 0.953 0.950 0.949 0.951 0.952 0.953

3200 0.954 0.950 0.950 0.945 0.950 0.951 0.947 0.949 0.952 0.950

6400 0.957 0.949 0.950 0.945 0.948 0.952 0.950 0.948 0.949 0.949

100 0.949 0.948 0.909 0.869 0.944 0.944 0.949 0.943 0.975 0.946

200 0.949 0.948 0.926 0.896 0.948 0.946 0.948 0.949 0.959 0.950

400 0.949 0.946 0.936 0.917 0.950 0.947 0.947 0.949 0.949 0.949

.8 800 0.952 0.952 0.943 0.935 0.947 0.949 0.950 0.952 0.950 0.950

1600 0.953 0.948 0.949 0.942 0.947 0.951 0.952 0.952 0.950 0.949

3200 0.948 0.949 0.945 0.945 0.949 0.950 0.950 0.950 0.952 0.950

6400 0.950 0.950 0.951 0.949 0.953 0.950 0.951 0.952 0.951 0.952

Table 4,95,β: Actual coverages (in 10,000 trials) of nominal 95% confidence intervals on β
Theory-based confidence intervals Simulation-based confidence intervals

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 0.987 0.999 1.000 0.897 0.948 0.990 0.983 0.959 0.935 0.969

200 0.989 0.997 1.000 0.897 0.962 0.990 0.987 0.972 0.949 0.977

400 0.989 0.996 1.000 0.915 0.974 0.990 0.985 0.977 0.959 0.982

.5 800 0.989 0.993 0.999 0.939 0.983 0.989 0.988 0.979 0.966 0.985

1600 0.992 0.993 0.998 0.953 0.985 0.991 0.991 0.984 0.971 0.987

3200 0.989 0.992 0.994 0.965 0.988 0.990 0.991 0.986 0.975 0.990

6400 0.990 0.988 0.991 0.977 0.989 0.990 0.989 0.989 0.981 0.989

100 0.990 0.998 1.000 0.903 0.945 0.991 0.985 0.963 0.936 0.970

200 0.990 0.998 1.000 0.920 0.964 0.989 0.984 0.975 0.955 0.976

400 0.991 0.996 1.000 0.924 0.975 0.991 0.987 0.976 0.964 0.982

.6 800 0.990 0.992 0.999 0.947 0.981 0.990 0.989 0.981 0.969 0.986

1600 0.991 0.993 0.998 0.961 0.986 0.991 0.990 0.987 0.976 0.988

3200 0.988 0.992 0.995 0.968 0.987 0.990 0.990 0.987 0.981 0.988

6400 0.990 0.990 0.992 0.980 0.989 0.990 0.989 0.988 0.985 0.991

100 0.988 0.998 1.000 0.910 0.947 0.992 0.985 0.966 0.940 0.970

200 0.990 0.999 1.000 0.925 0.965 0.991 0.984 0.977 0.959 0.978

400 0.991 0.996 1.000 0.946 0.976 0.991 0.985 0.978 0.970 0.981

.7 800 0.991 0.991 0.999 0.955 0.982 0.990 0.988 0.982 0.974 0.983

1600 0.989 0.991 0.996 0.971 0.987 0.990 0.988 0.987 0.978 0.989

3200 0.990 0.989 0.993 0.979 0.989 0.990 0.989 0.988 0.983 0.989

6400 0.990 0.991 0.992 0.984 0.988 0.990 0.990 0.989 0.985 0.989

100 0.988 0.998 1.000 0.924 0.950 0.990 0.988 0.969 0.941 0.974

200 0.989 0.997 1.000 0.940 0.969 0.991 0.985 0.979 0.962 0.977

400 0.990 0.996 1.000 0.950 0.976 0.992 0.987 0.980 0.969 0.982

.8 800 0.991 0.993 0.999 0.964 0.982 0.990 0.989 0.984 0.976 0.987

1600 0.993 0.991 0.996 0.976 0.985 0.993 0.989 0.986 0.979 0.988

3200 0.990 0.991 0.994 0.983 0.985 0.989 0.990 0.990 0.984 0.988

6400 0.990 0.988 0.991 0.988 0.989 0.989 0.989 0.988 0.988 0.991

Table 4,99,μ: Actual coverages (in 10,000 trials) of nominal 99% confidence intervals on μ
Theory-based confidence intervals Simulation-based confidence intervals

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 0.982 0.948 0.908 0.838 0.961 0.990 0.975 0.939 0.998 0.977

200 0.986 0.967 0.933 0.878 0.972 0.991 0.982 0.958 0.961 0.985

400 0.989 0.972 0.950 0.916 0.982 0.991 0.978 0.966 0.959 0.987

.5 800 0.988 0.979 0.965 0.942 0.985 0.989 0.983 0.972 0.968 0.988

1600 0.990 0.985 0.974 0.960 0.987 0.990 0.988 0.980 0.977 0.990

3200 0.990 0.989 0.981 0.972 0.988 0.990 0.989 0.983 0.981 0.991

6400 0.990 0.990 0.985 0.981 0.989 0.989 0.990 0.986 0.984 0.989

100 0.982 0.951 0.911 0.854 0.956 0.990 0.976 0.947 0.993 0.980

200 0.987 0.964 0.933 0.895 0.972 0.990 0.978 0.965 0.952 0.985

400 0.989 0.975 0.956 0.926 0.980 0.989 0.978 0.971 0.968 0.987

.6 800 0.989 0.981 0.967 0.951 0.986 0.989 0.984 0.978 0.976 0.988

1600 0.989 0.985 0.979 0.968 0.987 0.990 0.986 0.980 0.980 0.989

3200 0.991 0.986 0.984 0.976 0.988 0.990 0.988 0.985 0.986 0.990

6400 0.990 0.987 0.985 0.982 0.990 0.991 0.988 0.989 0.988 0.990

100 0.983 0.951 0.916 0.863 0.958 0.989 0.979 0.953 0.986 0.979

200 0.988 0.967 0.937 0.909 0.973 0.991 0.978 0.967 0.958 0.986

400 0.988 0.974 0.960 0.942 0.982 0.991 0.981 0.975 0.970 0.986

.7 800 0.989 0.983 0.974 0.961 0.986 0.989 0.984 0.977 0.979 0.987

1600 0.989 0.986 0.981 0.976 0.988 0.989 0.988 0.983 0.985 0.989

3200 0.988 0.989 0.987 0.983 0.990 0.988 0.990 0.986 0.986 0.990

6400 0.990 0.990 0.988 0.986 0.989 0.991 0.988 0.987 0.988 0.989

100 0.985 0.951 0.917 0.881 0.961 0.990 0.981 0.955 0.977 0.981

200 0.986 0.967 0.949 0.922 0.975 0.989 0.978 0.971 0.959 0.985

400 0.989 0.977 0.960 0.954 0.983 0.990 0.982 0.974 0.972 0.987

.8 800 0.991 0.984 0.974 0.971 0.983 0.990 0.985 0.979 0.980 0.990

1600 0.990 0.984 0.983 0.982 0.988 0.989 0.987 0.985 0.982 0.990

3200 0.991 0.988 0.987 0.985 0.988 0.990 0.990 0.987 0.987 0.990

6400 0.990 0.991 0.988 0.989 0.989 0.990 0.991 0.989 0.989 0.990

Table 4,99,σ: Actual coverages (in 10,000 trials) of nominal 99% confidence intervals on σ
Theory-based confidence intervals Simulation-based confidence intervals

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 0.984 0.972 0.936 0.927 0.983 0.988 0.981 0.997 0.988 0.987

200 0.987 0.988 0.956 0.935 0.986 0.990 0.985 0.991 0.994 0.988

400 0.989 0.990 0.971 0.949 0.986 0.988 0.988 0.988 0.995 0.989

.5 800 0.990 0.988 0.981 0.963 0.988 0.990 0.990 0.986 0.994 0.989

1600 0.990 0.989 0.987 0.972 0.991 0.990 0.990 0.988 0.992 0.992

3200 0.990 0.990 0.989 0.979 0.991 0.990 0.990 0.989 0.991 0.991

6400 0.990 0.992 0.990 0.985 0.989 0.988 0.990 0.990 0.990 0.989

100 0.982 0.974 0.935 0.924 0.983 0.988 0.984 0.994 0.988 0.986

200 0.984 0.986 0.958 0.944 0.984 0.990 0.989 0.992 0.992 0.989

400 0.988 0.988 0.972 0.954 0.989 0.989 0.988 0.990 0.992 0.992

.6 800 0.989 0.989 0.982 0.970 0.988 0.988 0.989 0.990 0.993 0.989

1600 0.991 0.988 0.986 0.976 0.990 0.990 0.990 0.989 0.992 0.990

3200 0.989 0.989 0.988 0.982 0.990 0.990 0.989 0.990 0.990 0.990

6400 0.989 0.990 0.990 0.988 0.989 0.989 0.989 0.991 0.990 0.989

100 0.983 0.975 0.938 0.928 0.981 0.987 0.987 0.992 0.983 0.986

200 0.987 0.986 0.958 0.948 0.986 0.989 0.988 0.994 0.986 0.988

400 0.988 0.989 0.974 0.961 0.987 0.989 0.989 0.992 0.987 0.987

.7 800 0.988 0.990 0.982 0.970 0.987 0.990 0.989 0.989 0.989 0.988

1600 0.989 0.990 0.986 0.982 0.990 0.989 0.991 0.990 0.988 0.990

3200 0.989 0.990 0.989 0.987 0.989 0.990 0.990 0.990 0.991 0.988

6400 0.991 0.989 0.990 0.989 0.989 0.990 0.990 0.990 0.991 0.990

100 0.980 0.975 0.940 0.933 0.983 0.987 0.987 0.988 0.975 0.982

200 0.987 0.984 0.964 0.946 0.985 0.990 0.988 0.992 0.981 0.986

400 0.990 0.987 0.977 0.965 0.988 0.991 0.988 0.991 0.984 0.988

.8 800 0.988 0.989 0.985 0.979 0.988 0.988 0.990 0.991 0.988 0.989

1600 0.989 0.989 0.986 0.984 0.990 0.989 0.990 0.990 0.990 0.990

3200 0.990 0.990 0.989 0.985 0.990 0.991 0.990 0.990 0.989 0.990

6400 0.990 0.992 0.991 0.988 0.991 0.989 0.991 0.992 0.990 0.992

Table 4,99,ρ: Actual coverages (in 10,000 trials) of nominal 99% confidence intervals on ρ
Theory-based confidence intervals Simulation-based confidence intervals

