Procedures for Estimation 

of Weibull Parameters

James W. Evans

David E. Kretschmann

David W. Green

United States Department of Agriculture

Forest

Service

Forest Products

Laboratory

General Technical Report

FPL–GTR–264

February

2019
Abstract

The primary purpose of this publication is to provide an 

overview of the information in the statistical literature on 

the different methods developed for fitting a Weibull 

distribution to an uncensored set of data and on any 

comparisons between methods that have been studied in the 

statistics literature. This should help the person using a 

Weibull distribution to represent a data set realize some 

advantages and disadvantages of some basic methods. It 

should also help both in evaluating other studies using 

different methods of Weibull parameter estimation and in 

discussions on American Society for Testing and Materials

Standard D5457, which appears to allow a choice for the 

method to estimate the parameters of a Weibull distribution 

from a data set. Because in D5457 the method to estimate 

parameters is to some extent optional, the resulting fitted 

distribution used to derive the reference resistance 

properties of wood-based materials and structural 

connections might result in different values developed under 

the standard. The maximum-likelihood method appears to 

be the method that should be used as the default, with other 

methods requiring some type of justification for their use in 

wood utilization research.

Keywords: Weibull distribution, estimation of parameters, 

maximum likelihood estimation, regression estimators, 

simple percentile estimators, method of moments

February 2019

Evans, James W.; Kretschmann, David E.; Green, David W. 2019. 

Procedures for estimation of Weibull parameters. General Technical Report 

FPL–GTR–264. Madison, WI: U.S. Department of Agriculture, Forest 

Service, Forest Products Laboratory. 17 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.

Contents

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

2 Background .................................................................. 1

3 Estimation Procedures.................................................. 1

4 Historical Comparisons of Individual 

Estimator Types ........................................................ 8

5 Other Methods of Estimating Parameters of 

the Weibull Distribution.......................................... 11

6 Discussion .................................................................. 12

7 Conclusion ................................................................. 13

8 References.................................................................. 14

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1

now President, American Lumber Standard Committee, Inc., Frederick, Maryland, USA

1 Introduction

Two-parameter and three-parameter Weibull distributions 

are widely used to represent the strength distribution of 

structural lumber and engineering-designed wood 

subassemblies. Current design practices for wood and 

related material are based on an estimated fifth percentile of 

the strength distribution. This deterministic method of 

design is known as allowable stress design (ASD). Although

the fifth percentile can be based on a distributional form, 

such as the Weibull distribution, the real interest in the 

Weibull distribution is occuring as wood construction 

practices in the United States and Canada are revised from 

deterministic procedures to reliability-based design (RBD) 

procedures. The two-parameter Weibull distribution is the 

underlying basis of the calculations in load and resistance 

factor design (LRFD), a subset of RBD that is discussed in 

American Society for Testing and Materials (ASTM)

D5457–04a (ASTM 2009). This design procedure starts 

with fitting a two-parameter Weibull distribution to either a 

complete data set or a set of data representing the lower tail 

of the distribution. The fit can be made using either 

maximum-likelihood methods or a regression-based 

estimation procedure. Initially, the standard also considered 

a method of moments estimate, but this was dropped before 

the standard was approved. Because of the sensitivity of 

reliability calculations to the distributional form and,

potentially, method of fit, the wood engineering community 

is interested in looking at (1) evaluating the effect of using a 

two-parameter instead of a three-parameter Weibull 

distribution, (2) the effect of estimation method, and (3) the 

effect of using censored data sets compared with using full 

data sets. Simulations such as those performed by Durrans

and others (1998) encouraged the wood community to 

address some of these issues. Unfortunately, there is a 

general lack of knowledge in the wood community of the 

extensive work available in the statistics literature on these 

issues, which could lead to needless work in areas in which

the information is already available. The primary purpose of 

this publication is to provide an overview of the basic 

information in the statistical literature on the different 

methods developed for fitting a Weibull distribution to an 

uncensored set of data and on any comparisons between 

methods that have been studied in the statistics literature.

This report is not meant to be a complete summary of all the 

literature available on the Weibull distribution. It is intended 

to provide basic information on the more well-known 

methods for estimating Weibull parameters from data. We 

start by giving background on the Weibull distribution.

2 Background

The “Weibull” distributional form (Eq. (1)) was first derived 

through an extreme-value approach by Fisher and Tippett 

(1928). As noted by Mann (1968), it became known as the 

Fisher–Tippett Type III distribution of smallest values or as 

the third asymptotic distribution of smallest (extreme) 

values. In 1939, a Swedish scientist, Waloddi Weibull 

(1939a), derived the same distribution in an analysis of 

breaking strengths using only certain practical requirements.

Several examples of its use were given by Weibull (1951).

Two other papers by Weibull (1939b, 1952) also used the 

distribution. Use of the distribution became common in 

reliability analyses after World War II, and the name 

Weibull became firmly associated with the distribution.

In its three-parameter form, the Weibull family is 

represented by the density function,

( ) {( ) } {( ) } 1 1 / exp / a a f x ab x c b x c b − −   = − −−    

( ; , 0) x cab > > (1)

where a is the shape parameter, b the scale parameter, and 

c the location parameter. A more common representation of 

the Weibull distribution in the wood community is to use the 

cumulative distribution function,

( ) 1 exp / {( ) }

a F x x c b   =− − −     (2)

The family of two-parameter Weibull distributions follows 

from Equation (1) when c = 0.

3 Estimation Procedures

Numerous methods of estimating Weibull parameters have 

been suggested by many authors. Estimation procedures are 

generally categorized into one of four major categories:

(1) method of moment estimators, (2) linear estimators, such 

as least-squares-type estimators, (3) estimators based on few 

order statistics, and (4) maximum-likelihood estimators.

There are also a few hybrid estimator systems that combine 

Procedures for Estimation 

of Weibull Parameters

James W. Evans, Mathematical Statistician

David E. Kretschmann, Research General Engineer (retired1)

David W. Green, Supervisory Research General Engineer Emeritus

USDA Forest Service, Forest Products Laboratory, Madison, Wisconsin, USA
General Technical Report FPL–GTR–264

2

estimators from more than one category. Zanakis (1979) 

considered more than 17 combinations of simple Weibull 

estimators in addition to the maximum-likelihood 

estimators.

3.1 Method of Moments

The method of moments method of estimation was 

introduced by Karl Pearson (1894, 1895). The procedure 

consists of equating as many population moments to sample 

moments as there are parameters to estimate. Mathematical 

support for this procedure comes from the principle of 

moments as discussed in detail in Kendall and Stuart (1969).

In essence, this principle says that two distributions that

have a finite number of lower moments in common will be 

approximations of one another. Thus, the distribution of the 

data is approximated by equating the moments of a 

distributional form to the data moments.

