United States

Department of

Agriculture

Forest Service

Forest

Products

Laboratory

Research

Paper

FPL-RP-493

Two- and ThreeParameter Weibull

Goodness-of-Fit Tests

James W. Evans

Richard A. Johnson

David W. Green
Abstract

Research Highlights

Extensive tables of goodness-of-fit critical values for the two- and three-parameter Weibull

distributions are developed through simulation for the Kolmogorov-Smirnov statistic, the

Anderson-Darling statistic, and Shapiro-Wilk-type correlation statistics. Approximating formulas

for the critical, values are derived and compared with values in the tables. Power studies using

several different distributional forms show that the Anderson-Darling statistic is the most

sensitive to lack of fit of a two-parameter Weibull and correlation statistics of the Shapiro-Wilk

type the most sensitive to departures from a three-parameter Weibull.

Keywords: Goodness-of-fit statistics, Weibull distribution, lumber strength, modulus of elasticity

This paper presents the results of a study to develop and evaluate goodness-of-fit tests for the

two- and three-parameter Weibull distributions. The study was initiated because of discrepancies

in published critical values for two-parameter Weibull distribution goodness-of-fit tests, the lack

of any critical values for a Shapiro-Wilk-type correlation statistic, and the lack of general

three-parameter Weibull distribution goodness-of-fit tests. The results of the study will be used

by Forest Products Laboratory (FPL) scientists to evaluate the goodness-of-fit of Weibull

distributions to experimental data. This will allow evaluation of distributional forms that may be

used in reliability-based design procedures.

Through computer simulation, extensive tables of critical values for three standard

goodness-of-fit statistics are developed for both the two- and three-parameter Weibull

distributions. The statistics used are the Kolmogorov-Smirnov D statistic, the Anderson-Darling

A2 statistic, and a Shapiro-Wilk-type correlation statistic. The critical values for the statistics are

modeled through regression to provide equations to estimate the critical values. The equations

allow computer programs to evaluate the goodness-of-fit of data sets containing up to 400

observations. The abilities of the tests to detect poor fits (the “power of the tests”) are studied

using both the equations and the exact critical values of the test statistics. Finally, invariance

properties of the statistics are proved. These invariance properties show that the scope of the

simulation is adequate for all problems that the tests might be applied to.

Four general conclusions are readily apparent from the results of this study:

1. Of the three statistics considered, the Anderson-Darling A2 statistic appears to be the statistic

of choice for testing the goodness-of-fit of a two-parameter Weibull distribution to a set of

data. The choice might be different if censored data were used since the correlation statistic

has the potential advantage of being easily modified for type I and type II censored

observations.

2. For testing the three-parameter Weibull goodness-of-fit, the correlation statistics

appear to be the best.

3. The critical value approximations appear to be very good for the range of sample sizes

considered.

4. The power of the tests is very dependent upon the true distributional form of the data.

November 1989

Evans, James W.; Johnson, Richard A.; Green, David W. Two- and three-parameter Weibull goodness-of-fit

tests. Res. Pap. FPL-RP-493. Madison, WI: U.S. Department of Agriculture, Forest Service, Forest

Products Laboratory; 1989. 27 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 53705-2398. Laboratory publications are sent to more

than 1,000 libraries in the United States and elsewhere.

The Forest Products Laboratory is maintained in Madison, Wisconsin, by the U.S. Department of

Agriculture, Forest Service, in cooperation with the University of Wisconsin.
Two- and ThreeParameter Weibull

Goodness-of-Fit Tests

James W. Evans, Supervisory Mathematical Statistician

USDA Forest Service

Forest Products Laboratory

Madison, Wisconsin

Richard A. Johnson, Professor

Department of Statistics

University of Wisconsin

Madison. Wisconsin

David W. Green, Research General Engineer

USDA Forest Service

Forest Products Laboratory

Madison, Wisconsin

Background and

Introduction

Two-parameter and three-parameter Weibull distributions are widely used to represent

the strength distribution of structural lumber and engineering-designed wood

subassemblies. As wood construction practices in the United States and Canada are

revised from deterministic to reliability-based design procedures, assessing the

goodness-of-fit of these Weibull distributional forms becomes increasingly important.