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 0.989 0.995 0.997 0.981 0.965 0.990 0.984 0.962 0.959 0.972

200 0.988 0.993 0.998 0.977 0.970 0.989 0.987 0.971 0.963 0.978

400 0.991 0.992 0.997 0.972 0.978 0.990 0.988 0.978 0.966 0.982

.5 800 0.990 0.990 0.996 0.974 0.985 0.990 0.989 0.984 0.969 0.983

1600 0.990 0.992 0.993 0.972 0.986 0.990 0.991 0.989 0.975 0.989

3200 0.988 0.990 0.990 0.979 0.989 0.989 0.989 0.987 0.979 0.989

6400 0.989 0.989 0.991 0.982 0.989 0.990 0.989 0.991 0.983 0.991

100 0.988 0.995 0.997 0.982 0.962 0.988 0.984 0.964 0.959 0.971

200 0.989 0.994 0.996 0.975 0.971 0.990 0.986 0.975 0.963 0.976

400 0.989 0.994 0.997 0.974 0.977 0.991 0.989 0.978 0.970 0.981

.6 800 0.991 0.991 0.997 0.974 0.984 0.991 0.990 0.984 0.974 0.985

1600 0.990 0.992 0.994 0.978 0.988 0.990 0.991 0.988 0.978 0.986

3200 0.991 0.989 0.992 0.980 0.989 0.991 0.988 0.987 0.985 0.988

6400 0.990 0.991 0.991 0.985 0.989 0.990 0.990 0.990 0.987 0.989

100 0.987 0.996 0.998 0.982 0.956 0.990 0.986 0.965 0.958 0.969

200 0.992 0.995 0.996 0.979 0.968 0.991 0.985 0.977 0.964 0.976

400 0.989 0.993 0.995 0.979 0.975 0.990 0.989 0.982 0.971 0.979

.7 800 0.992 0.992 0.995 0.979 0.982 0.991 0.990 0.983 0.976 0.984

1600 0.989 0.991 0.993 0.983 0.987 0.990 0.989 0.987 0.982 0.988

3200 0.991 0.992 0.990 0.986 0.989 0.990 0.990 0.988 0.984 0.991

6400 0.993 0.990 0.991 0.986 0.988 0.991 0.990 0.989 0.987 0.989

100 0.986 0.996 0.997 0.985 0.953 0.989 0.984 0.967 0.955 0.971

200 0.989 0.994 0.996 0.985 0.970 0.990 0.985 0.978 0.965 0.976

400 0.990 0.993 0.995 0.983 0.976 0.991 0.987 0.980 0.972 0.982

.8 800 0.991 0.992 0.992 0.984 0.981 0.991 0.989 0.983 0.979 0.984

1600 0.991 0.990 0.992 0.985 0.985 0.991 0.988 0.986 0.980 0.987

3200 0.992 0.991 0.991 0.988 0.985 0.992 0.992 0.990 0.986 0.989

6400 0.988 0.990 0.991 0.989 0.989 0.989 0.990 0.990 0.989 0.992

Table 4,99,γ: Actual coverages (in 10,000 trials) of nominal 99% confidence intervals on γ
Theory-based confidence intervals Simulation-based confidence intervals

ρ n full .1,.9 .2,.8 .4,.8 .4,1 full .1,.9 .2,.8 .4,.8 .4,1

100 0.989 0.992 0.987 0.960 0.989 0.985 0.977 0.971 0.989 0.988

200 0.991 0.993 0.986 0.944 0.988 0.988 0.984 0.974 0.986 0.990

400 0.990 0.993 0.984 0.951 0.989 0.986 0.989 0.978 0.985 0.989

.5 800 0.990 0.992 0.986 0.962 0.989 0.990 0.989 0.981 0.982 0.989

1600 0.989 0.991 0.988 0.970 0.990 0.990 0.990 0.987 0.987 0.990

3200 0.989 0.990 0.991 0.977 0.991 0.990 0.989 0.987 0.987 0.991

6400 0.990 0.991 0.990 0.983 0.989 0.990 0.990 0.990 0.988 0.989

100 0.991 0.990 0.979 0.933 0.987 0.986 0.977 0.977 0.996 0.987

200 0.989 0.990 0.979 0.938 0.987 0.987 0.984 0.977 0.992 0.989

400 0.990 0.992 0.983 0.950 0.990 0.989 0.988 0.980 0.990 0.991

.6 800 0.990 0.991 0.985 0.968 0.990 0.989 0.988 0.985 0.989 0.990

1600 0.992 0.991 0.989 0.976 0.992 0.991 0.990 0.989 0.989 0.991

3200 0.990 0.989 0.989 0.982 0.990 0.990 0.989 0.987 0.990 0.990

6400 0.990 0.990 0.990 0.987 0.991 0.990 0.989 0.991 0.991 0.991

100 0.990 0.988 0.971 0.926 0.985 0.984 0.980 0.982 0.999 0.989

200 0.990 0.990 0.974 0.941 0.986 0.987 0.987 0.983 0.996 0.988

400 0.989 0.987 0.979 0.960 0.987 0.988 0.989 0.986 0.991 0.989

.7 800 0.991 0.989 0.982 0.971 0.989 0.989 0.988 0.987 0.990 0.990

1600 0.991 0.989 0.987 0.983 0.989 0.991 0.989 0.988 0.991 0.990

3200 0.992 0.990 0.990 0.986 0.991 0.991 0.990 0.988 0.991 0.991

6400 0.991 0.989 0.990 0.988 0.989 0.989 0.989 0.989 0.990 0.990

100 0.991 0.986 0.966 0.928 0.986 0.986 0.984 0.989 0.998 0.989

200 0.992 0.988 0.977 0.957 0.990 0.986 0.986 0.986 0.998 0.990

400 0.991 0.990 0.981 0.967 0.990 0.989 0.988 0.989 0.989 0.991

.8 800 0.990 0.990 0.985 0.980 0.987 0.988 0.989 0.989 0.991 0.988

1600 0.989 0.989 0.989 0.986 0.988 0.989 0.989 0.990 0.990 0.989

3200 0.990 0.988 0.990 0.986 0.989 0.989 0.989 0.991 0.990 0.989

6400 0.991 0.991 0.990 0.989 0.991 0.990 0.992 0.989 0.990 0.991

Table 4,99,β: Actual coverages (in 10,000 trials) of nominal 99% confidence intervals on β
Actual Theory based Simulation based

Parameter coverage ρ Full .4,.8 Full .4,.8

.5 < 100 2184 < 100 2737

.6 < 100 1347 < 100 1972

.88 – .92 .7 < 100 697 < 100 1517

.8 < 100 446 < 100 1050

μ .5 < 100 5815 < 100 > 12000

.6 < 100 2388 < 100 4884

.89 – .91 .7 106 1360 < 100 4028

.8 < 100 806 < 100 3626

.5 < 100 2794 < 100 214

.6 < 100 1816 < 100 185

.88 – .92 .7 < 100 656 < 100 121

.8 < 100 414 < 100 114

σ .5 < 100 9381 < 100 4693

.6 111 5565 < 100 2034

.89 – .91 .7 < 100 1109 < 100 1373

.8 106 706 < 100 1261

.5 < 100 3386 < 100 131

.6 < 100 2250 < 100 100

.88 – .92 .7 < 100 1111 < 100 < 100

.8 < 100 800 < 100 < 100

ρ .5 < 100 10530 < 100 202

.6 < 100 6658 < 100 165

.89 – .91 .7 < 100 2499 < 100 203

.8 < 100 1389 < 100 247

.5 < 100 < 100 < 100 1240

.6 < 100 < 100 < 100 912

.88 – .92 .7 < 100 < 100 < 100 976

.8 < 100 170 < 100 824

γ .5 < 100 429 < 100 5565

.6 < 100 436 < 100 3017

.89 – .91 .7 < 100 150 < 100 2462

.8 < 100 1670 < 100 1947

.5 < 100 2886 < 100 < 100

.6 < 100 1988 < 100 < 100

.88 – .92 .7 < 100 1088 < 100 < 100

.8 < 100 678 < 100 < 100

β .5 < 100 8529 < 100 < 100

.6 < 100 4949 < 100 < 100

.89 – .91 .7 < 100 2414 < 100 < 100

.8 < 100 1274 < 100 < 100

Table 5: Approximate sample sizes needed to obtain actual confidence interval coverages that lie between

.88 and .92 (“wide”) or between .89 and .91 (“narrow”) for nominal two-sided, 90% confidence intervals

on the parameter
Bias (%) rmse (%)