To see how this could be done with the three-parameter 

Weibull distribution, and hence the two-parameter Weibull 

as a special case, let the population kth moments about the 

origin be given by

( ) k M x f x dx k

+∞

= ∫

−∞

(3)

For k equal to 1, this is just the expected value or mean of X, 

denoted E(X). The kth moment about the mean for the 

population (also called the kth central moment (CMk)), can 

be denoted by

CM ( 1 ) ( ) k

k x M f x dx

+∞

= − ∫

−∞

(4)

For a three-parameter Weibull distribution, the first three 

central moments are

CM 0 1 = (5)

( ) 2 2 CM2 21 = Γ −Γ b (6)

( ) 3 3 CM 3 2 3 3 12 1 = Γ − ΓΓ + Γ b (7)

where

Γ =Γ + j   1 ( j a/ )   (8)

and Гj is the well-known gamma function. The first three 

moments about the origin are

M cb 1 1 = +Γ (9)

2 2

2 21 M c b cb = + Γ+ Γ 2 (10)

33 2 2

3 321 M c b cb c b =+ + + Γ3Γ3Γ (11)

The corresponding sample moments are

( )

1

k

i

k

n x

m

n i

= ∑

=

and

( 1 ) cm

1

k

i

k

n x m

n i

− = ∑

=

where n is the number of observations. Note that m1 is the 

mean of the observations and cm2 is the variance of the 

observations. Johnson and Kotz (1970) and Talreja (1981) 

pointed out that Equations (6) and (7), along with their 

corresponding sample moments, are sufficient to estimate 

the shape and scale parameters of the Weibull distribution.

This is done by equating the measure of population 

skewness given by

( )

3

3/2

2

CM

CM

which is a function only of a (the shape parameter) to the 

corresponding sample skewness

( )

3

3/2

2

cm

cm

Because the population skewness is a function of the shape 

parameter, finding the value of a that gives the sample 

skewness requires either a table of a values and the 

corresponding skewness as is given in Johnson and Kotz 

(1970) or a computer program that will search for a value of 

a that gives the sample skewness. Given a, equating CM2 to 

cm2 (the sample variance) gives an estimate of b (the scale 

parameter). Finally noting that 

M cb 1 1 = +Γ

and equating this to m1 (the mean of the observations) gives 

an estimate of the location parameter for the three-parameter 

Weibull. These are sometimes called the method of 

moments estimates (Johnson and Kotz 1970). Note that it is 

possible for the estimate of the location parameter c to 

exceed the smallest observation. Dubey (1966b) proposed 

an alternative estimator:

( ) (1/ )

1 1 / a cX b n = −Γ     (12)

For the two-parameter Weibull distribution, estimation of 

the shape and scale parameters could be done as before and 

the location parameter taken as zero. Saylor (1977) 

proposed an alternative. When c = 0 is known, Equations (6)

and (8) are sufficient to estimate the shape and scale 
Procedures for Estimation of Weibull Parameters

3

parameters. Specifically,

( )

1

1/2 CM2

M

is a function of only the shape parameter. When equated to 

the sample moments

( )

1

1/2

1

1 mean

COV CM standard deviation

M = = (13)

an estimate of the shape parameter can be obtained. The 

scale parameter can come from equating the mean (m1) to 

the first moment about the origin (M1).

Note that these choices of moment estimators are not 

unique. Other choices would also generate estimates of the 

parameters for either the two- or three-parameter Weibull 

distribution. However, in general, method of moment 

estimation has used the smallest moments to provide 

estimates.

When only a two-parameter Weibull distribution is to be 

considered, a transformation to the extreme-value 

distribution is often used to generate moment estimators as 

follows:

If X ~ Weibull(a,b), i.e., if X has a Weibull distribution 

given by

( ) 1 exp / ( )

a F x x b =− −   

then the variable T = ln(X) will have an extreme-value 

distribution with cumulative distribution function

F t( ) =− − − 1 exp exp /   {(t l s ) }   (14)

The parameters l and s are location and scale parameters that 

can be estimated by the first two moments m1 and cm2 of the 

data on this scale (i.e., a mean and variance of the ti = ln (xi) 

values). The estimation is generally done one of three ways:

Method 1:

(1/2)

2 cm / n s S =

and

lm Ys = −1 ( n / )

where Sn and Yn are numerical constants depending upon the 

sample size n and are found in Gumbel (1960).

Method 2:

( ) [ ] 2

2 2 s nn = −     cm / 1 / ZM

and

1 l m Zs = −

where

Z p i i = − ln ln 1/ 1 {     ( ) } ,

p in i = + / 1 ( ) ,

2

2

2

1 1

ZM / /

n n

i i

i i

Z Z nn

= =

    = −          

∑ ∑ , and

Z is the mean of the Zi values.

Mann (1968) noted that a simplified form of method 2 can 

be derived from the fact that Z and ZM2 are asymptotically 

equal to Euler’s constant h = 0.5772... and π/ 6 , 

respectively. The resulting estimates are

Method 3:

{ [ ]}

1/2

2

6 cm / 1

π s n n  

=  −    

and

lm s = −i η

These simplified estimates were used in Menon (1963).

Given estimates of l and s, estimates of the original Weibull 

parameters can be obtained from the relationships

s a =1/

and

l b = ln( )

This approach is also often called the method of moments 

estimators for the Weibull distribution and will differ 

slightly in general from the earlier procedures. To help 

differentiate moment estimators on the extreme-value scale 

from those on the Weibull scale, some authors (Mann 1968)

refer to them as modified moment estimators.

3.2 Least Squares for Two-Parameter 

Weibull Distributions

The origin of the method of least squares as an estimation 

technique for Weibull parameters is less clear than that for 

the method of moments. This is in part because of the fact 

that several variations of the least squares approach have 

been proposed for parameter estimation. The general theory 

of using least squares to obtain estimates of the parameters 

of a location- and scale-dependent distribution was given by 

Lloyd (1952, 1962). Although this is different from the 

approach in the current ASTM standard, it is important to 

look at the procedure to understand the proposed procedures 

better. This procedure can be called a weighted least squares 

procedure.
General Technical Report FPL–GTR–264

4

We will limit our discussion initially to a two-parameter 

Weibull distribution. Recall that if X has a Weibull 

distribution given by

( ) 1 exp / ( )

a F x x b =− −   

then the variable T = ln(X) will have an extreme-value 

distribution with cumulative distribution function

F t( ) =− − − 1 exp exp /   {(t l s ) }   (15)

where l and s are location and scale parameters, 

respectively. Lloyd noted that for any location scale 

distribution, if we transform the observations by

( )/ V Tls r r = − (16a)

to get standardized variables, we get observations of a 

standardized variable whose distribution is parameter free 

and thus completely known. Let V(r) denote the rth order 

statistic from a sample of such standarized variables. Then,

its expectation for mean, variance, and covariances can

denoted by

( (r r ) ) ( ) EV A =

Var (r rr ) ( )   V W=  

and

( ) ( ) ( ) Cov , i j ij   VV W=  

where i is row and j is column. Then using Equation (16a) 

and solving for our original order statistics, we can show 

that

(r r ) ( ) E T l sA   = +   (16b)

Equation (16b) is in the format of our usual least squares 

regression model. If we let D be an n × 2 matrix with ones 

in the first column and the expected values of the order 

statistics, A(r), in the second column, then

(r) E T DB   =  

where

l

B

s

  =    

is a 2 × 1 vector of parameters to estimate. Note this is in the 

form E(Y) = XB, which is the usual form for our regression 

model. From the previous equations, we see

2 Var T sW ( )r   =  

where W is the n × n variance–covariance matrix with terms 

W(rr) on the diagonal and W(rs) in the rth row and sth column 

of the off diagional. Then, the least squares estimate of our 

parameters B are

( )

1 1 1

est r B dW d dW T − − − =   ′   ′

the ordinary weighted least squares regression. To use it for 

the extreme-value distribution and hence for the Weibull 

through transformation, we would take the ordered 

observations T(r), use the expected value of the order 

statistics to get the matrix D and, for the distribution in 

question, find the appropriate weights W. The resulting 

estimates will be best linear unbiased estimates and thus 

have very good properties. For example, the estimate of the 

center of a rectangular distribution from this method will 

have smaller variance than using the mean of the 

observations (Lloyd 1952, 1962).