In its three-parameter form, the family is represented by the density function

(1)

where a is the shape parameter, b the scale parameter, and c the location parameter.

The family of two-parameter Weibull distributions follows from Equation (1) when

c = 0.

Goodness-of-fit tests for the two-parameter Weibull distributions have received

considerable attention. Mann and others (1973), Smith and Bain (1976), Stephens

(1977), Littell and others (1979), Chandra and others (1981), Tiku and Singh (1981),

and Wozniak and Warren (1984) have all discussed aspects of this problem. Mann and

others (1973) and Tiku and Singh (1981) proposed new statistics to test the

goodness-of-fit of two-parameter Weibull distributions. Smith and Bain (1976)

proposed a test statistic analogous to the Shapiro-Francis statistic for testing

normality. The Smith and Bain T statistic was based on the sample correlation between

the order statistics of a sample and the expected value of the order statistics under the

assumption that the sample comes from a two-parameter Weibull distribution. For

complete samples and two levels of censoring, they provided critical values for samples
containing 8, 20, 40, 60, or 80 observations. Stephens (1977) produced tables of the

asymptotic critical values of the Anderson-Darling A

2

 statistic and the Cramervon Mises W 2

 statistic for various significance levels. Littell and others (1979) compared

the Mann, Scheuer, and Fertig S statistic, the Smith and Bain T statistic, the modified

Kolmogorov-Smirnov D statistic, the modified Cramer-von Mises W2 statistic, and the

modified Anderson-Darling A

2

 statistic through a series of power studies for sample

size n = 10 to 40. They also calculated critical values for the D, W2, and A

2

 statistics

for n = 10, 15, . . . , 40. Chandra and others (1981) calculated critical values for the

Kolmogorov-Smirnov D statistic for n = 10, 20, 50, and infinity for three situations.

The three-parameter Weibull distribution has received considerably less attention,

probably because the critical values depend upon the unknown shape parameter.

Woodruff and others (1983) produced tables of critical values for a modified

Kolmogorov-Smirnov test for a Weibull distribution with unknown location and scale

parameters and a known shape parameter. Their tables included n = 5 to 15, 20, 25,

and 30 for shape parameters of 0.5, 1.0, . . . , 4.0. Bush and others (1983) produced

similar tables for the Cramer-von Mises and Anderson-Darling statistics, together with

a relationship between the critical value and the inverse shape parameter. No

asymptotic results or Monte Carlo determinations for n > 30 appear in the literature.

Despite this extensive literature, we encountered difficulties when we studied the

goodness-of-fit of two- and three-parameter Weibull distributions to numerous data

sets consisting of 80 to 400 observations of various lumber strength properties. For the

three-parameter Weibull tests, critical values are not published for the sample sizes

involved, and it is not clear how estimated shape parameters would affect the critical

values derived by assuming a known shape parameter. For the two-parameter Weibull

tests, these difficulties included a lack of published critical values for sample sizes

larger than 50 and some apparent inconsistencies in published critical values. These

apparent inconsistencies can be seen in our Tables 1 and 2. In Table 1 the Chandra and

others (1981) and the Littell and others (1979) critical values for the KolmogorovSmirnov D statistic show generally good agreement. (We scaled Littell’s values to

compare with Chandra’s by multiplying the test statistic critical value by n

1/2.)

However, Littell and others (1979) might have obtained a smoother set of values if they

had used a larger simulation. Some inconsistencies at the 0.01 level of significance are

also evident. Since Littell and others (1979) reported the means of 10 runs of 1,000

simulations and Chandra and others (1981) performed one run of 10,000, the latter

values might be expected to be smoother at the 0.01 level. Wozniak and Warren (1984)

calculated critical values for the Anderson-Darling statistic for the two-parameter

Weibull distribution when parameters are estimated by maximum likelihood. A

comparison of their results with those of Stephens (1977) and Littell (Table 2) shows

some variability in the second decimal of the critical values. For example, the Stephens

critical value at the 0.05 level for n = 40 is the same as the Wozniak and Warren

0.05-level value for n = 15. The Littell value for n = 15 is even larger than the

Wozniak value.