ρ n Full .1,.9 .2,.8 .4,.8 .4,1 Full .1,.9 .2,.8 .4,.8 .4,1

100 0.49 0.56 0.55 0.75 0.63 4.10 4.20 4.41 4.20 4.05

200 0.24 0.31 0.28 0.45 0.37 2.87 2.98 3.18 3.03 2.84

400 0.12 0.16 0.16 0.30 0.21 2.03 2.09 2.22 2.17 1.98

.5 800 0.04 0.06 0.10 0.15 0.07 1.43 1.46 1.59 1.54 1.41

1600 0.03 0.04 0.03 0.08 0.04 1.01 1.04 1.10 1.09 0.99

3200 0.02 0.01 0.01 0.05 0.02 0.72 0.73 0.77 0.78 0.71

6400 0.01 0.01 0.00 0.02 0.02 0.51 0.52 0.56 0.55 0.50

100 0.49 0.51 0.45 0.58 0.57 3.69 3.76 3.97 3.80 3.64

200 0.23 0.23 0.25 0.38 0.30 2.59 2.61 2.77 2.74 2.51

400 0.07 0.13 0.15 0.22 0.13 1.82 1.86 1.99 1.97 1.77

.6 800 0.06 0.07 0.06 0.10 0.07 1.27 1.31 1.40 1.41 1.27

1600 0.03 0.02 0.02 0.07 0.04 0.89 0.93 0.98 1.00 0.87

3200 0.01 0.01 0.02 0.03 0.02 0.64 0.66 0.69 0.71 0.63

6400 0.01 0.01 0.02 0.02 0.01 0.45 0.46 0.49 0.50 0.44

100 0.34 0.34 0.34 0.47 0.44 3.08 3.14 3.29 3.32 3.11

200 0.17 0.20 0.18 0.16 0.24 2.18 2.21 2.35 2.40 2.19

400 0.08 0.11 0.08 0.17 0.10 1.54 1.59 1.66 1.72 1.56

.7 800 0.03 0.05 0.04 0.06 0.04 1.10 1.11 1.17 1.23 1.09

1600 0.01 0.02 0.02 0.03 0.03 0.76 0.78 0.82 0.85 0.77

3200 0.01 0.00 0.02 0.02 0.01 0.54 0.55 0.58 0.61 0.54

6400 0.01 0.01 0.00 0.01 0.01 0.38 0.39 0.41 0.43 0.39

100 0.27 0.25 0.20 0.30 0.32 2.42 2.47 2.52 2.73 2.50

200 0.17 0.11 0.12 0.19 0.17 1.70 1.73 1.78 1.96 1.77

400 0.05 0.06 0.06 0.09 0.08 1.20 1.21 1.25 1.41 1.25

.8 800 0.06 0.03 0.02 0.04 0.04 0.84 0.86 0.90 0.99 0.87

1600 0.02 0.01 0.02 0.04 0.02 0.60 0.61 0.63 0.70 0.62

3200 0.01 0.01 0.01 0.01 0.01 0.42 0.43 0.44 0.49 0.45

6400 0.00 0.00 0.01 0.00 0.01 0.30 0.30 0.31 0.35 0.31

Table 6: Fifth percentile estimates
Theory based Simulation based

ρ n Full .1,.9 .2,.8 .4,.8 .4,1 Full .1,.9 .2,.8 .4,.8 .4,1

100 0.701 0.702 0.695 0.682 0.693 0.711 0.775 0.890 0.942 0.743

200 0.714 0.705 0.712 0.697 0.702 0.717 0.727 0.833 0.911 0.727

400 0.728 0.723 0.719 0.707 0.717 0.730 0.732 0.772 0.876 0.725

.5 800 0.744 0.742 0.723 0.723 0.736 0.744 0.745 0.749 0.826 0.741

1600 0.742 0.736 0.741 0.731 0.738 0.742 0.738 0.755 0.789 0.740

3200 0.738 0.745 0.747 0.729 0.738 0.740 0.747 0.753 0.763 0.741

6400 0.746 0.740 0.747 0.740 0.735 0.746 0.743 0.753 0.755 0.737

100 0.694 0.688 0.690 0.694 0.687 0.704 0.780 0.895 0.946 0.738

200 0.709 0.717 0.710 0.703 0.715 0.717 0.738 0.828 0.911 0.734

400 0.733 0.720 0.713 0.716 0.726 0.737 0.733 0.772 0.867 0.733

.6 800 0.734 0.733 0.722 0.726 0.732 0.735 0.739 0.754 0.813 0.737

1600 0.734 0.739 0.741 0.729 0.739 0.733 0.743 0.757 0.787 0.736

3200 0.742 0.737 0.743 0.736 0.740 0.743 0.740 0.752 0.765 0.743

6400 0.743 0.744 0.738 0.741 0.744 0.744 0.744 0.745 0.753 0.743

100 0.700 0.703 0.699 0.693 0.693 0.712 0.785 0.912 0.952 0.742

200 0.716 0.711 0.708 0.724 0.712 0.723 0.737 0.835 0.918 0.736

400 0.728 0.719 0.726 0.715 0.728 0.734 0.737 0.782 0.868 0.740

.7 800 0.735 0.733 0.733 0.733 0.736 0.741 0.742 0.763 0.817 0.739

1600 0.745 0.741 0.730 0.739 0.735 0.745 0.747 0.749 0.783 0.735

3200 0.743 0.752 0.740 0.735 0.747 0.742 0.755 0.750 0.759 0.747

6400 0.746 0.743 0.747 0.743 0.743 0.745 0.744 0.753 0.756 0.745

100 0.697 0.696 0.702 0.703 0.698 0.712 0.793 0.934 0.967 0.749

200 0.706 0.721 0.715 0.711 0.714 0.715 0.751 0.846 0.919 0.738

400 0.735 0.726 0.726 0.723 0.719 0.741 0.743 0.783 0.868 0.731

.8 800 0.728 0.734 0.740 0.732 0.733 0.729 0.744 0.769 0.804 0.736

1600 0.734 0.739 0.735 0.730 0.737 0.735 0.748 0.755 0.772 0.739

3200 0.747 0.736 0.741 0.740 0.735 0.749 0.743 0.752 0.760 0.738

6400 0.748 0.749 0.743 0.747 0.740 0.749 0.753 0.749 0.758 0.743

Table 7: Actual coverages of nominal one-sided, lower 75% confidence intervals on the 5th percentile of

the .4,.8 PTW population

Actual Theory based Simulation based

coverage ρ Full .4,.8 Full .4,.8

.5 441 1910 403 2653

.6 566 1391 405 2665

.73 – .77 .7 454 753 297 2287

.8 586 815 421 1702

.5 1203 8121 1087 4106

.6 2183 5895 2134 4064

.74 – .76 .7 1248 5375 914 3728

.8 1655 3188 1294 3694

Table 8: Approximate sample sizes needed to obtain actual confidence interval coverages that lie between

.73 and .77 (“wide”) or between .74 and .76 (“narrow”) for nominal one-sided, lower 75% confidence

intervals on the 5th percentile of the .4,.8 PTW population
Median pBr ratio frequencies

ρ n pBr ratio [0, .01] (.01, .02] (.02, .1] (.1, .2] (.2, .5] (.5, 1) (1, 2) [2, 5) [5, 10) [10, 50) [50, 100) [100, ∞)

tr 1.200 0.0000 0.0000 0.0017 0.0147 0.1447 0.2646 0.2744 0.2019 0.0542 0.0336 0.0035 0.0067

100 full 1.141 0.0000 0.0000 0.0007 0.0132 0.1408 0.2805 0.3069 0.2094 0.0372 0.0112 0.0001 0.0000

tr 1.100 0.0000 0.0000 0.0001 0.0015 0.1022 0.3382 0.3638 0.1696 0.0191 0.0048 0.0006 0.0001

200 full 1.058 0.0000 0.0000 0.0000 0.0009 0.0938 0.3664 0.3885 0.1429 0.0072 0.0003 0.0000 0.0000

tr 1.048 0.0000 0.0000 0.0000 0.0001 0.0414 0.4163 0.4515 0.0892 0.0014 0.0001 0.0000 0.0000

400 full 1.032 0.0000 0.0000 0.0000 0.0001 0.0360 0.4329 0.4681 0.0628 0.0001 0.0000 0.0000 0.0000

tr 1.016 0.0000 0.0000 0.0000 0.0000 0.0113 0.4691 0.4944 0.0252 0.0000 0.0000 0.0000 0.0000

.5 800 full 1.011 0.0000 0.0000 0.0000 0.0000 0.0060 0.4778 0.5030 0.0132 0.0000 0.0000 0.0000 0.0000

tr 1.013 0.0000 0.0000 0.0000 0.0000 0.0009 0.4777 0.5195 0.0019 0.0000 0.0000 0.0000 0.0000

1600 full 1.009 0.0000 0.0000 0.0000 0.0000 0.0003 0.4827 0.5163 0.0007 0.0000 0.0000 0.0000 0.0000

tr 1.002 0.0000 0.0000 0.0000 0.0000 0.0000 0.4950 0.5049 0.0001 0.0000 0.0000 0.0000 0.0000

3200 full 1.005 0.0000 0.0000 0.0000 0.0000 0.0000 0.4856 0.5144 0.0000 0.0000 0.0000 0.0000 0.0000

tr 1.003 0.0000 0.0000 0.0000 0.0000 0.0000 0.4911 0.5089 0.0000 0.0000 0.0000 0.0000 0.0000

6400 full 1.002 0.0000 0.0000 0.0000 0.0000 0.0000 0.4915 0.5085 0.0000 0.0000 0.0000 0.0000 0.0000

tr 1.257 0.0000 0.0000 0.0044 0.0265 0.1565 0.2288 0.2411 0.2053 0.0700 0.0523 0.0057 0.0094

100 full 1.142 0.0000 0.0000 0.0030 0.0208 0.1523 0.2677 0.2603 0.2139 0.0588 0.0228 0.0004 0.0000

tr 1.100 0.0000 0.0000 0.0001 0.0057 0.1263 0.3186 0.3194 0.1866 0.0327 0.0100 0.0002 0.0004

200 full 1.079 0.0000 0.0000 0.0000 0.0033 0.1161 0.3375 0.3568 0.1700 0.0145 0.0018 0.0000 0.0000

tr 1.051 0.0000 0.0000 0.0000 0.0001 0.0677 0.3958 0.4078 0.1217 0.0062 0.0007 0.0000 0.0000

400 full 1.026 0.0000 0.0000 0.0000 0.0001 0.0592 0.4200 0.4349 0.0851 0.0007 0.0000 0.0000 0.0000

tr 1.024 0.0000 0.0000 0.0000 0.0000 0.0210 0.4537 0.4793 0.0459 0.0001 0.0000 0.0000 0.0000

.6 800 full 1.021 0.0000 0.0000 0.0000 0.0000 0.0141 0.4616 0.5011 0.0232 0.0000 0.0000 0.0000 0.0000

tr 1.012 0.0000 0.0000 0.0000 0.0000 0.0023 0.4823 0.5091 0.0063 0.0000 0.0000 0.0000 0.0000

1600 full 1.006 0.0000 0.0000 0.0000 0.0000 0.0003 0.4892 0.5087 0.0018 0.0000 0.0000 0.0000 0.0000

tr 1.006 0.0000 0.0000 0.0000 0.0000 0.0000 0.4854 0.5143 0.0003 0.0000 0.0000 0.0000 0.0000