For the extreme-value distribution, and hence the Weibull 

distribution, however, the appropriate weights were only

calculated for very limited sample sizes (generally less than 

n = 25). Some approximating methods existed for 

generating weights, but these procedures generally stop at 

n = 100 because of computational difficulties.

Recently, several papers have extended this type of 

procedure. In a recent paper, van Zyl and Schall (2012) 

proposed using

( ( ) )

( ( ) )

1 1 log

. .

. .

. .

1 log n

x

X

x

   

 

 

= 

 

 

 

 

W ww = diag( 1 , ..., n )

and

' 1 ) ˆ

θ (X WX X Wy − = ′

where

( ) 2 1 1 log 1 r

nn r n r

w

r n

− +  −+    =         +

for r = 1, . . . n. Then ( 1 ) ˆ ˆ b a = −θ / ˆ and 2 ˆ θ

ˆ a = . The paper 

includes simulations where a = 1.0 and b = 1.0 and then 

where a = 1.5 and b = 1.0. These are not typical values for 

lumber properties. But results for large sample sizes 

(n = 100 and n = 250) show very close estimates to the 

true values.
Procedures for Estimation of Weibull Parameters

5

Several methods exist for obtaining other least-squares-type 

estimators. Using graphical plotting techniques, Kimball

(1960) presented several possible estimators based on the 

extreme-value scale parameter, which is an inverse of the 

Weibull shape parameter, including a linear approximation 

to an unbiased version of the maximum-likelihood estimator 

of the inverse of the Weibull shape parameter. These 

methods are based on small complete samples. Other 

estimators include the “unbiased nearly best linear”

estimators of Blom (1958, 1962), a weighted-least-squares 

estimator of White (1965), and for uncensored data sets 

only, the estimators of McCool (1965) and Downton (1966).

Also similar in some ways to least squares estimates are 

nonlinear regression estimates, one of which is discussed in 

Berger and Lawrence (1974).

The methods contained in the ASTM standard are another 

variation that has been proposed by several authors. That 

method essentially ignores the weights (i.e., W becomes the 

identity matrix, and the procedure reduces to ordinary 

unweighted least squares). This procedure (which we will 

call ordinary least squares (OLS)) is related to the concept 

of probability plotting, which is often used for visually 

assessing the goodness-of-fit of a distribution to a data set.

If we plot the ordered observations versus the approximate 

expected value of the order statistics, the data should lie on a 

straight line. For the extreme-value distribution, we would 

plot T(i) versus ln ln 1( )i   − − p   where p in i = + / 1 ( ) . Then 

the slope of the line would be s and the intercept l, which

could then be transformed to the Weibull parameter 

estimates. Alternatively for the two-parameter Weibull, we 

could also plot ln( X(i) ) versus ln ln 1( )i   − − p   . The 

disagreement as to which variable should be considered the 

independent and which the dependent variable will be 

discussed later. The slope of this line would be (1/a), and 

the intercept would be ln(b).

Using regression procedures to estimate Weibull parameters 

from this graphical approach is not a new procedure. Miller 

and Freund (1965) presented the procedure in their textbook 

without a reference. Gumbel (1954) used it for the extremevalue distribution without a reference. Chernoff and 

Lieberman (1954), in looking at probability plots for the 

normal distribution, state the assumption “Let us assume 

that the visually fitted line is a very good approximation to 

the line that would be obtained by minimizing the sum of 

squared deviations (in the x direction) from the line.” This 

assumption is further pursued in Chernoff and Lieberman 

(1956). Kimball (1960) discusses these two earlier papers 

and then applies “least squares’ theory”, i.e., miminizing the 

sum of squared deviations, in getting estimates for the 

extreme-value distribution.

Other variations to the basic regression technique include 

different choices for pi and the use of robust regression 

methods, such as L1 regression, also called absolute 

deviation regression because we minimize the sum of the 

absolute values of the deviations. Lawrence and Shier 

(1981) include L1 regression in their comparison of 

estimation techniques, and numerous papers have proposed 

and evaluated other choices for pi.

3.3 Simple Percentile Estimators

Zanakis (1979) in a paper doing a simulation study presents 

some simplified parameter estimates for the three-parameter 

distribution in a practical way for simulations and scientists.

As discussed earlier, if we assume that data come from a 

Weibull distribution with the cumulative distribution 

function of Equation (2), then the population percentile xp

given by p = F(xp) can be determined by

( ) 1/ ln 1 a

p x cb p =+ − −     (17)

The paper uses in its simulation an estimate of the percentile 

of the ith ordered observation using

pin i = + / 1 ( )

and the corresponding 100pi percent sample percentile ti by 

the y[npi] where [ ] denotes rounding up (if npi is not an 

integer). In computer simulations, such rules are needed.

There is no evidence to suggest other standard estimates of 

the percentile of the ith order statistic or different rules 

related to calculating ti values would make a difference.

Given three such sample percentiles from the data, which 

we denote as ti, tj, and tk, which correspond to approximate 

cumulative probabilities pi, pj, and pk, it is possible to 

estimate the Weibull parameters by setting up three 

approximate equations from Equation (17) and then solving

system of approximate equations for a, b, and c using

( ) 1/ ln 1 a

s s t cb p =+ − −    

s i jk = , , (18)

With a little algebra, it follows that for a

( ) ( )

( ) ( )

1/ 1/

1/ 1/

ln 1 ln 1

ln 1 ln 1

a a

k j k j

a a j i j i

t t p p

t t p p

  − − −− −   −     = −   − − −− −      

If we further choose pj such that

( ) { ( ) ( ) }

1/2

ln 1 ln 1 ln 1 j ik − − =− − − − p pp       (19)

or

{ ( ) ( ) } 1/2 1 exp ln 1 ln 1 j i k p pp =− − − −     (20)
General Technical Report FPL–GTR–264

6

the estimate of the shape parameter aa� simplifies to

( )

(1 ) ˆ 1 ln 1 ln / ln

2 ln

k k j

i j i

p t t

a

p t t

    − −   −   =       − −   −      

(21)

Then our estimated scale ˆ

b and location cˆ parameters from 

Equation (18) are

( ) { ( ) ( ) } 1/ ˆ 1/ ˆ / ˆ ln 1 ln 1 a a

s r s r btt p = − − − −− −     p     (22)

where (r s i jk < ∈) ( , , ) and

{ ( )}

1/ ˆ ln 1 ˆ ˆ a

s s ct b p =− − − (23)

Note, the estimate ˆ

b is simplified if c is known. Then 

( ) ( ) 1/ ˆ 1 ˆ / ln a

s s btc p =− − −    

s i jk ∈(, , ) (24)

As is the case with any location estimate, it must be less 

than or equal to the smallest observation, and as with other 

estimation procedures, there is a possibility that our estimate 

will not meet this requirement and have to be modified.