In this paper, we (1) develop extensive and definitive two-parameter Weibull

distribution goodness-of-fit critical values for the Shapiro-Wilk-type correlation

coefficient statistics, the Anderson-Darling A

2

 statistic, and the Kolmogorov-Smirnov

D statistic, (2) develop and evaluate formulas to yield a smoothed estimate of the

critical values for the statistics, and (3) extend these three statistics for use as

goodness-of-fit statistics with a three-parameter Weibull distribution.

2
Shapiro-Wilk-Type

Correlation Statistic

Studies such as the Monte-Carlo study of Shapiro and others (1968) have consistently

shown that for testing goodness-of-fit of normal distributions, the Shapiro-Wilk

statistic has superior power to other statistics in detecting that the data comes from a

wide range of other distributions. To develop a similar goodness-of-fit test for Weibull

distributions, we modify a simplified form of this statistic first suggested by Shapiro

and Francia (1972) but with approximate “scores” suggested by Filliben (1975).

Let X(1) , X(2), . . . , X( n )

 denote an ordered sample of size n from the population of

interest. Specifically, for the two- and three-parameter Weibull distribution, we

consider the test statistic where

(2)

and

is a median score, in the spirit of Filliben, except that these scores depend upon the

maximum likelihood estimate of the Weibull shape parameter a. The corresponding

Q-Q plot of (X( i ), m w , i) should resemble a straight line if the underlying population is

Weibull. (A Q-Q plot is an abbreviated notation for a Quantile-Quantile probability

plot, where corresponding percentiles of one distribution are plotted against the

percentiles of the other, as discussed in Wilk and Gnanadesikan (1968).) The statistic

 is the squared correlation coefficient of this plot. Note that the choice of m w, i

follows the approximate scores suggested by Filliben (1975) and is slightly different

from that chosen by Smith and Bain (1976).

If a variable X has the two-parameter Weibull distribution, the variable Y = ln X has

an extreme value distribution. Calculating goodness-of-fit on this scale has advantages.

Since the extreme value distributions are defined by location and scale parameters, the

critical values for the correlation statistic are not dependent on the true shape

parameter. Thus, for two-parameter Weibull distributions, we propose the

correlation-type statistic where

(3)

and

Other

Goodness-of-Fit

Statistics

We include two other well-studied goodness-of-fit statistics for comparative purposes.

The modified Komogorov-Smirnov D statistic is given by

where Fn(x) is the empirical distribution function of the sample and F(x;b,a) is the

fitted distribution.

3
The modified Anderson-Darling A

2

 statistic is given by

where U( i) = F(X(i);b,a) and X( i ) is the ith-order statistic.

Unlike the statistic, neither D nor A

2

 is affected by the choice of scale. Thus, we

can use either X or ln X in the calculations.

Two-Parameter Simulation Results

Weibull

Distributions The results of David and Johnson (1948) imply that the distributions of D and A 2

 do

not depend upon the values of b and a. Similarly, the distribution of does not

depend upon the values of b and a. These and some other invariance properties of the

statistics are shown in the Appendix. Therefore, without loss of generality, the

distributions of D, A2, and can be obtained assuming b = 1 and a = 3.6.

To calculate the critical values of the statistics, we used IMSL (1979) to generate a

fixed sample U(1) , U(2), . . . , U( n )

 of n order statistics from a uniform distribution.