3200 full 1.004 0.0000 0.0000 0.0000 0.0000 0.0000 0.4907 0.5093 0.0000 0.0000 0.0000 0.0000 0.0000

tr 1.004 0.0000 0.0000 0.0000 0.0000 0.0000 0.4867 0.5133 0.0000 0.0000 0.0000 0.0000 0.0000

6400 full 1.002 0.0000 0.0000 0.0000 0.0000 0.0000 0.4935 0.5065 0.0000 0.0000 0.0000 0.0000 0.0000

Table 9, .1, .9: Frequencies for pBr,true/pBr,est for bivariate Gaussian–Weibulls with generating correlations .5 and .6 (tr: truncated (on

the Gaussian) bivariate Gaussian–Weibull, pl = .1, pu = .9; full: full bivariate Gaussian–Weibull)
Median pBr ratio frequencies

ρ n pBr ratio [0, .01] (.01, .02] (.02, .1] (.1, .2] (.2, .5] (.5, 1) (1, 2) [2, 5) [5, 10) [10, 50) [50, 100) [100, ∞)

tr 1.232 0.0000 0.0000 0.0113 0.0458 0.1772 0.2024 0.1962 0.1902 0.0820 0.0689 0.0104 0.0156

100 full 1.164 0.0000 0.0002 0.0067 0.0391 0.1722 0.2289 0.2293 0.2080 0.0748 0.0386 0.0018 0.0004

tr 1.132 0.0000 0.0000 0.0007 0.0175 0.1584 0.2714 0.2678 0.2096 0.0494 0.0230 0.0014 0.0008

200 full 1.093 0.0000 0.0000 0.0007 0.0115 0.1439 0.3046 0.3125 0.1895 0.0327 0.0046 0.0000 0.0000

tr 1.061 0.0000 0.0000 0.0000 0.0026 0.1141 0.3444 0.3650 0.1544 0.0173 0.0021 0.0001 0.0000

400 full 1.044 0.0000 0.0000 0.0000 0.0006 0.0899 0.3795 0.4012 0.1244 0.0042 0.0002 0.0000 0.0000

tr 1.035 0.0000 0.0000 0.0000 0.0000 0.0512 0.4175 0.4434 0.0871 0.0007 0.0001 0.0000 0.0000

.7 800 full 1.012 0.0000 0.0000 0.0000 0.0000 0.0351 0.4519 0.4628 0.0501 0.0001 0.0000 0.0000 0.0000

tr 1.014 0.0000 0.0000 0.0000 0.0000 0.0113 0.4691 0.4996 0.0199 0.0001 0.0000 0.0000 0.0000

1600 full 1.004 0.0000 0.0000 0.0000 0.0000 0.0037 0.4913 0.4954 0.0096 0.0000 0.0000 0.0000 0.0000

tr 1.005 0.0000 0.0000 0.0000 0.0000 0.0004 0.4918 0.5055 0.0023 0.0000 0.0000 0.0000 0.0000

3200 full 1.007 0.0000 0.0000 0.0000 0.0000 0.0001 0.4847 0.5147 0.0005 0.0000 0.0000 0.0000 0.0000

tr 1.003 0.0000 0.0000 0.0000 0.0000 0.0000 0.4932 0.5068 0.0000 0.0000 0.0000 0.0000 0.0000

6400 full 1.003 0.0000 0.0000 0.0000 0.0000 0.0000 0.4917 0.5083 0.0000 0.0000 0.0000 0.0000 0.0000

tr 1.333 0.0002 0.0012 0.0488 0.0744 0.1625 0.1471 0.1550 0.1587 0.0891 0.1069 0.0204 0.0357

100 full 1.228 0.0000 0.0007 0.0307 0.0599 0.1695 0.1834 0.1814 0.1936 0.0854 0.0820 0.0070 0.0064

tr 1.120 0.0000 0.0000 0.0126 0.0468 0.1885 0.2131 0.2157 0.1803 0.0789 0.0546 0.0057 0.0038

200 full 1.118 0.0000 0.0000 0.0050 0.0325 0.1675 0.2477 0.2547 0.2076 0.0610 0.0239 0.0001 0.0000

tr 1.068 0.0000 0.0000 0.0010 0.0134 0.1640 0.2905 0.2876 0.1940 0.0364 0.0127 0.0004 0.0000

400 full 1.049 0.0000 0.0000 0.0000 0.0070 0.1443 0.3248 0.3339 0.1708 0.0171 0.0021 0.0000 0.0000

tr 1.043 0.0000 0.0000 0.0000 0.0019 0.1059 0.3654 0.3772 0.1406 0.0082 0.0008 0.0000 0.0000

.8 800 full 1.037 0.0000 0.0000 0.0000 0.0004 0.0720 0.3988 0.4271 0.0997 0.0017 0.0003 0.0000 0.0000

tr 1.019 0.0000 0.0000 0.0000 0.0000 0.0423 0.4427 0.4513 0.0632 0.0005 0.0000 0.0000 0.0000

1600 full 1.015 0.0000 0.0000 0.0000 0.0000 0.0228 0.4602 0.4843 0.0327 0.0000 0.0000 0.0000 0.0000

tr 1.012 0.0000 0.0000 0.0000 0.0000 0.0075 0.4779 0.4987 0.0159 0.0000 0.0000 0.0000 0.0000

3200 full 1.007 0.0000 0.0000 0.0000 0.0000 0.0027 0.4871 0.5054 0.0048 0.0000 0.0000 0.0000 0.0000

tr 1.003 0.0000 0.0000 0.0000 0.0000 0.0004 0.4937 0.5051 0.0008 0.0000 0.0000 0.0000 0.0000

6400 full 0.997 0.0000 0.0000 0.0000 0.0000 0.0001 0.5053 0.4945 0.0001 0.0000 0.0000 0.0000 0.0000

Table 9, .1, .9 continued: Frequencies for pBr,true/pBr,est for bivariate Gaussian–Weibulls with generating correlations .7 and .8 (tr:

truncated (on the Gaussian) bivariate Gaussian–Weibull, pl = .1, pu = .9; full: full bivariate Gaussian–Weibull)
Median pBr ratio frequencies

ρ n pBr ratio [0, .01] (.01, .02] (.02, .1] (.1, .2] (.2, .5] (.5, 1) (1, 2) [2, 5) [5, 10) [10, 50) [50, 100) [100, ∞)

tr 1.259 0.0000 0.0000 0.0017 0.0192 0.1596 0.2364 0.2354 0.1851 0.0650 0.0686 0.0122 0.0168

100 full 1.150 0.0000 0.0000 0.0003 0.0112 0.1393 0.2833 0.3106 0.2064 0.0393 0.0096 0.0000 0.0000

tr 1.130 0.0000 0.0000 0.0000 0.0043 0.1226 0.3087 0.3030 0.1815 0.0439 0.0310 0.0033 0.0017

200 full 1.079 0.0000 0.0000 0.0000 0.0014 0.0877 0.3596 0.4008 0.1422 0.0079 0.0004 0.0000 0.0000

tr 1.063 0.0000 0.0000 0.0000 0.0000 0.0662 0.3871 0.3878 0.1358 0.0183 0.0047 0.0001 0.0000

400 full 1.031 0.0000 0.0000 0.0000 0.0000 0.0371 0.4333 0.4683 0.0611 0.0002 0.0000 0.0000 0.0000

tr 1.034 0.0000 0.0000 0.0000 0.0000 0.0194 0.4483 0.4657 0.0646 0.0019 0.0001 0.0000 0.0000

.5 800 full 1.008 0.0000 0.0000 0.0000 0.0000 0.0059 0.4826 0.4996 0.0119 0.0000 0.0000 0.0000 0.0000

tr 1.013 0.0000 0.0000 0.0000 0.0000 0.0023 0.4797 0.5057 0.0123 0.0000 0.0000 0.0000 0.0000

1600 full 1.009 0.0000 0.0000 0.0000 0.0000 0.0003 0.4837 0.5156 0.0004 0.0000 0.0000 0.0000 0.0000

tr 1.004 0.0000 0.0000 0.0000 0.0000 0.0000 0.4923 0.5069 0.0008 0.0000 0.0000 0.0000 0.0000

3200 full 1.004 0.0000 0.0000 0.0000 0.0000 0.0000 0.4871 0.5129 0.0000 0.0000 0.0000 0.0000 0.0000

tr 1.003 0.0000 0.0000 0.0000 0.0000 0.0000 0.4907 0.5093 0.0000 0.0000 0.0000 0.0000 0.0000

6400 full 1.003 0.0000 0.0000 0.0000 0.0000 0.0000 0.4856 0.5144 0.0000 0.0000 0.0000 0.0000 0.0000

tr 1.290 0.0000 0.0000 0.0083 0.0393 0.1702 0.2089 0.1897 0.1828 0.0715 0.0820 0.0152 0.0321

100 full 1.132 0.0000 0.0000 0.0025 0.0219 0.1582 0.2620 0.2716 0.2068 0.0558 0.0209 0.0003 0.0000

tr 1.117 0.0000 0.0000 0.0002 0.0117 0.1531 0.2840 0.2653 0.1805 0.0554 0.0403 0.0057 0.0038

200 full 1.063 0.0000 0.0000 0.0000 0.0031 0.1175 0.3411 0.3572 0.1644 0.0153 0.0014 0.0000 0.0000

tr 1.076 0.0000 0.0000 0.0000 0.0007 0.1044 0.3540 0.3440 0.1599 0.0260 0.0099 0.0005 0.0006

400 full 1.046 0.0000 0.0000 0.0000 0.0000 0.0507 0.4084 0.4498 0.0898 0.0013 0.0000 0.0000 0.0000

tr 1.037 0.0000 0.0000 0.0000 0.0000 0.0438 0.4259 0.4310 0.0953 0.0038 0.0001 0.0001 0.0000

.6 800 full 1.017 0.0000 0.0000 0.0000 0.0000 0.0139 0.4663 0.4959 0.0239 0.0000 0.0000 0.0000 0.0000

tr 1.017 0.0000 0.0000 0.0000 0.0000 0.0087 0.4725 0.4902 0.0285 0.0001 0.0000 0.0000 0.0000