Dubey (1967a) studied the two-parameter Weibull simple 

percentile estimators. When only two parameters need to be 

estimated, only two percentiles are needed. If we call them tk

and ti, the equation for the shape parameter simplifies to

( )

( )

ln 1

ln / ln

ln 1

k k

i i

p t

a

p t

    − −     =    

      − −  

(25)

Dubey showed that the asymptotic variance of this estimator 

is minimized when pi = 0.16731 and pk = 0.97366. This 

means we should take the sample 17 and 97 percentiles. An 

estimate of the shape parameter based on these two sample 

percentiles was shown by Dubey to be 66% efficient 

compared with the maximum-likelihood estimate of the 

shape parameter. Normally, 66% efficient is not a good 

value. But it is simple to calculate when in a field situation 

and can be used to give an indication of the shape of the 

distribution. Using these two values of pi and pk in Equation 

(20) would give pj = 0.5578.

Several other estimators based on simple percentiles have 

been proposed. For any three-parameter Weibull 

distribution, it is relatively simple to show that the 63rd 

population percentile is equal to c + b (the sum of the 

location and scale parameters), i.e.,

63 x cb = + (26)

Thus, for the two-parameter Weibull distribution, the scale 

parameter can be estimated by the 63rd sample percentile.

For the three-parameter Weibull, an estimate of either the 

scale or location parameter can be inserted into this 

relationship to get an estimate of the other parameter.

If the shape parameter is small (0 < a ≤ 2), Dubey (1966a) 

recommended estimating the location parameter with the 

smallest order statistic. Dubey (1967b) also proposed a 

simple estimate of the location parameter based on the first, 

second, and nth order statistic. This estimator

( ( ) ( ) ( ) ) ( ( ) ( ) ( ) ) 2

1 21 2 / 2 n n c XX X X X X = − +− (27)

is very close to the first order statistic.

Under the assumption that a good location parameter 

estimate such as Equation (26) is available, Hassanein 

(1972) created some asymptotically unbiased shape and 

scale estimators for which Mann and Fertig (1977) obtained 

small sample unbiased corrections that were evaluated in 

Zanakis (1979). Using the fact that the parameters of the 

extreme-value distribution and the Weibull distribution are 

related, Bain (1972) proposed for censored samples a 

simple, unbiased estimator of the scale parameter that is 

equivalent to the inverse of the shape parameter of the 

Weibull distribution. These latter estimators had high 

efficiency for censored samples but zero asymptotic relative 

efficiency for complete samples. Examples of further use of 

these estimators with real data has been relatively difficult 

to find. Englehardt and Bain (1977) developed confidence 

bounds on Weibull reliability and inference procedures 

using simplified estimators.

3.4 Maximum-Likelihood Estimation

As pointed out in Harter (1970), the use of the method of 

maximum likelihood dates back to Gauss (1809), who 

employed it for some particular problems. However, its 

general use was first proposed by Fisher (1912). Fisher 

(1921) began the study of the properties of maximumlikelihood estimators (MLE), which has been continued by 

numerous researchers. The major justification of maximumlikelihood estimates is usually its large-sample efficiency.

Under mild regularity conditions, the MLE of a single 

parameter from singly censored samples has been shown by 

Halperin (1952) to be consistent (i.e., converges to the true 

parameter value as sample size increases), to be 

asymptotically normally distributed, and to have minimum 

variance for large samples. Many of these properties can be 

extended to the case of several parameters and more general 

censoring.

With all these advantages, one could well ask why not 

always use maximum-likelihood estimates for the 

parameters of a Weibull distribution. The problems appear

to center around three areas of concern: (1) potential 

problems in calculating Weibull parameter estimates, (2) the 

bias of the estimates for small samples, and (3) the possible 

existence of more efficient and simpler estimates for small 

sample size. These problems are lessened in many lumber 

strength properties because the sample size is often quite
Procedures for Estimation of Weibull Parameters

7

large. But, in lumber property studies, many issues can arise 

from the sampling procedure used.

Efforts to derive MLEs of the parameters of a threeparameter Weibull distribution have received considerable 

attention in statistical literature. Leone and others (1960) 

showed the MLE of the scale parameter was a function of 

the shape and location parameters and the sample values.

MLEs for the shape and location were found through an 

iterative simultaneous solution of two equations involving a, 

c, and the sample values. Numerous advances followed, 

such as allowing various censoring situations (Harter and 

Moore 1965, 1967; Cohen 1965, 1973; Wingo 1972, 1973).

Lemon (1975) discusses some of these earlier advances and 

then gives a procedure based on the simultaneous solution 

of two iterative equations that allow both single left and 

progressive censoring. In getting three-parameter estimates 

from two equations, Lemon gives two formulas for the 

location parameter depending if the shape parameter is less 

than 1 or greater than or equal to 1. Zanakis and Kyparisis 

(1986) discuss these advances and the computational 

procedures that have been used to try to solve some of the 

iteration problems that can develop.

As previously mentioned, MLEs of Weibull parameters 

have been developed by several researchers. To understand 

these estimators, it is best to begin with the two-parameter 

Weibull distribution. In its two-parameter form, the Weibull 

family is represented by the density function

( ) ( ) (( ) ) 1 1 / exp / a a f x ab x b x b − −   = −

   

( 0; 0; 0) xab ≥>> (28)

where a is the shape parameter and b the scale parameter.

Consider a random sample of n observations. The likelihood 

function of this sample is

( )

( )( )

1

1

1

, . . . ; , 

 / / exp 

n

n a a i i

i

Lx x a b

x ab x b

b

−

=

      =   −        ∏ (29)

If we take the logarithm of L and differentiate with respect 

to a and b and then set the derivatives equal to zero, we get 

two equations:

1 1

ln 1 ln ln

n n a

i ii a

L n x xx a a b∂

=+ −

∂ ∑ ∑

2

1

ln 1 

n a

a a i

L n x b b b

∂

=− +

∂ ∑

Eliminating b in the two equations and simplifying gives

1

1 1

ln 1 1 ln

n a n i i

n i a

i

x x

x

x a n

 

  − =    

∑ ∑ ∑

which then can be solved through iterative procedures to 

give an estimate of the shape parameter, denoted by aˆ .

With aˆ determined, then

ˆ

1

 / ˆ n a

i b xn =∑

where the symbol (^) denotes that this is an estimate.

Computer routines to produce three-parameter Weibull 

estimates often show inconsistent convergence. Numerical 

problems exist, which can make difficult the direct 

calculation of maximum-likelihood estimates using 

optimization routines that maximize the likelihood. When 

the shape parameter is less than or equal to 2, the 

information matrix of the Weibull distribution has 

singularities, which means an iterative method that uses 

second derivatives is likely to be unstable. [This is discussed 

in Harter (1970) and further in Smith (1985, 1994) with 

references to Wingo (1973), Rockette and others (1974),

Lemon (1975), and Cohen (1975)] In fact, Weibull MLEs 

are regular if, and only if, at least one of three conditions 

hold:

(1) a > 2, 

(2) c is known, or

(3) the sample is censored from below (Harter 1970).