Then we transformed to the Weibull order statistics,

where b = 1 and a = 3.6. The sample X(1), X(2), . . . , X( n ) was used to calculate

maximum likelihood estimates of b and a. Finally we calculated D, A2, and for the

sample of size n. Values of n = 10, 1.5, . . . , 50; n = 60, 70, . . . , 100; n = 120,

140, . . . , 200; and n = 240, 280, . . . , 400 were used. Concern over the apparent

variability of results for the Anderson-Darling critical values led us to try several

preliminary simulations with 1,000 to 10,000 replications. Results confirmed that the

type of variability seen in Table 2 could occur from one simulation to the next.

Because we were interested in smoothing out irregularities found in other tables, we

generated 50,000 values for each goodness-of-fit statistic. These 50,000 values were

ranked, and the 80th, 85th, 90th, 95th, and 99th percentiles were determined. The

values in Table 3 are the results of this simulation. For the Kolmogorov-Smirnov

D statistic, as in Chandra and others (1981), we again multiplied the critical value by

n

1/2

. By repeating the simulation with a different set of 50,000 replications for various

sample sizes, we determined that extending the simulation to 50,000 replications

reduces the variability in the Anderson-Darling A2 statistic critical values from the

second to the third decimal point. This is important for n > 50 because the critical

values become much closer together.

Critical Value Approximations

In an effort to smooth the critical values in Table 3 and simplify future use of the test

statistics, we modeled the critical values as functions of sample size. The modeling

resulted in the following separate equations for the critical values of the

Kolmogorov-Smirnov D statistic at the 0.10, 0.05, and 0.01 levels of significance:

4
These curves fit the tabled values reasonably well, as shown in Figure 1. We made no

attempt to force these curves to give the asymptotic values of Chandra and others

(1981). Those asymptotic values were estimated by an extrapolation procedure

described in their paper. Since the critical values in Table 3 are slightly above the

estimated asymptotic values for large samples, we decided to fit the data of Table 3

without forcing the curve through Chandra’s estimate.

Stephens’ (1977) modification of the asymptotic critical values of the Anderson-Darling

A

2

 statistic for use with finite samples requires multiplying the value found by the test

statistic by (1 + 0.2/n

1/2). The resulting value can then be compared to the asymptotic

critical values shown in Table 2. Results of this procedure are plotted in Figure 2. Here

the critical values in Table 3 are converted and compared to the asymptotic values.

Figure 2 shows that the approximation is very good.

Modeling the critical values of the correlation statistic produces the following

separate equations for the 0.10, 0.05, and 0.01 levels of significance:

These curves are plotted with the critical values in Figure 3.

Power Studies

To evaluate our results, we conducted a power study of the three statistics, D, A2, and

using four different distributions:

1. Uniform distribution on 0 to 1

2. Truncated normal with a mean of 1.4 and standard deviation of 0.35 (The

distribution is truncated so that no values less than 0.00001 are allowed.)

3. Lognormal distribution where In X has a mean of 1.6 and standard deviation of 0.4

4. Gamma distribution with shape parameter equal to 2 and scale parameter equal to 1

The procedure involved generating 5,000 pseudorandom samples of size n from each of

the four alternative distributions considered. We then calculated each of the three test

statistics and compared them to their critical values from Table 3 and the

5
approximations to the critical values obtained from the equations for D, A2

, and

In each case, we counted the number of rejections of the null hypothesis. We repeated

this procedure for sample sizes of n = 20, 50, 80, 100, and 200.

Results of the power study are presented in Tables 4 to 6, which show that the

Anderson-Darling A

2

 statistic is generally superior to (that is, has a larger power than)

both the Kolmogorov-Smirnov D and the correlation statistics for the alternatives

presented. Neither D nor appears to be very powerful across this group of

alternative distributions. One other feature illustrated in Tables 4 to 6 is the close

agreement in power between the actual critical values and the values from the

approximation formulas, which attests to the accuracy of the approximations.
Three-Parameter

Weibull

Distributions

Simulation Results

Developing critical values for the three-parameter Weibull distribution goodness-of-fit

critical statistics is more difficult because they depend upon the unknown shape

parameter. Selecting the scale on which to perform the tests is also a problem. In the

two-parameter case, using In X(i) instead of X(i) made the correlation statistic

independent of the shape parameter. In the three-parameter case, the choice of which

scale to use is not obvious. For the Kolmogorov-Smirnov D and the Anderson-Darling

A

2

 statistics, the two scales produce the same results. Therefore, we investigated D, A2,

and using a = 2.0, 2.8, 3.6, 4.4, and 5.2. We chose b = 1.0 and c = 2.0.