1600 full 1.007 0.0000 0.0000 0.0000 0.0000 0.0008 0.4870 0.5100 0.0022 0.0000 0.0000 0.0000 0.0000

tr 1.010 0.0000 0.0000 0.0000 0.0000 0.0004 0.4812 0.5154 0.0030 0.0000 0.0000 0.0000 0.0000

3200 full 0.999 0.0000 0.0000 0.0000 0.0000 0.0000 0.5022 0.4978 0.0000 0.0000 0.0000 0.0000 0.0000

tr 1.009 0.0000 0.0000 0.0000 0.0000 0.0000 0.4779 0.5220 0.0001 0.0000 0.0000 0.0000 0.0000

6400 full 1.003 0.0000 0.0000 0.0000 0.0000 0.0000 0.4920 0.5080 0.0000 0.0000 0.0000 0.0000 0.0000

Table 9, .2, .8: Frequencies for pBr,true/pBr,est for bivariate Gaussian–Weibulls with generating correlations .5 and .6 (tr: truncated (on

the Gaussian) bivariate Gaussian–Weibull, pl = .2, pu = .8; full: full bivariate Gaussian–Weibull)
Median pBr ratio frequencies

ρ n pBr ratio [0, .01] (.01, .02] (.02, .1] (.1, .2] (.2, .5] (.5, 1) (1, 2) [2, 5) [5, 10) [10, 50) [50, 100) [100, ∞)

tr 1.301 0.0000 0.0002 0.0232 0.0626 0.1759 0.1706 0.1634 0.1571 0.0762 0.0936 0.0239 0.0533

100 full 1.159 0.0000 0.0000 0.0090 0.0402 0.1702 0.2280 0.2302 0.2078 0.0713 0.0413 0.0018 0.0002

tr 1.153 0.0000 0.0000 0.0033 0.0343 0.1814 0.2323 0.2150 0.1863 0.0660 0.0596 0.0090 0.0128

200 full 1.081 0.0000 0.0000 0.0005 0.0101 0.1406 0.3117 0.3091 0.1930 0.0291 0.0059 0.0000 0.0000

tr 1.081 0.0000 0.0000 0.0000 0.0078 0.1458 0.3080 0.2941 0.1809 0.0438 0.0177 0.0012 0.0007

400 full 1.042 0.0000 0.0000 0.0000 0.0010 0.0827 0.3881 0.3973 0.1266 0.0041 0.0002 0.0000 0.0000

tr 1.030 0.0000 0.0000 0.0000 0.0006 0.0830 0.3976 0.3685 0.1370 0.0115 0.0018 0.0000 0.0000

.7 800 full 1.011 0.0000 0.0000 0.0000 0.0000 0.0296 0.4608 0.4623 0.0473 0.0000 0.0000 0.0000 0.0000

tr 1.021 0.0000 0.0000 0.0000 0.0000 0.0322 0.4503 0.4546 0.0624 0.0004 0.0001 0.0000 0.0000

1600 full 1.006 0.0000 0.0000 0.0000 0.0000 0.0045 0.4883 0.4977 0.0095 0.0000 0.0000 0.0000 0.0000

tr 1.019 0.0000 0.0000 0.0000 0.0000 0.0037 0.4697 0.5109 0.0157 0.0000 0.0000 0.0000 0.0000

3200 full 1.004 0.0000 0.0000 0.0000 0.0000 0.0001 0.4917 0.5079 0.0003 0.0000 0.0000 0.0000 0.0000

tr 1.006 0.0000 0.0000 0.0000 0.0000 0.0000 0.4893 0.5099 0.0008 0.0000 0.0000 0.0000 0.0000

6400 full 1.005 0.0000 0.0000 0.0000 0.0000 0.0000 0.4874 0.5126 0.0000 0.0000 0.0000 0.0000 0.0000

tr 1.361 0.0011 0.0036 0.0762 0.0811 0.1559 0.1271 0.1200 0.1282 0.0709 0.1061 0.0284 0.1014

100 full 1.258 0.0000 0.0009 0.0301 0.0591 0.1680 0.1830 0.1765 0.1965 0.0867 0.0851 0.0081 0.0060

tr 1.178 0.0000 0.0003 0.0270 0.0685 0.1874 0.1745 0.1633 0.1619 0.0742 0.0952 0.0180 0.0297

200 full 1.109 0.0000 0.0000 0.0051 0.0330 0.1709 0.2468 0.2542 0.2066 0.0577 0.0253 0.0004 0.0000

tr 1.071 0.0000 0.0000 0.0035 0.0392 0.1875 0.2433 0.2254 0.1815 0.0622 0.0493 0.0039 0.0042

400 full 1.048 0.0000 0.0000 0.0001 0.0073 0.1370 0.3294 0.3311 0.1734 0.0196 0.0021 0.0000 0.0000

tr 1.028 0.0000 0.0000 0.0000 0.0071 0.1592 0.3179 0.3035 0.1708 0.0317 0.0093 0.0004 0.0001

.8 800 full 1.029 0.0000 0.0000 0.0000 0.0005 0.0705 0.4064 0.4223 0.0976 0.0027 0.0000 0.0000 0.0000

tr 1.028 0.0000 0.0000 0.0000 0.0004 0.0869 0.3914 0.3973 0.1166 0.0071 0.0003 0.0000 0.0000

1600 full 1.017 0.0000 0.0000 0.0000 0.0000 0.0231 0.4549 0.4896 0.0323 0.0001 0.0000 0.0000 0.0000

tr 1.016 0.0000 0.0000 0.0000 0.0000 0.0293 0.4562 0.4594 0.0550 0.0001 0.0000 0.0000 0.0000

3200 full 1.005 0.0000 0.0000 0.0000 0.0000 0.0024 0.4901 0.5035 0.0040 0.0000 0.0000 0.0000 0.0000

tr 1.010 0.0000 0.0000 0.0000 0.0000 0.0035 0.4822 0.5039 0.0104 0.0000 0.0000 0.0000 0.0000

6400 full 1.006 0.0000 0.0000 0.0000 0.0000 0.0001 0.4879 0.5120 0.0000 0.0000 0.0000 0.0000 0.0000

Table 9, .2, .8 continued: Frequencies for pBr,true/pBr,est for bivariate Gaussian–Weibulls with generating correlations .7 and .8 (tr:

truncated (on the Gaussian) bivariate Gaussian–Weibull, pl = .2, pu = .8; full: full bivariate Gaussian–Weibull)
Median pBr ratio frequencies

ρ n pBr ratio [0, .01] (.01, .02] (.02, .1] (.1, .2] (.2, .5] (.5, 1) (1, 2) [2, 5) [5, 10) [10, 50) [50, 100) [100, ∞)

tr 1.214 0.0000 0.0000 0.0009 0.0151 0.1594 0.2439 0.2503 0.1695 0.0642 0.0721 0.0137 0.0109

100 full 1.124 0.0000 0.0000 0.0008 0.0124 0.1426 0.2906 0.2914 0.2103 0.0406 0.0113 0.0000 0.0000

tr 1.095 0.0000 0.0000 0.0000 0.0024 0.1211 0.3294 0.2834 0.1665 0.0506 0.0416 0.0034 0.0016

200 full 1.075 0.0000 0.0000 0.0001 0.0011 0.0916 0.3556 0.4057 0.1377 0.0078 0.0004 0.0000 0.0000

tr 1.074 0.0000 0.0000 0.0000 0.0002 0.0607 0.3954 0.3537 0.1532 0.0286 0.0082 0.0000 0.0000

400 full 1.037 0.0000 0.0000 0.0000 0.0000 0.0366 0.4291 0.4734 0.0605 0.0004 0.0000 0.0000 0.0000

tr 1.034 0.0000 0.0000 0.0000 0.0000 0.0214 0.4485 0.4231 0.1006 0.0060 0.0004 0.0000 0.0000

.5 800 full 1.019 0.0000 0.0000 0.0000 0.0000 0.0063 0.4675 0.5119 0.0143 0.0000 0.0000 0.0000 0.0000

tr 1.021 0.0000 0.0000 0.0000 0.0000 0.0035 0.4691 0.4881 0.0391 0.0002 0.0000 0.0000 0.0000

1600 full 1.010 0.0000 0.0000 0.0000 0.0000 0.0003 0.4791 0.5199 0.0007 0.0000 0.0000 0.0000 0.0000

tr 1.009 0.0000 0.0000 0.0000 0.0000 0.0001 0.4864 0.5056 0.0079 0.0000 0.0000 0.0000 0.0000

3200 full 1.007 0.0000 0.0000 0.0000 0.0000 0.0000 0.4801 0.5199 0.0000 0.0000 0.0000 0.0000 0.0000

tr 1.002 0.0000 0.0000 0.0000 0.0000 0.0000 0.4949 0.5048 0.0003 0.0000 0.0000 0.0000 0.0000

6400 full 1.003 0.0000 0.0000 0.0000 0.0000 0.0000 0.4871 0.5129 0.0000 0.0000 0.0000 0.0000 0.0000

tr 1.166 0.0000 0.0000 0.0050 0.0382 0.1978 0.2114 0.1843 0.1482 0.0727 0.0931 0.0218 0.0275

100 full 1.141 0.0000 0.0000 0.0014 0.0231 0.1589 0.2614 0.2719 0.2100 0.0506 0.0222 0.0004 0.0001

tr 1.103 0.0000 0.0000 0.0002 0.0117 0.1706 0.2806 0.2292 0.1688 0.0687 0.0604 0.0052 0.0046

200 full 1.071 0.0000 0.0000 0.0001 0.0043 0.1121 0.3385 0.3617 0.1657 0.0158 0.0018 0.0000 0.0000

tr 1.056 0.0000 0.0000 0.0000 0.0008 0.1259 0.3463 0.2932 0.1717 0.0409 0.0203 0.0007 0.0002

400 full 1.038 0.0000 0.0000 0.0000 0.0001 0.0528 0.4148 0.4378 0.0936 0.0009 0.0000 0.0000 0.0000

tr 1.024 0.0000 0.0000 0.0000 0.0000 0.0654 0.4155 0.3696 0.1340 0.0138 0.0017 0.0000 0.0000