(Note: ASTM D5457 uses a two-parameter Weibull and 

thus meets criteria (2). This means computational problems 

and problems with asymptotic properties of MLE holding 

disappear.) For the shape parameter less than or equal to 1, 

the smallest observation becomes a hyper-efficient estimator 

of the location parameter (Dubey 1966b), but no true MLE 

for the location and scale exist. Usually in this case, the 

smallest observation or some other estimate of the location 

parameter is used and a pseudo MLE for the shape and scale 

is produced by assuming c is known and reducing the 

problem to a two-parameter Weibull estimation problem.

For 1 < a < 2, MLEs exist but problems with the 

information matrix make computational problems possible 

and make the asymptotic variance–covariance matrix 

meaningless. For a = 2 (the Rayleigh distribution), the 

determinant of the information matrix is zero and 

computational problems are possible.

Some concern is periodically expressed about the bias in the 

MLEs for small samples. It is well known that for small 

sample sizes, the MLE of the shape parameter can be quite 

biased. That may or may not be a problem for reliability 

calculations. For complete samples, Thoman and others

(1969) present a table of unbiasing factors for the shape 

parameter for sample sizes from N = 5 to 120. The factors 

are multiplied by the MLE of the shape parameter to get an 

unbiased estimate. The following selected entries from the 
General Technical Report FPL–GTR–264

8

tables give an indication of the amount of bias for different 

sample sizes (N):

N 5 6 7 8 10 14 28 44 64 120

Factor 0.669 0.752 0.792 0.820 0.859 0.901 0.951 0.970 0.980 0.990

This bias means that you should multiply the shape

parameter by the bias number. The biasing number

approaching 1 shows that the estimate is asymptotically 

consistent (because MLEs are consistent, and for consistent 

estimators, the bias will approach zero as the sample size 

increases). Similar tables are given in Billmann and others

(1972) in which either 25% or 50% of the largest 

observations were censored for sample sizes N = 40, 60, 80, 

100, 120. These results show that the greater the censoring, 

the greater the bias.

In spite of the potential problem of bias of the shape 

parameter, two separate studies indicated that functions 

estimated using estimates of the shape and scale parameters

often show little bias. Al-Baidhani and Sinclair (1987) 

compared several estimators of percentiles of the 

distribution. Using sample sizes N = 10, 25; shape parameter 

a = 0.5, 1.0, 2.0, 4.0; and scale parameter b = 0.1, 1.0, 10.0, 

100.0, they compared the estimation procedures on the 

parameter estimates and on their ability to predict the 95th 

percentile and the 99th percentile. In estimating percentiles, 

the authors concluded the “MLE is clearly the best method 

for all the values of (the shape parameter)”. Billman and 

others (1972) looked at the problem of estimating the 

reliability at time t, which for a two-parameter Weibull 

distribution is estimating

( ) exp ( )

a

R t t

b

  = −   

They concluded “It is fortunate that in spite of skewness, 

censoring and other difficulties, the MLE of R t( ) for 

reasonable values of R t( ) has negligible bias and its 

variance is very close to the Rao-Cramer lower bound for 

the variance of a regular unbiased estimator of R t( ).”

Because in LRFD we are estimating functions of the shape 

and scale parameters, three options appear possible. We can 

use the unbiasing factors, assume that the compensation the 

scale parameter makes to the shape parameter will negate 

any effect of the bias when estimating functions of both, or 

do simulations to check if there is a problem.

The potential problem of the possible existence of more 

efficient and simpler estimates for small sample sizes is not 

so easily answered. There are certainly some more efficient 

estimators for small samples than the MLE, as has been 

shown in numerous studies. However, our evaluation of the 

literature indicates that they are not method of moment 

estimators or OLS estimators. More work needs to be done 

to identify which of these other estimation procedures works

well as a set and produces efficient estimators for Weibull 

distribution characteristics such as the 5th percentile and the 

functions of the parameters needed in LRFD calculation 

procedures.

Any discussion of maximum-likelihood estimation of 

Weibull parameters would be incomplete without at least 

mentioning the existence of some simple, closed form 

approximations for MLEs of the parameters of the Weibull 

distribution. Bain (1972) and Engelhardt and Bain (1973, 

1974) developed simple estimators for the parameters of the 

extreme-value distribution. These relatively efficient, 

unbiased estimates can easily be converted through the 

relationship between the extreme-value distribution and the 

Weibull distribution to provide estimates of the Weibull 

distribution parameters. Their work showed that the simple 

estimators were related to maximum-likelihood estimators.

4 Historical Comparisons of 

Individual Estimator Types

Dubey (1966a) looked at the asymptotic (large sample) 

efficiency of moment estimators for the Weibull location 

and scale parameters assuming the shape parameter was 

known. This efficiency (defined as a ratio of the MLE 

generalized variance to the moment estimator generalized 

variance) for shape parameters from a = 2.1 to a = 100 had 

values ranging from 12% to 93%. A few specific values are 

shown in Table 1.

Talreja (1981) generated five random samples of size n = 10 

from each of two two-parameter Weibull distributions and 

used the method of moments described first to estimate 

three-parameter Weibull distribution estimates and twoparameter Weibull distribution estimates. Talreja concluded 

that the method could not be relied on for estimating all 

three parameters, especially for his example where the shape 

Table 1—A few specific values for shape parameters

Shape

Efficiency of location and scale

(%)

2.1 11.98

2.6 54.63

3.1 76.60

3.6 87.18

4.1 91.75

4.6 93.18

5.1 92.96

5.6 91.89

6.1 90.39

6.6 88.72

7.1 87.00

10.0 78.26

20.0 63.96

100.0 51.75
Procedures for Estimation of Weibull Parameters

9

parameter was large (a = 10). He also noted that in fitting a 

two-parameter Weibull, if a three-parameter Weibull 

distribution was used to generate the data (i.e., the location 

parameter was not zero), errors in estimating the parameter 

increased linearly with the true value of the location 

parameter, to the point that errors were more than 100%

when the true location parameter was equal to or greater 

than the true scale parameter.

Saylor (1977) also used a very small simulation with shape 

parameters less than 2 to compare method of moment 

estimators with maximum-likelihood estimates for a threeparameter Weibull. As an estimate of the location 

parameter, Saylor used Dubey’s estimate given in Equation 

(12) and the estimation procedure based on the mean and 

coefficient of variation previously described. This required 

iteration back and forth between the two methods until 

estimates remained constant. Saylor concluded that the 

location parameter estimate of Dubey was more efficient 

than the maximum-likelihood estimate of the location 

parameter and suggested use of Dubey’s estimate, 

iteratively, with the MLE for scale and shape. He also noted 

that the moment estimators appeared to be more efficient 

than MLEs for sample sizes of less than 50. However, he 

cautioned that further experimentation with other types of 

generators and larger Monte Carlo sample sizes was needed.

Newby (1984) used the method of moment estimators for 

the three-parameter Weibull derived using the mean, 

variance, and skewness. He compared efficiency of the 

method of moment estimators to the MLEs as a function of 

the true shape alone. He could only look at shape parameters 

greater than 2 because the asymptotic information matrix of 

the maximum-likelihood estimators has singularites when 

the shape is less than 2 (note this is why the use of iterative 

methods to get MLEs that are based on second derivatives 

are often unstable). Newby’s results are summarized in the 

table he produced (Table 2).

Mann (1968) looked at several estimators for the parameters 

of the two-parameter Weibull distribution including these 

modified moment estimators. Mann concluded that these 

estimators gave poor results for estimating s (and hence a) 

for small as well as large samples compared with other 

better estimation procedures including maximum likelihood.