Invariance properties derived in the Appendix show that the critical values of the

statistics depend only upon the shape parameter.

To calculate the critical values of the statistics, IMSL (1979) routines were again used

to generate a fixed sample U(1) , U(2) , . . . , U(n) of n order statistics from a uniform

distribution. Then we transformed to the Weibull order statistics,

for the different combinations of a, b, and c. Maximum likelihood estimates of a, b,

and c were calculated for the sample X(1), X(2), . . . , X(n),. Finally we calculated D,

A

2

, and for the sample of size n. Values of n = 20, 40, . . . , 200 were used.

Because of the increased computer time required to calculate the maximum likelihood

estimates for the three-parameter Weibull distribution compared with the

two-parameter case, it was possible to do only 10,000 replications. Thus, for each

sample size and choice of a, b, and c, 10,000 data sets were created and the

goodness-of-fit statistics evaluated. For each statistic, the 10,000 values were ranked

and the 80th, 85th, 90th, 95th and 99th percentiles were determined (Tables 4 to 10).

Again the results for the Kolmogorov-Smirnov statistic have been multiplied by n

1/2

.

7
Critical Value Approximations

We wanted to smooth the results and develop formulas that would approximate the

critical values. Because the critical values vary slightly for different shape parameters,

two models were tried for each statistic. One model ignores the differences in values

due to shape and just expresses the general trend due to sample size. For this simple

model, we determined that for the Kolmogorov-Smirnov D statistic and the

Anderson-Darling A

2

 statistic, the two-parameter models could be modified by

subtracting a constant from the two-parameter model for the critical value. For the

correlation statistics, producing an entirely new model was better. A second model,

more complex, attempts to model the apparent quadratic nature of the effect of shape

on the critical values. This refinement was added to the simple model developed for

each of the three statistics. Thus, two models were created to predict the critical values

of the Kolmogorov-Smirnov, Anderson-Darling, and correlation (original and extreme

value scales) statistics.

Kolmogorov-Smirnov Models –

1. Two-parameter model plus a shift

2. Two-parameter model plus a shift and an adjustment for the estimated shape

parameter values (labeled SHAPE in the formulas)

Testing is done by multiplying the Kolmogorov-Smirnov D statistic by n

1/2 and

comparing the result to the critical values from the formulas. We reject the hypothesis

that a three-parameter Weibull fits the data if n

1/2 (D) is larger than the critical value.

Anderson-Darling Models –

1. Two-parameter model plus a shift

8
2. Two-parameter model plus a shift and an adjustment for shape

Testing is done by multiplying the Anderson-Darling A2 statistic by (1.0 + 0.2/n

1/2)

and comparing to the critical values from the formulas. We reject the hypothesis that a

three-parameter Weibull fits the data if A

2

(1 + 0.2/n

1/2) is larger than the critical

value.

Correlation Models – Original Scale –

1. New models

2. New models plus an adjustment for shape

Testing is done by comparing calculated using X(i) to the critical values from the

formulas. We reject the hypothesis that a three-parameter Weibull fits the data if is

less than the critical value.

Correlation Models-Extreme Value Scale –

1. New models

9
2. New models plus an adjustment for shape

Testing is done by comparing calculated using ln X( i)

to the critical values from the

formulas. We reject the hypothesis that a three-parameter Weibull fits the data if is

less than the critical value.

Power Studies

We compared the four statistics and their approximations for critical values in a power

study using two Weibull distributions and two alternative distributions:

We chose Weibull distributions for use in the three-parameter case to evaluate the

difference between the simple and complex models for critical values. This was not

necessary in the two-parameter case, because the statistics did not depend on the true

shape parameter.