.6 800 full 1.024 0.0000 0.0000 0.0000 0.0000 0.0131 0.4578 0.5005 0.0286 0.0000 0.0000 0.0000 0.0000

tr 1.017 0.0000 0.0000 0.0000 0.0000 0.0192 0.4639 0.4491 0.0656 0.0022 0.0000 0.0000 0.0000

1600 full 1.010 0.0000 0.0000 0.0000 0.0000 0.0006 0.4810 0.5167 0.0017 0.0000 0.0000 0.0000 0.0000

tr 1.007 0.0000 0.0000 0.0000 0.0000 0.0023 0.4872 0.4944 0.0160 0.0001 0.0000 0.0000 0.0000

3200 full 1.005 0.0000 0.0000 0.0000 0.0000 0.0000 0.4885 0.5115 0.0000 0.0000 0.0000 0.0000 0.0000

tr 1.003 0.0000 0.0000 0.0000 0.0000 0.0000 0.4943 0.5048 0.0009 0.0000 0.0000 0.0000 0.0000

6400 full 1.002 0.0000 0.0000 0.0000 0.0000 0.0000 0.4940 0.5060 0.0000 0.0000 0.0000 0.0000 0.0000

Table 9, .4, .8: Frequencies for pBr,true/pBr,est for bivariate Gaussian–Weibulls with generating correlations .5 and .6 (tr: truncated (on

the Gaussian) bivariate Gaussian–Weibull, pl = .4, pu = .8; full: full bivariate Gaussian–Weibull)
Median pBr ratio frequencies

ρ n pBr ratio [0, .01] (.01, .02] (.02, .1] (.1, .2] (.2, .5] (.5, 1) (1, 2) [2, 5) [5, 10) [10, 50) [50, 100) [100, ∞)

tr 1.166 0.0001 0.0002 0.0290 0.0772 0.2036 0.1578 0.1314 0.1277 0.0664 0.1101 0.0299 0.0666

100 full 1.215 0.0000 0.0000 0.0064 0.0357 0.1681 0.2196 0.2384 0.2113 0.0743 0.0431 0.0028 0.0003

tr 1.028 0.0000 0.0000 0.0063 0.0544 0.2146 0.2167 0.1709 0.1544 0.0700 0.0779 0.0168 0.0180

200 full 1.085 0.0000 0.0000 0.0002 0.0122 0.1445 0.3052 0.3177 0.1866 0.0284 0.0052 0.0000 0.0000

tr 1.070 0.0000 0.0000 0.0003 0.0145 0.1922 0.2680 0.2373 0.1788 0.0589 0.0432 0.0038 0.0030

400 full 1.033 0.0000 0.0000 0.0001 0.0012 0.0854 0.3923 0.3980 0.1178 0.0049 0.0003 0.0000 0.0000

tr 1.029 0.0000 0.0000 0.0000 0.0024 0.1358 0.3471 0.3152 0.1584 0.0303 0.0105 0.0002 0.0001

.7 800 full 1.019 0.0000 0.0000 0.0000 0.0000 0.0318 0.4498 0.4681 0.0503 0.0000 0.0000 0.0000 0.0000

tr 1.002 0.0000 0.0000 0.0000 0.0000 0.0623 0.4358 0.3882 0.1082 0.0047 0.0008 0.0000 0.0000

1600 full 1.011 0.0000 0.0000 0.0000 0.0000 0.0049 0.4787 0.5073 0.0091 0.0000 0.0000 0.0000 0.0000

tr 1.012 0.0000 0.0000 0.0000 0.0000 0.0141 0.4722 0.4715 0.0419 0.0002 0.0001 0.0000 0.0000

3200 full 1.002 0.0000 0.0000 0.0000 0.0000 0.0002 0.4954 0.5043 0.0001 0.0000 0.0000 0.0000 0.0000

tr 1.001 0.0000 0.0000 0.0000 0.0000 0.0012 0.4974 0.4942 0.0072 0.0000 0.0000 0.0000 0.0000

6400 full 1.001 0.0000 0.0000 0.0000 0.0000 0.0000 0.4976 0.5024 0.0000 0.0000 0.0000 0.0000 0.0000

tr 1.123 0.0024 0.0103 0.1234 0.0990 0.1455 0.1055 0.0866 0.0868 0.0581 0.0999 0.0356 0.1469

100 full 1.222 0.0000 0.0003 0.0328 0.0616 0.1695 0.1822 0.1820 0.1926 0.0894 0.0753 0.0072 0.0071

tr 1.113 0.0000 0.0008 0.0646 0.0986 0.1809 0.1337 0.1244 0.1215 0.0685 0.1056 0.0294 0.0720

200 full 1.104 0.0000 0.0000 0.0057 0.0329 0.1775 0.2470 0.2554 0.2023 0.0550 0.0230 0.0011 0.0001

tr 1.059 0.0000 0.0000 0.0226 0.0707 0.2051 0.1866 0.1638 0.1606 0.0758 0.0792 0.0148 0.0208

400 full 1.061 0.0000 0.0000 0.0001 0.0062 0.1321 0.3247 0.3397 0.1753 0.0198 0.0021 0.0000 0.0000

tr 1.036 0.0000 0.0000 0.0025 0.0286 0.2029 0.2528 0.2296 0.1842 0.0574 0.0379 0.0030 0.0011

.8 800 full 1.028 0.0000 0.0000 0.0000 0.0003 0.0751 0.4012 0.4164 0.1053 0.0017 0.0000 0.0000 0.0000

tr 1.036 0.0000 0.0000 0.0000 0.0055 0.1460 0.3308 0.3170 0.1657 0.0283 0.0067 0.0000 0.0000

1600 full 1.024 0.0000 0.0000 0.0000 0.0000 0.0223 0.4520 0.4924 0.0333 0.0000 0.0000 0.0000 0.0000

tr 1.009 0.0000 0.0000 0.0000 0.0002 0.0784 0.4146 0.3872 0.1141 0.0054 0.0001 0.0000 0.0000

3200 full 1.006 0.0000 0.0000 0.0000 0.0000 0.0027 0.4900 0.5030 0.0043 0.0000 0.0000 0.0000 0.0000

tr 1.001 0.0000 0.0000 0.0000 0.0000 0.0229 0.4757 0.4617 0.0395 0.0002 0.0000 0.0000 0.0000

6400 full 1.005 0.0000 0.0000 0.0000 0.0000 0.0001 0.4898 0.5100 0.0001 0.0000 0.0000 0.0000 0.0000

Table 9, .4, .8 continued: Frequencies for pBr,true/pBr,est for bivariate Gaussian–Weibulls with generating correlations .7 and .8 (tr:

truncated (on the Gaussian) bivariate Gaussian–Weibull, pl = .4, pu = .8; full: full bivariate Gaussian–Weibull)
Median pBr ratio frequencies

ρ n pBr ratio [0, .01] (.01, .02] (.02, .1] (.1, .2] (.2, .5] (.5, 1) (1, 2) [2, 5) [5, 10) [10, 50) [50, 100) [100, ∞)

tr 1.148 0.0000 0.0000 0.0004 0.0082 0.1404 0.2857 0.2975 0.2056 0.0466 0.0148 0.0007 0.0001

100 full 1.150 0.0000 0.0000 0.0008 0.0142 0.1426 0.2761 0.3025 0.2122 0.0411 0.0104 0.0001 0.0000

tr 1.064 0.0000 0.0000 0.0000 0.0004 0.0848 0.3724 0.3832 0.1470 0.0112 0.0010 0.0000 0.0000

200 full 1.066 0.0000 0.0000 0.0000 0.0008 0.0910 0.3652 0.3896 0.1452 0.0077 0.0005 0.0000 0.0000

tr 1.030 0.0000 0.0000 0.0000 0.0000 0.0332 0.4376 0.4619 0.0666 0.0007 0.0000 0.0000 0.0000

400 full 1.031 0.0000 0.0000 0.0000 0.0000 0.0368 0.4339 0.4671 0.0619 0.0003 0.0000 0.0000 0.0000

tr 1.019 0.0000 0.0000 0.0000 0.0000 0.0046 0.4698 0.5124 0.0132 0.0000 0.0000 0.0000 0.0000

.5 800 full 1.019 0.0000 0.0000 0.0000 0.0000 0.0064 0.4677 0.5143 0.0116 0.0000 0.0000 0.0000 0.0000

tr 1.005 0.0000 0.0000 0.0000 0.0000 0.0002 0.4910 0.5082 0.0006 0.0000 0.0000 0.0000 0.0000

1600 full 1.005 0.0000 0.0000 0.0000 0.0000 0.0004 0.4906 0.5087 0.0003 0.0000 0.0000 0.0000 0.0000

tr 1.004 0.0000 0.0000 0.0000 0.0000 0.0000 0.4896 0.5104 0.0000 0.0000 0.0000 0.0000 0.0000

3200 full 1.004 0.0000 0.0000 0.0000 0.0000 0.0000 0.4893 0.5107 0.0000 0.0000 0.0000 0.0000 0.0000

tr 1.002 0.0000 0.0000 0.0000 0.0000 0.0000 0.4922 0.5078 0.0000 0.0000 0.0000 0.0000 0.0000

6400 full 1.002 0.0000 0.0000 0.0000 0.0000 0.0000 0.4921 0.5079 0.0000 0.0000 0.0000 0.0000 0.0000

tr 1.141 0.0000 0.0000 0.0007 0.0134 0.1657 0.2601 0.2734 0.2028 0.0567 0.0259 0.0012 0.0001

100 full 1.143 0.0000 0.0000 0.0017 0.0176 0.1682 0.2513 0.2776 0.2084 0.0538 0.0210 0.0003 0.0001

tr 1.086 0.0000 0.0000 0.0000 0.0031 0.1089 0.3423 0.3585 0.1691 0.0156 0.0025 0.0000 0.0000

200 full 1.086 0.0000 0.0000 0.0000 0.0045 0.1134 0.3358 0.3631 0.1693 0.0125 0.0014 0.0000 0.0000

tr 1.036 0.0000 0.0000 0.0000 0.0000 0.0515 0.4188 0.4316 0.0964 0.0016 0.0001 0.0000 0.0000

400 full 1.037 0.0000 0.0000 0.0000 0.0000 0.0566 0.4127 0.4376 0.0923 0.0008 0.0000 0.0000 0.0000

tr 1.025 0.0000 0.0000 0.0000 0.0000 0.0117 0.4568 0.5039 0.0276 0.0000 0.0000 0.0000 0.0000