With such a computationally easy method of estimating 

parameters available, compared with maximum-likelihood 

procedures, a natural question is why is a regression 

procedure not more widely used. There are several reasons.

A subtle reason deals with the loss of unbiasedness when 

unweighted estimates replace weighted estimates. Bias in 

estimates does not eliminate their use. Statistics is full of 

biased estimates being used in place of unbiased. The 

commonly used estimate of a standard deviation is one 

example. The argument for use of a biased estimate is 

usually that the bias is small and the variance so much 

smaller than the unbiased estimate that we will usually be 

closer to the true value than with the unbiased estimate. This 

argument is incorporated in the concept of mean square 

error (MSE) of an estimator. Mathematically,

MSE = Variance + Bias2

The square root of the MSE of an estimator is similar to a 

standard deviation except it incorporates bias. Use of a 

biased estimator usually involves showing that the MSE or 

square root of the MSE is smaller for that estimator than for 

other estimators. Thus, in evaluating least squares estimates 

of Weibull parameters, we must look at their MSE.

Kimball (1960) looked at the OLS estimator for small 

samples and different choices of pi for the extreme-value 

distribution. His results for s are shown in Table 3.

Thus, the choice of pi appears important. In the current 

ASTM standard, pi = (i – 0.3)/(n + 0.4) is used.

Properties of the OLS procedure for complete samples 

versus other estimation techniques have been studied by 

several researchers in addition to Kimball. Berger and 

Lawrence (1974) compared the OLS estimator with 

( ) ln i   X

  considered as the fixed points (independent 

variables) and ln ln 1 (− − ( p(i) )) as the random variables, as 

was done in Miller and Freund (1965) and as they said was 

“common industry practice”. This was compared with using 

nonlinear regression to fit the model on the original scale for 

sample size n = 50, shape parameters 0.5 to 4.0 by 0.5, and 

scale parameters 2, 4, 6, 8, and 10. 100 replications of each 

combination were simulated. Berger and Lawrence only 

considered pi = i/(n + 1). Their results showed that both 

procedures produced results with large MSE. They 

speculated that the reversal of the variables from 

Table 2—Joint asymptotic efficiency 

of estimators for the Weibull

Shape 

parameter

Efficiency of moment estimators

(%)

2.1 3

3.0 41

4.0 71

5.0 85

6.0 89

7.0 89

8.0 88

9.0 86

10.0 83

Table 3—Kimball (1960) results for s

Value of i p Bias Mean square error

1. i/(n + 1) 0.2284 0.3011

2. [i – (3/8)]/[n + (1/4)] 0.0220 0.1754

3. [i – (1/2)]/n –0.0547 0.1543
General Technical Report FPL–GTR–264

10

considering ( ) ln i   X

  as the dependent variable and 

ln ln 1 (− − ( p(i) )) as the independent variable contributed to 

the large variability.

Lawrence and Shier (1981) extended the earlier work of 

Berger and Lawrence (1974). Both the earlier OLS 

estimator with ( ) ln i   X

  considered as the independent 

variable and ln ln 1 (− − ( p(i) )) as the dependent variable and 

the OLS with the roles reversed (the ASTM procedure) were 

compared with least absolute deviation (L1) regression in 

which the sum of the absolute values of the deviates is 

minimized. The L1 regressions considered both roles for the 

variables as did the OLS estimators. The values of pi

considered were

(1) (i – 0.5)/n

(2) (i – 0.375)/(n + 0.25)

(3) (i – 0.3)/(n + 0.4) (ASTM procedure)

Sample sizes N = 20, 30, 40, 50, 100; shape parameters of 

0.5, 1.0, 3.0, 5.0, 10.0; and a scale parameter of 1.0 were 

considered. Again, 100 replications of each combination 

were generated. Lawrence and Shier (1981) concluded that 

using ( ) ln i   X

  as the independent variable and 

ln ln 1 (− − ( p(i) )) as the dependent variable produced more 

efficient estimates of the shape parameter. This is how the 

ASTM procedure is done. They also concluded that L1

regression was 10% to 30% more efficient in estimating the 

shape than was the OLS estimator. Estimation results for the 

scale parameter were mixed with no apparent winner. They 

also concluded that the number 2 definition of pi was the 

most consistent performer.

Engeman and Keefe (1982) compared the simple moment 

estimates of Menon, maximum-likelihood estimates, 

ordinary least squares estimates, and the “unbiased nearly 

best linear” estimates of Blom (1958, 1962) for the twoparameter Weibull for sample size n = 25; true shape 

parameters 0.5, 1, 2, and 4; and true scale parameters 0.1, 1, 

10, and 100. They used i p = i/(n + 1) for the OLS 

estimator. The number of replications of this simulation 

were not reported. To standardize the MSEs of the 

estimators for the different parameterizations, they reported 

the ratio of the Cramer–Rao lower bounds for the variance 

of unbiased estimates of the two-parameter Weibull to the 

observed MSE of the estimator. The Cramer–Rao lower 

bounds are calculated from

Var(a) = 0.608a2

/n

and

Var(b) = 1.109(b/a)

2

/n

The average relative efficiencies they reported were as 

follows:

Blom

OLS

Menon

MLE

Shape parameter a 0.8453

0.6275

0.4840

0.6862

Scale parameter b 1.0207

0.9903

1.0187

1.0483

These results show the relative efficiency of the simple 

moment estimators to be substantially below those of all 

other estimators of the shape parameter. For this sample 

size, the best estimate of shape was that of Blom and for 

scale the MLE. The MLE was slightly better than the OLS 

for both parameters.

Shier and Lawrence (1984) compared the OLS estimator 

and the L1 estimator with a number of reweighting schemes 

based on the L1 residuals. Using pi = (i – 0.5)/n; sample 

sizes N = 20, 30, 40, 50, 60; shape parameters a = 0.5, 1.0, 

3.0, 5.0, 10.0; and scale parameter b = 1, they generated 100 

replications for each combination. They again considered

( ) ln i   X

  as the independent variable. Their results showed 

the OLS estimator to be “clearly inferior to all of these five 

L1-based procedures. ... On average, the five procedures are 

some 14% more efficient than the (OLS) procedure”.

Al-Baidhani and Sinclair (1987) compared a generalized 

least squares (GLS) estimate that takes into account the 

variance–covariance matrix, the OLS estimator, the MLE, a 

method based on Hazard plotting position, and two mixed 

methods. Using pi = i/(n + 1); sample sizes n = 10, 25; shape 

parameter a = 0.5, 1.0, 2.0, 4.0; and scale parameter b = 0.1, 

1.0, 10.0, 100.0, they compared the estimation procedures 

on the parameter estimates and on their ability to predict the 

95th percentile and the 99th percentile. These percentiles 

were chosen to look at problems engineers might have in 

predicting the size of floods that occur only once in 

100 years. The number of replications was 1,000. Results 

showed that the highest average relative efficiency for the 

shape parameter was the GLS estimator, whereas the OLS 

estimator had the lowest average relative efficiency. For the 

scale parameter, the MLE was better than the OLS for every 

case considered and either best or second best in every case 

compared with all the estimators. In estimating percentiles, 

the authors concluded that “MLE is clearly the best method 

for all the values of (the shape parameter)”. Looking at 
Procedures for Estimation of Weibull Parameters

11

parameter and percentile estimation combined, they 

concluded “the worst of these methods in every situation 

was OLS”.