The procedure involved generating 2,000 pseudorandom samples of size n from each of

the four distributions considered. We then calculated each of the four test statistics and

compared them with the critical values obtained from the models presented in the

previous section. In each case we counted the number of rejections of the null

hypothesis. This procedure was repeated for n = 20, 50, 80, 100, and 200.

Results of the power study are presented in Tables 11 to 13, which show that nothing

was gained by using the more complex critical value formulas. The approximation

models for critical values do produce the correct percentage of rejections for the two

Weibull distributions, indicating the adequacy of the models. For the non-Weibull

distributions, the correlation statistics and are the most powerful of the test

statistics. The Anderson-Darling A

2

 statistic is slightly more powerful than the

Kolmogorov-Smirnov D statistic.

Comparing the two-parameter results with the three-parameter results shows that for a

given sample size, the Anderson-Darling statistic is much more powerful in the

10
Conclusions

References

two-parameter case than in the three-parameter case. Conversely, the correlation test is

more powerful in the three-parameter case than in the two-parameter case.

Four general conclusions are readily apparent from the results of this study:

1. Of the three statistics considered, the Anderson-Darling A2 statistic appears to be

the statistic of choice for testing the goodness-of-fit of a two-parameter Weibull

distribution to a set of data because of the larger power it displayed in the power

studies. The choice might be different if censored data were used since the

correlation statistic has the potential advantage of being easily modified for type I

and type II censored observations.

2. For testing the three-parameter Weibull goodness-of-fit, the correlation statistics

and appear to be the best because of their greater power against the alternative

distributions considered.

3. The critical value approximations appear to be very good for the range of sample

sizes considered. For the two-parameter Weibull distribution critical values, this is

seen in the close agreement of the plotted curves and the critical values as shown in

Figures 1 to 3. For the three-parameter Weibull distribution critical values, the

quality of the approximations can be seen in the power study where powers for the

Weibull distributions tested were near the significance level of the test.

4. The power of the tests is very dependent upon the true distributional form of the

data. For example, in the two-parameter Weibull distribution power study, power

against the uniform distribution was much higher than against the truncated normal.

Bush, J.G.; Woodruff, B.W.; Moore, A.H.; Dunne, E.J. 1983. Modified Cramervon Mises and Anderson-Darling tests for Weibull distributions with unknown location

and scale parameters. Communications in Statistics, Part A-Theory and Methods. 12:

2465-2476.

Chandra, M.; Singpurwalla, N.D.; Stephens, M.A. 1981. Kolmogorov statistics for

tests of fit for the extreme-value and Weibull distribution. Journal of the American

Statistics Association. 76: 729-731.

David, F.N.; Johnson, N.L. 1948. The probability integral transformation when

parameters are estimated from the sample. Biometrika. 35:182-190.

Filliben, J. 1975. The probability plot correlation coefficient test for normality.

Technometrics. 17:111-117.

IMSL. 1979. Reference Manual. Houston, TX: International Mathematical and

Statistical Libraries, Inc.

Lawless, J.F. 1982. Statistical Models and Methods for Lifetime Data. New York:

John Wiley and Sons.

Lemon, G. 1975. Maximum likelihood estimation for the three-parameter Weibull

distribution, based on censored samples. Technometrics. 17:247-254.

11
Littell, R.D.; McClave, J.R.; Offen, W.W. 1979. Goodness-of-fit test for the

two-parameter Weibull or extreme-value distribution with unknown parameters.

Communications in Statistics, Part B-Simulation and Computation. 8: 257-269.

Mann, N.R.; Scheuer, E.M.; Fertig, K.W. 1973. A new goodness-of-fit test for the two

parameter Weibull or extreme-value distribution with unknown parameters.

Communications in Statistics. 2:383-400.

Shapiro, S.S.; Francia, R.S. 1972. Approximate analysis of variance tests for

normality. Journal of the American Statistics Association. 67: 215-216.