.6 800 full 1.025 0.0000 0.0000 0.0000 0.0000 0.0135 0.4544 0.5067 0.0254 0.0000 0.0000 0.0000 0.0000

tr 1.011 0.0000 0.0000 0.0000 0.0000 0.0011 0.4806 0.5160 0.0023 0.0000 0.0000 0.0000 0.0000

1600 full 1.011 0.0000 0.0000 0.0000 0.0000 0.0011 0.4804 0.5168 0.0017 0.0000 0.0000 0.0000 0.0000

tr 1.003 0.0000 0.0000 0.0000 0.0000 0.0000 0.4923 0.5076 0.0001 0.0000 0.0000 0.0000 0.0000

3200 full 1.003 0.0000 0.0000 0.0000 0.0000 0.0000 0.4920 0.5079 0.0001 0.0000 0.0000 0.0000 0.0000

tr 1.003 0.0000 0.0000 0.0000 0.0000 0.0000 0.4915 0.5085 0.0000 0.0000 0.0000 0.0000 0.0000

6400 full 1.003 0.0000 0.0000 0.0000 0.0000 0.0000 0.4915 0.5085 0.0000 0.0000 0.0000 0.0000 0.0000

Table 9, .4, 1: Frequencies for pBr,true/pBr,est for bivariate Gaussian–Weibulls with generating correlations .5 and .6 (tr: truncated (on

the Gaussian) bivariate Gaussian–Weibull, pl = .4, pu = 1; full: full bivariate Gaussian–Weibull)
Median pBr ratio frequencies

ρ n pBr ratio [0, .01] (.01, .02] (.02, .1] (.1, .2] (.2, .5] (.5, 1) (1, 2) [2, 5) [5, 10) [10, 50) [50, 100) [100, ∞)

tr 1.171 0.0000 0.0000 0.0037 0.0340 0.1754 0.2353 0.2324 0.1969 0.0733 0.0449 0.0032 0.0009

100 full 1.175 0.0000 0.0000 0.0073 0.0377 0.1732 0.2292 0.2360 0.2029 0.0716 0.0402 0.0012 0.0007

tr 1.082 0.0000 0.0000 0.0002 0.0083 0.1443 0.3078 0.3125 0.1833 0.0346 0.0090 0.0000 0.0000

200 full 1.082 0.0000 0.0000 0.0005 0.0121 0.1444 0.3035 0.3165 0.1843 0.0323 0.0064 0.0000 0.0000

tr 1.042 0.0000 0.0000 0.0000 0.0004 0.0826 0.3876 0.3969 0.1268 0.0055 0.0002 0.0000 0.0000

400 full 1.043 0.0000 0.0000 0.0000 0.0006 0.0870 0.3824 0.4013 0.1243 0.0042 0.0002 0.0000 0.0000

tr 1.019 0.0000 0.0000 0.0000 0.0000 0.0264 0.4567 0.4658 0.0509 0.0002 0.0000 0.0000 0.0000

.7 800 full 1.019 0.0000 0.0000 0.0000 0.0000 0.0297 0.4532 0.4694 0.0475 0.0002 0.0000 0.0000 0.0000

tr 1.007 0.0000 0.0000 0.0000 0.0000 0.0042 0.4861 0.4991 0.0106 0.0000 0.0000 0.0000 0.0000

1600 full 1.007 0.0000 0.0000 0.0000 0.0000 0.0046 0.4852 0.5004 0.0098 0.0000 0.0000 0.0000 0.0000

tr 1.003 0.0000 0.0000 0.0000 0.0000 0.0000 0.4928 0.5069 0.0003 0.0000 0.0000 0.0000 0.0000

3200 full 1.004 0.0000 0.0000 0.0000 0.0000 0.0000 0.4928 0.5069 0.0003 0.0000 0.0000 0.0000 0.0000

tr 1.005 0.0000 0.0000 0.0000 0.0000 0.0000 0.4831 0.5169 0.0000 0.0000 0.0000 0.0000 0.0000

6400 full 1.005 0.0000 0.0000 0.0000 0.0000 0.0000 0.4828 0.5172 0.0000 0.0000 0.0000 0.0000 0.0000

tr 1.261 0.0000 0.0001 0.0256 0.0572 0.1694 0.1857 0.1857 0.1807 0.0887 0.0875 0.0105 0.0089

100 full 1.269 0.0001 0.0004 0.0310 0.0591 0.1636 0.1832 0.1866 0.1867 0.0896 0.0843 0.0092 0.0062

tr 1.117 0.0000 0.0000 0.0034 0.0279 0.1858 0.2421 0.2436 0.2031 0.0632 0.0299 0.0009 0.0001

200 full 1.118 0.0000 0.0000 0.0053 0.0312 0.1842 0.2378 0.2469 0.2052 0.0628 0.0259 0.0006 0.0001

tr 1.057 0.0000 0.0000 0.0002 0.0063 0.1320 0.3317 0.3353 0.1695 0.0214 0.0036 0.0000 0.0000

400 full 1.059 0.0000 0.0000 0.0004 0.0078 0.1335 0.3282 0.3378 0.1700 0.0197 0.0026 0.0000 0.0000

tr 1.017 0.0000 0.0000 0.0000 0.0005 0.0694 0.4157 0.4146 0.0979 0.0019 0.0000 0.0000 0.0000

.8 800 full 1.018 0.0000 0.0000 0.0000 0.0006 0.0721 0.4130 0.4175 0.0955 0.0013 0.0000 0.0000 0.0000

tr 1.007 0.0000 0.0000 0.0000 0.0000 0.0189 0.4740 0.4704 0.0367 0.0000 0.0000 0.0000 0.0000

1600 full 1.007 0.0000 0.0000 0.0000 0.0000 0.0203 0.4724 0.4720 0.0353 0.0000 0.0000 0.0000 0.0000

tr 1.006 0.0000 0.0000 0.0000 0.0000 0.0013 0.4890 0.5038 0.0059 0.0000 0.0000 0.0000 0.0000

3200 full 1.006 0.0000 0.0000 0.0000 0.0000 0.0015 0.4885 0.5045 0.0055 0.0000 0.0000 0.0000 0.0000

tr 1.004 0.0000 0.0000 0.0000 0.0000 0.0000 0.4918 0.5080 0.0002 0.0000 0.0000 0.0000 0.0000

6400 full 1.004 0.0000 0.0000 0.0000 0.0000 0.0000 0.4917 0.5081 0.0002 0.0000 0.0000 0.0000 0.0000

Table 9, .4, 1 continued: Frequencies for pBr,true/pBr,est for bivariate Gaussian–Weibulls with generating correlations .7 and .8 (tr:

truncated (on the Gaussian) bivariate Gaussian–Weibull, pl = .4, pu = 1; full: full bivariate Gaussian–Weibull)
0 1000 2000 3000 4000 5000 6000

0.75 0.80 0.85 0.90 0.95

Sample size

Coverage

full data CI

.4,.8 data CI

  	

 
  
   

 


   
  ! "  #   


 

 
   

  $%&

  

" 
 #   
  

 
   '()


 *  $+  $%&

   " 



 
    
  
 

 "  



    ,  )) 
 

 "  
 


     ) 
 

 " 
 
 
  

 
   

  '()   

0 1000 2000 3000 4000 5000 6000

0.80 0.85 0.90 0.95

Sample size

Coverage

full data CI

.4,.8 data CI

  	

 
  
   

 


   
  ! "  #   


 

 
   

  $%&

  

" 
 #   
  

 
   '(!


 )  $*  $%&

   " 



 
    
  
 

 "  



      !! 
 

 "  
 


   +  ! 
 

 " 
 
 
  

 
   

  '(!   

0 1000 2000 3000 4000 5000 6000

0.70 0.75 0.80 0.85 0.90 0.95

Sample size

Coverage

.4,.8 data CI

full data CI

  	

 
  
  
  




 
    !"#  $

  $
 %

 & '' " 
 (   
  

 


   
 )  *+  *,#

 

 " $- (   
  

 
 

 

  *,#

   " 
 %
 
  

 
 ' 
 

 "  %
 
    '

 ' 
 

 "  %
 
    '. 

' 
 

 " 
 
 
    $


    

   

0 1000 2000 3000 4000 5000 6000

0.65 0.70 0.75 0.80

Sample size

Coverage

full data CI

.4,.8 data CI

  	

 
  
  
  



 
    !"#  $

  $
 %  &

'' " $ (   
  

 
 

 

  )*#

   " 
 (  


  

 
    
 +  ),

 )*#

   " 
 %
 
   


 ' 
 

 "  %
 
    ' 

'- 
 

 "  %
 
    '.  '


 

 " 
 
 
    $
 

 

     

)3/6WDWLVWLFV*URXS

(VWLPDWLRQRIDOHIWDQGULJKWWUXQFDWHGRQWKH

*DXVVLDQELYDULDWH*DXVVLDQ:HLEXOO

GLVWULEXWLRQ

'LVFODLPHURI:DUUDQWLHV

:HOFRPHWRWKHOHIWDQGULJKWWUXQFDWHGRQWKH*DXVVLDQELYDULDWH*DXVVLDQ:HLEXOOHVWLPDWLRQ

SURJUDP

:HDOVRKDYHSURJUDPVWKDWZLOOKDQGOHWKHIROORZLQJNLQGVRIGDWD

)XOOXQWUXQFDWHGELYDULDWH*DXVVLDQ:HLEXOOGDWD

/HIWWUXQFDWHGRQWKH*DXVVLDQELYDULDWH*DXVVLDQ:HLEXOOGDWD

5LJKWWUXQFDWHGRQWKH*DXVVLDQELYDULDWH*DXVVLDQ:HLEXOOGDWD

'RFXPHQWDWLRQIRUWKHSURJUDPFDQEHIRXQGLQVHFWLRQRI86'$)RUHVW3URGXFWV/DERUDWRU\

5HVHDUFK3DSHU[[[0D[LPXP/LNHOLKRRG(VWLPDWLRQRIWKH3DUDPHWHUVRID%LYDULDWH*DXVVLDQ

:HLEXOO'LVWULEXWLRQIURP0DFKLQH6WUHVV5DWHG'DWDSGI

$VFXUUHQWO\ZULWWHQWKLVSURJUDPFDQKDQGOHDWPRVWELYDULDWHREVHUYDWLRQV,IWKLVLVLQVXIILFLHQW

IRU\RXUSXUSRVHVSOHDVHFRQWDFW6WHYH9HUULOODWRUDWsverrill@fs.fed.us

%HIRUHSURFHHGLQJZLWKWKHDQDO\VLV\RXPXVWSURYLGHWKHGDWDILOH,I\RXKDYHDOUHDG\GRQHVR\RX

PD\SURFHHGZLWKWKHIRUPEHORZ

:KDWLVWKHQDPHRIWKHGDWDILOH"