Durrans and others (1997), using the two-parameter and 

three-parameter Weibull distribution on censored data, 

looked at several methods of estimating quantiles and 

parameters that would be used in quality control in 

describing structural lumber failure strengths. They 

investigated MLEs and regression estimators. On the basis 

of their simulations, they concluded that “the method of 

maximum likelihood is nearly universally superior to a 

regression method . . .”. The advantage of regression 

methods is that they will always lead to a solution, whereas

maximum-likelihood methods may not converge in some 

cases.

When the results of all these studies are taken together, the 

ASTM standard appears to have several weaknesses. It does

not appear to use the most efficient choice of pi. They are 

less efficient than other regression type estimators that are 

easily calculated. The direct comparisons to MLEs showed 

superiority of the MLE for even a small sample size.

5 Other Methods of Estimating

Parameters of the Weibull 

Distribution

Previously, the most common methods of estimation of 

Weibull parameters have been discussed. Each of the 

methods covered has had extensive discussion in statistical 

literature. There are a few other methods that have been 

raised that should be discussed. These methods might 

benefit from more study, but they help provide a broader 

view than just the most common methods.

5.1 Idea of Mixing Types of Methods

Initially, estimation of Weibull parameters was generally 

restricted to picking one method of estimation and then 

using that method to estimate all the parameters of the 

Weibull distribution. There are several sound reasons for 

considering this approach that will be covered next. It has 

already been mentioned that in some cases the estimate of 

the location parameter for a three-parameter Weibull 

distribution can be above the lowest order statistic. For some 

people who are dealing with strength properties of materials, 

this can be a problem. Many times, computer programs 

calculating maximum-likelihood estimates of Weibull 

parameters will substitute the lowest order statistic for the 

location parameter, which follows Dubey’s (1966a) 

recommendation when the shape parameter is small 

(0 < a ≤ 2). When the estimated shape parameter is greater 

than 2, Dubey’s (1967b) estimate in Equation (26) is 

sometimes used.

The idea of mixing types of estimators was studied in 

Zanakis (1979). In a simulation study, Zanakis studied how 

various simple estimators for the shape, scale, and location 

parameters compared regarding their accuracy in a 

maximum-likelihood sense. He concludes that the best 

estimators are Equation (27) for location, Equation (26) for 

scale where cˆ is put in c, and Equation (25) for shape where 

pi = 0.16731 and pk = 0.97366. He furthermore stated that 

these three estimates proved to be more accurate than 

maximum-likleihood estimates for all three parameters, 

especially when a < 2 (Zanakis 1977, 1978).

Some programs used to calculate maximum-likelihood 

estimates benefit from an initial estimate of the parameters.

When this is the case, an initial estimate of the parameters 

using Zanakis’ recommendations might prove useful.

5.2 Minimum Chi-Square Estimation

As an alternative to maximum-likelihood estimation, 

Berkson (1980) proposed using minimum chi-square 

estimation in response to a previous publication (Efron 

1975). Berkson (1980) suggested that minimum chi-square 

estimation “yields the same estimating equations as MLE”.

A spirited discussion followed this paper. Recently, a paper 

by Barbiero (2016) looked at estimating the parameters of a 

two-parameter discrete Weibull distribution. Discrete 

Weibull distributions occur when observations can only be 

found in categories, such as when measurements must be 

rounded off into units or a range of units, perhaps because of

the inability to get more accurate measurements. The paper 

looks at least-squares methods and minimum chi-square 

methods. Using a simulation, Barbiero (2016) concludes 

“The simulation results indicate that the performance of the 

three point estimators based on the least-square method 

is ... overall better than that derived through the minimum 

chi-square method.”

5.3 Generalized Least Squares Estimation

Weighted regression and ordinary least squares regression, 

which were previously covered, are the most common 

methods of regression used to estimate the parameters of a 

Weibull distribution. Two other methods will be briefly 

mentioned in this section and the following section. They 

are GLS estimation and least absolute deviation estimation.

GLS is a variation of regression that is sometimes used 

when a correlation is expected between the residuals of the 

model. It was first described by Aitken (1934). The GLS 

procedure is a more generalized version of a weighted least 

squares procedure in which the weights are not only on the 

diagonal of the weighting matrix. Engeman and Keefe 

(1982) described how to use GLS to estimate the parameters 

of the two-parameter Weibull distribution. Using a 

simulation study with very small sample sizes, they found 

that the GLS estimate of the shape parameter was better than 

ordinary least squares, maximum likelihood, and the 
General Technical Report FPL–GTR–264

12

estimation procedure of Menon (1963). Maximum 

likelihood provided the best estimate of the scale parameter.

Kantar (2015) compared GLS, weighted least squares, and 

maximum likelihood for the Weibull, log-logistic, and 

Pareto distributions. Again, a simulation found that GLS 

provided the best estimate of the shape parameter. Kantar 

(2015) also provided a review of some other papers using 

GLS and some real life examples.

5.4 Least Absolute Deviation Estimation

Once regression estimates of Weibull parameters became 

popular, the logical thing to do was to look at different 

regression techniques to see if they would solve some 

potential problems. The use of minimizing the sum of the 

least absolute deviations

1

 

n

i i

i

S yy

=

= − ∑

This is one of several methods called robust regression 

because they have favorable properties when in the typical 

regression analysis, the residuals are not i.i.d.

(independently and identically distributed) normally 

distributed with the same mean and standard deviation. As 

with GLS, this method is trying to handle a deviation from 

the assumption of independence of the residuals. This 

method of least absolute deviation regression, also 

sometimes called L1 regression, is trying to handle residuals 

that might have skewed distributions (i.e., not normally 

distributed). Lawrence and Shier (1981) compared least 

square estimates (sometimes called L2 regression) and least 

absolute deviation regression estimation of Weibull 

parameters. They concluded that in some cases L1

regression can be superior to least squares. Again in 1984, 

Shier and Lawrence (1984) compared robust regression 

procedures on the estimation of Weibull parameters. In that

paper, they also compared using simulations with several 

reweighting schemes based on L1 residuals.

6 Discussion

Initially, estimation of Weibull parameters was generally 

restricted to picking one method of estimation and then 

using that method to estimate all the parameters of the 

Weibull distribution. There are several sound reasons for 

considering the approach that will be covered next. But the 

general increase in simple percentile estimates might change 

this approach as methods become more numerous. You can 

now mix and match estimators. For example, a simple 

percentile estimator of the location parameter followed by 

maximum-likelihood estimates of the shape and scale 

parameters might sound like a good idea because the 

maximum-likelihood estimation of the location parameter 

would often cause nonconvergence of the shape and scale 

parameters, especially when the shape parameter was near 2 

or less than 2. For example, in the U.S. in-grade study of 

Hem-Fir maximum-likelihood shape parameter values for 

13 grade–size combinations of specimens with sample sizes 

ranging from 20 to 428 at 15% moisture content, the shape 

parameter estimates for the three-parameter Weibull 

distribution ranged in value from 1.38 to 4.15 (Evans and 

Green 1988). There is also the important question of how 

the estimation procedure might affect the estimate of 

properties such as the 75% tolerance limit of the fifth 

percentile of a strength distribution for modulus of rupture,

which is used to quantify the strengths of various wood 

species used in wood construction. Therefore, the problem 

becomes not only which method of estimating Weibull 

parameters is best but also which method might provide the 

best estimate of distributional properties and possibly 

properties such as confidence limits on these property 

estimates or goodness-of-fit tests on the fitted distribution.