Shapiro, S.S.; Wilk, M.B.; Chen, H. 1968. A comparative study of various tests for

normality. Journal of the American Statistics Association. 63: 1343-1372.

Smith, R.M.; Bain, L.J. 1976. Correlation type goodness-of-fit statistics with censored

sampling. Communications in Statistics, Part A-Theory and Methods. 5:119-132.

Stephens, M.A. 1977. Goodness-of-fit for the extreme value distribution. Biometrika.

64: 583-588.

Tiku, M.L.; Singh, M. 1981. Testing the two parameter Weibull distribution.

Communications in Statistics, Part A-Theory and Methods. 10:907-917.

Wilk, M.B.; Gnanadesikan, R. 1968. Probability plotting methods for the analysis of

data. Biometrika. 55:1-19.

Woodruff, B.W.; Moore, A.H.; Dunne, E.J.; Cortes, R. 1983. A modified

Kolmogorov-Smirnov test for Weibull distributions with unknown location and scale

parameters. IEEE Transactions on Reliability. 2:209-212.

Wozniak, P.J.; Warren, W.G. 1984. Goodness of fit for the two-parameter Weibull

distribution. Presented at the American Statistical Association National Meeting,

Philadelphia, PA.
Table 1 –Two-parameter Weibull critical values for the Kolmogorov-Smirnov D statistic based on

maximum likelihood estimates

Critical values at various significance levels

Sample

size 0.20

Littell and others (1979)a

0.15 0.10 0.05 0.01

Chandra and others (1981)

0.10 0.05 0.025 0.01

a Values multiplied by n1/2 to convert to the scale of Chandra and others (1981)

b These are theoretical asymptotic values given by Chandra and others (1981).

Table 2–Two-parameter Weibull critical values for the Anderson-Darling A2 statistic based on

maximum likelihood estimates

Sample

size

Critical values at various significance levels

Wozniak and Warren (1984) Stephens (1977) Littell and others (1979)

0.10 0.05 0.01 0.10 0.05 0.01 0.10 0.05 0.01

a Theoretical asymptotic values.

13
Table 3–Two-parameter Weibull critical values for D, A2

, and

Sample

size

Critical values at various significance levels

Statistic 0.20 0.15 0.10 0.05 0.01

14
Table 3–Two-parameter Weibull critical values for D, A2

, and –con.

Sample

size Statistic

Critical values at various significance levels

0.20 0.15 0.10 0.05 0.01

Table 4–Simulated powers of two-parameter Weibull test statisticsa-α = 0.10

Sample D A2

Distribution size T F T F T F

a Values from table (T) and formula (F).

15
Table 5–Simulated powers of two-parameter Weibull test statisticsa –α = 0.05

Sample D

Distribution size T F

A2

T F

a Values from table (T) and formula (F)

Table 6–Simulated powers of two-parameter Weibull test statisticsa – α = 0.01

Sample D A2

Distribution size T F T F T F

a Values from table (T) and formula (F).

16
Table 7–Three-parameter Weibull critical values for the Kolmogorov-Smirnov

D statistic

Sample

size Shape

Critical values at various significance levels

0.20 0.15 0.10 0.05 0.01

17
Table 8–Three-parameter Weibull critical values for the Anderson-Darling

A2statistic

Sample

size Shape

Critical values at various significance levels

0.20 0.15 0.10 0.05 0.01

18
Table 9–Three-parameter Weibull critical values for the correlation test

statistic

Sample

size Shave

Critical values at various significance levels

0.20 0.15 0.10 0.05 0.01

19
Table 10–Three-parameter Weibull critical values for the correlation test

statistic

Sample

size Shape

Critical values at various significance levels

0.20 0.15 0.10 0.05 0.01

20
Table 11 – Simulated powers of three-parameter Weibull test statisticsa – α = 0.10

Simple D A2

Distribution size S C S C S C S C

a Values from simple (S) or complex (C) formula.