:KDWLVWKHQDPHRIWKHUHVXOWVILOH"

(VWLPDWLRQRIDOHIWDQGULJKWWUXQFDWHGRQWKH*DXVVLDQELYDULDWH*D KWWSZZZISOIVIHGXVILWBWUBJDXVVBZHLEKWPO

RI $0

Figure 5: Page 1 of sample input to the fit tr gauss weib Web program.
7KHQDPHVKRXOGEHXQLTXHWR\RXWRSUHYHQWWKHILOHIURPEHLQJDFFLGHQWDOO\RYHUZULWWHQE\DQRWKHU

XVHU

6LQFH\RXZLOOVHHDQKWPOGRFXPHQWWKDWSUHVHQWVWKHUHVXOWVDVVRRQDV\RXH[HFXWHWKHSURJUDP\RX

PD\QRWZDQWDQDVFLLFRS\+RZHYHUZHGRSURYLGHWKLVRSWLRQ

7KHDVFLLILOHZLOOEHZULWWHQLQWKHSXEGDWDDQRQ\PRXVIWSGLUHFWRU\,IDILOHE\WKHVDPHQDPHDOUHDG\

H[LVWVLQWKHGLUHFWRU\WKHQHZUHVXOWVZLOOEHDSSHQGHGWRWKHH[LVWLQJILOH&KHFNWKHDQRQ\PRXVIWS

OLQNIRUGLUHFWLRQVDERXWUHWULHYLQJWKHILOH

:KDWLVWKHVDPSOHVL]H"

RUIHZHU

:KDWDUHWKHORZHUDQGXSSHUWUXQFDWLRQYDOXHV"

+RZPDQ\WULDOVLQWKHVLPXODWLRQ"

RUIHZHU

,PSRUWDQW7KHUHVSRQVHZLOOQRWEHLPPHGLDWH%HFDXVHVLPXODWLRQVDUHEHLQJUXQWKHUHZLOOEHDGHOD\

EHIRUHWKHUHVXOWVDSSHDU$QDSSUR[LPDWHIRUPXODIRUWKHQXPEHURIVHFRQGVQHHGHGWRSHUIRUPWKH

VLPXODWLRQVLVQ[1ZKHUHQLVWKHVDPSOHVL]HDQG1LVWKHQXPEHURIWULDOV)RUH[DPSOHWKH

WLPHQHHGHGWRUXQWULDOVRIVDPSOHVRIVL]HLVDSSUR[LPDWHO\VHFRQGV*LYHQHTXDO

VDPSOHVL]HVWULDOVIRUOHVVWUXQFDWHGGDWDVHWVHJDGDWDVHWWDNHOHVVWLPHWRFRPSOHWHWKDQWULDOV

IRUDPRUHWUXQFDWHGGDWDVHWHJDGDWDVHW

:KDWLVWKHLVWDUWYDOXHIRUWKHUDQGRPQXPEHUJHQHUDWRU"

HJ

)RUIXUWKHULQIRUPDWLRQSOHDVHFRQWDFW6WHYH9HUULOODWsverrill@fs.fed.usRU

(VWLPDWLRQRIDOHIWDQGULJKWWUXQFDWHGRQWKH*DXVVLDQELYDULDWH*D KWWSZZZISOIVIHGXVILWBWUBJDXVVBZHLEKWPO

RI $0

Figure 6: Page 2 of sample input to the fit tr gauss weib Web program.
'LVFODLPHURI:DUUDQWLHV

7KHUHVXOWVIURPWKLVUXQZLOOEHDSSHQGHGWRWKHILOH/export/home/ftp/pub/data/delme

7RVHHKRZWRGRZQORDGWKLVILOHUHDGWKLVGHVFULSWLRQRIDQRQ\PRXVIWS

$OWHUQDWLYHO\\RXVKRXOGEHDEOHWRGRD6DYH$VRU3ULQWLQ\RXUEURZVHU

7KHPD[LPXPOLNHOLKRRGHVWLPDWHVRIPXVLJPDUKRJDPPDDQGEHWDDUH

(((((

3DUDPHWHU

0/(

(VWLPDWH

&RQILGHQFH,QWHUYDOV

6LPXODWLRQEDVHG 7KHRU\EDVHG

PX ( ( ( ( (

VLJPD ( ( ( ( (

UKR ( ( ( ( (

JDPPD ( ( ( ( (

EHWD ( ( ( ( (

$FWXDOFRYHUDJHVRIQRPLQDOFRQILGHQFHLQWHUYDOV

3DUDPHWHU 7\SHRI

&RQILGHQFH,QWHUYDO

6LPXODWLRQ(VWLPDWH

RI$FWXDO&RYHUDJH

&RQILGHQFH,QWHUYDO

RQ$FWXDO&RYHUDJH

PX

VLPXODWLRQEDVHG  >@

WKHRU\EDVHG  >@

VLJPD

VLPXODWLRQEDVHG  >@

WKHRU\EDVHG  >@

KWWSZZZISOIVIHGXVFJLELQILWBWUBJDXVVBZHLESO

RI $0

Figure 7: Page 1 of sample output from the fit tr gauss weib Web program.
UKR

VLPXODWLRQEDVHG  >@

WKHRU\EDVHG  >@

JDPPD

VLPXODWLRQEDVHG  >@

WKHRU\EDVHG  >@

EHWD

VLPXODWLRQEDVHG  >@

WKHRU\EDVHG  >@

3DUDPHWHU

0/(

(VWLPDWH

&RQILGHQFH,QWHUYDOV

6LPXODWLRQEDVHG 7KHRU\EDVHG

PX ( ( ( ( (

VLJPD ( ( ( ( (

UKR ( ( ( ( (

JDPPD ( ( ( ( (

EHWD ( ( ( ( (

$FWXDOFRYHUDJHVRIQRPLQDOFRQILGHQFHLQWHUYDOV

3DUDPHWHU 7\SHRI

&RQILGHQFH,QWHUYDO

6LPXODWLRQ(VWLPDWH

RI$FWXDO&RYHUDJH

&RQILGHQFH,QWHUYDO

RQ$FWXDO&RYHUDJH

PX

VLPXODWLRQEDVHG  >@

WKHRU\EDVHG  >@

VLJPD

VLPXODWLRQEDVHG  >@

WKHRU\EDVHG  >@

UKR

VLPXODWLRQEDVHG  >@

WKHRU\EDVHG  >@

JDPPD

VLPXODWLRQEDVHG  >@

WKHRU\EDVHG  >@

KWWSZZZISOIVIHGXVFJLELQILWBWUBJDXVVBZHLESO

RI $0

Figure 8: Page 2 of sample output from the fit tr gauss weib Web program.
EHWD

VLPXODWLRQEDVHG  >@

WKHRU\EDVHG  >@

3DUDPHWHU

0/(

(VWLPDWH

&RQILGHQFH,QWHUYDOV

6LPXODWLRQEDVHG 7KHRU\EDVHG

PX ( ( ( ( (

VLJPD ( ( ( ( (

UKR ( ( ( ( (

JDPPD ( ( ( ( (

EHWD ( ( ( ( (

$FWXDOFRYHUDJHVRIQRPLQDOFRQILGHQFHLQWHUYDOV

3DUDPHWHU 7\SHRI

&RQILGHQFH,QWHUYDO

6LPXODWLRQ(VWLPDWH

RI$FWXDO&RYHUDJH

&RQILGHQFH,QWHUYDO

RQ$FWXDO&RYHUDJH

PX

VLPXODWLRQEDVHG  >@

WKHRU\EDVHG  >@

VLJPD

VLPXODWLRQEDVHG  >@

WKHRU\EDVHG  >@

UKR

VLPXODWLRQEDVHG  >@

WKHRU\EDVHG  >@

JDPPD

VLPXODWLRQEDVHG  >@

WKHRU\EDVHG  >@

EHWD

VLPXODWLRQEDVHG  >@

WKHRU\EDVHG  >@

KWWSZZZISOIVIHGXVFJLELQILWBWUBJDXVVBZHLESO

RI $0

Figure 9: Page 3 of sample output from the fit tr gauss weib Web program.
3DUDPHWHU

0/(

(VWLPDWH

&RQILGHQFH,QWHUYDOV

6LPXODWLRQEDVHG 7KHRU\EDVHG

PX ( ( ( ( (

VLJPD ( ( ( ( (

UKR ( ( ( ( (

JDPPD ( ( ( ( (

EHWD ( ( ( ( (

$FWXDOFRYHUDJHVRIQRPLQDOFRQILGHQFHLQWHUYDOV

3DUDPHWHU 7\SHRI

&RQILGHQFH,QWHUYDO

6LPXODWLRQ(VWLPDWH

RI$FWXDO&RYHUDJH

&RQILGHQFH,QWHUYDO

RQ$FWXDO&RYHUDJH

PX

VLPXODWLRQEDVHG  >@

WKHRU\EDVHG  >@

VLJPD

VLPXODWLRQEDVHG  >@

WKHRU\EDVHG  >@

UKR

VLPXODWLRQEDVHG  >@

WKHRU\EDVHG  >@

JDPPD

VLPXODWLRQEDVHG  >@

WKHRU\EDVHG  >@

EHWD

VLPXODWLRQEDVHG  >@

WKHRU\EDVHG  >@

KWWSZZZISOIVIHGXVFJLELQILWBWUBJDXVVBZHLESO

RI $0

Figure 10: Page 4 of sample output from the fit tr gauss weib Web program.