Most earlier work looked at parameter estimation. But, it is 

time to consider all the benefits that different methods 

provide.

Looking first at the benefits of using all estimators of the 

same kind, we consider the case for MLEs for all the 

parameters of a two-parameter Weibull distribution, which 

have most of the benefits that a method of estimation can 

provide. Thoman and others (1969) discussed four major 

properties that are important:

(1) Exact confidence limits for the parameters are available.

(2) A table of unbiasing factors is available for several 

sample sizes.

(3) Tests of hypotheses about the parameters and the power 

of the tests are developed.

(4) Guidelines of when sample sizes are large enough to

assume large sample theory are available.

Johnson and Haskell (1983) considered parameter estimates 

of the three-parameter Weibull and showed the consistency 

of the shape parameter estimate when the parameter was

greater than 1. They also studied the joint distribution of the 

estimators and determined that estimates of the 5th 

percentile of the population from the sample 5th percentile 

required sample sizes of more than 70 to assume that 

asymptotic normality applied. Evans and others (1989) 

provided goodness-of-fit tests for both two- and threeparameter fits of Weibull distributions to data. Johnson and 

others (2003) provided confidence intervals on ratio 

estimates of Weibull 5th percentiles that would be useful in 

evaluating dry–green ratios.

Other major individual types of estimators have fewer 

properties that are discussed in statistical literature. Mixtures 

of parameter estimator types generally have even fewer 

known properties, and that can be a problem when trying to 

discuss what these types of sets of estimators show. That 

does not mean that they are useless because they can 

provide important insights into data. However, the lack of 
Procedures for Estimation of Weibull Parameters

13

major statistical properties can limit the usefulness of 

estimates with a mixture of methods for the different 

parameters when used on a data set.

There are a few papers that have looked at properties that 

other estimation procedures might have in comparison with

MLEs. They generally compare the estimators on issues 

such as bias, variation, ease of calculation, and other 

measures that do not include the major properties previously 

discussed here. It might be useful for people looking at tail 

properties of a distributional fit to have a study that 

compares various methods. In wood engineering, some data 

sets used for grades of visual lumber and machine-stressrated lumber may be from whole populations in which lower 

or higher grades were taken out in the process of creating 

multiple grades.

MLEs, for some combinations of parameter values of the 

three-parameter Weibull distribution, do not satisfy the 

regularity conditions, which give estimates of the standard 

asymptotic properties, resulting in possible inconsistent 

estimates or even failure to exist. In these cases, some 

researchers such as Cohen and Whitten (1982) have 

suggested using some modified estimators. The conditions

for which MLEs are regular were previously covered. In 

cases for which the conditions don’t apply, Cohen and 

Whitten note that the mean μx, variance σx

2

, median Mex,

and the skewness α3:x are 

μx 1 =+Γ c b

22 2 σ 21 2 x = Γ −Γ =

b   CM  

( )

1/ Me ln 2 a

x = +c b

[ ]

3

3: 3/2

2

CM

CM

α x =

where Γk is defined in Equation (11) and the central 

moments are defined in Equations (6) and (7). Although the 

Weibull distribution is usually thought of as being positively 

skewed, it is really negatively skewed when a > 

3.6023494257197 (Cohen 1973). This is basically what was 

covered in the section on method of moments. Cohen

suggests using modified moment estimators and shows

through simulation under different conditions that the 

modified moment estimators may prove useful in some 

cases. Cohen concludes that for a ≥ 2.2156 (α3 < 0.5) MLEs

perform well with respect to bias and variance. For a = 4.5,

both MLE and moment estimators perform well with the 

MLEs taking considerably more time to calculate. (With 

modern increase in computer power, this is not the case 

anymore, but in the field, the moment estimators might be 

useful as initial estimates when computers are not 

available). When a < 2.2156, using modified moment 

estimators of a may prove useful with respect to bias and 

error. Cohen offers three different modifications and 

evaluates them through simulations and ease of calculation.

Gross and Lurie (1977) compared two estimators from Bain 

and Antle (1967) of the shape parameter with the MLE of 

the shape for the two-parameter Weibull distribution when 

a < 1. The comparison was based on bias, standard 

deviation, and root mean square error. This simulation 

showed that the Bain and Antle estimators were better for 

small samples and the MLE was superior for larger samples.

(For wood engineers, a < 1 is unlikely unless there are 

outliers that have extremely large values.)

Verrill and others (2012) compared regression estimators 

and MLEs for the parameters of a two-parameter Weibull 

distribution. The four coefficients of variation considered 

were 0.1, 0.2, 0.3, and 0.4. The corresponding shape values 

were 12.154, 5.7974, 3.7138, and 2.6956. They concluded 

that regression estimators were competitive for sample sizes 

of 15. But for sample sizes of 30 and larger, the MLE 

estimates were superior, and they were even better for the 

shape parameter.

7 Conclusion

Procedures for estimating the parameters of a Weibull 

distribution from a data set have been widely studied in the 

statistical literature and are still a major source of study.

Any review of such procedures is bound to be incomplete.

There are many more studies of the aspects of this problem 

than can be summarized in a short overview such as this 

report. However, a general overview of the procedures and 

their properties as found in the statistical literature can form 

a background for a more specific comparison of the 

procedures with uncensored data, which is often 

encountered in wood utilization research. To summarize a 

general interpretation of the information presented, the 

authors would like to stress two points. There are a wide 

variety of procedures and adaptations of procedures that are 

available to estimate the parameters of a Weibull 

distribution. Any comparison of these methods for use on 

wood engineering has to assume that the data truly has a 

Weibull distribution, which may not be true for censored 

data sets such as machine-stress-rated data in which

different grades that overlap machine-stress-rated data may 

have been removed before grading the remaining specimens.

Recent research has suggested that these grades of lumber 

may not follow a Weibull distribution. The second possible 

conclusion is that, although there is no uniformly “best” 

method of parameter estimation, the use of maximumlikelihood methods for data that wood scientists are likely to 

encounter appears to be the method that wood scientists 

should be using as a default. Other methods should require

some type of justification for use in wood utilization 

research.
General Technical Report FPL–GTR–264

14

Based on this conclusion of the advantage of maximumlikelihood estimates for Weibull data and the distinct 

possibility that lumber grade data might be a truncated data 

set, ASTM D5457 appears to have several potential 

weaknesses that should be studied further. Because the 

choice for estimating parameters of a Weibull distribution 

from a data set is optional in D5457 and the resulting fitted 

distribution is used to derive the reference resistance of 

wood-based materials and structural connections, there 

might be an issue in determining the properties of the values 

developed. This could be an important research topic to 

further explore. Also, for small sample sizes for tests of 

glulam beams or wood timbers, other methods that don’t 

have the bias that MLEs can have with small sample sizes

may be useful to check on the estimates prescribed in D5457 

using plots of the data verses predicted measurements from 

the distributional estimates.

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