Table 12 – Simulated powers of three-parameter Weibull test statisticsa – α = 0.05

Simple D A 2

Distribution size S C S C S C S C

a Values from simple (S) or complex (C) formula

2 1
Table 13–Simulated powers of three-parameter Weibull test statisticsa – α = 0.01

Simple D A2

Distribution size S C S C

a Values from simple (S) or complex (C) formula.

S C S C

22
Appendix –

Some Invariance

Properties of

the Statistics

That is, the uniform variables are estimated using

The estimated uniform variables are the same whichever scale is used. Equation (A. 1)

holds even if the underlying population is not Weibull.

A similar conclusion holds under any given monotonic transformation.

Three-Parameter Case-In the three-parameter case, the estimate of the uniform

variables takes the form

On the log scale, we estimate

Consequently,

In this Appendix, we verify a number of invariance considerations that pertain to this

study. We present results related to the scale choice, null distributions, and power.

Invariance of D and A2 Under a Change to the Log Scale

Two-Parameter Case- In the two-parameter case, the estimate of the population

distribution function is

where and are the maximum likelihood estimates of the scale and shape

parameters, respectively. Under a change to the log scale, the estimate becomes

The estimated uniform variables are the same whichever scale is used. Because the

uniform variables have the same values, A2 and D are the same under both the original

and log scales.

23
Appendix –

Some Invariance

Properties of

the Statistics

That is, the uniform variables are estimated using

The estimated uniform variables are the same whichever scale is used. Equation (A. 1)

holds even if the underlying population is not Weibull.

A similar conclusion holds under any given monotonic transformation.

Three-Parameter Case-In the three-parameter case, the estimate of the uniform

variables takes the form

On the log scale, we estimate

Consequently,

In this Appendix, we verify a number of invariance considerations that pertain to this

study. We present results related to the scale choice, null distributions, and power.

Invariance of D and A2 Under a Change to the Log Scale

Two-Parameter Case- In the two-parameter case, the estimate of the population

distribution function is

where and are the maximum likelihood estimates of the scale and shape

parameters, respectively. Under a change to the log scale, the estimate becomes

The estimated uniform variables are the same whichever scale is used. Because the

uniform variables have the same values, A2 and D are the same under both the original

and log scales.

23
Invariance When the Underlying Distribution Is Weibull

Two-Parameter Case – On the log scale

is distributed free of the location parameter log b and scale parameter a

– 1 (David and

Johnson 1948; Lawless 1982, p. 147). From Equation (A.l) the distribution of

estimated uniform observations does not depend on a and b, and hence the A

2

and D

statistics do not depend on a and b.

The result in Equation (A.3) depends on a property of the maximum likelihood

estimator (see Lawless (1982), p. 147) that

That is, if X is transformed to b0 X

a0 , the maximum likelihood estimates of location

and scale for the extreme value distribution obey Equation (A.4).

The statistic, on the log scale, has a distribution that is free of the parameters since

is a function of the standard exponential variables Zi = (X( i)/b)

a

and is the

correlation of these variables with the scores log mw,i where the 1/ factor can be

ignored in the calculation of correlation.

Three-Parameter Case-For the three-parameter case, Lemon (1975) gives the pivotal

functions

where the equality in distribution is for the values of (a,b,c) indicated in the subscript.

In other words, these quantities depend only on the underlying population shape

parameter. Because

24
where the distribution of each term depends only on a, the estimated uniform variables

in Equation (A.2) and hence the distributions of A2 and D depend only on a, not the

other parameters.

In the case of the R w statistic, on the original scale,

which is a function of the standard Weibull with parameters a10. Since Rw is the

correlation of these variables with the scores mw , i

 and the scores only depend on , and

hence a, the distribution of Rw depends only on a.

For the log scale,

Similar to Equation (A.6),

has a distribution that depends only on a. Since Rw is the correlation of the variables

(A.7) with the scores ln mw ,i, its distribution also depends only on a.

Invariance and Power

Because of the properties (Eq. (A.4)) of the maximum likelihood estimators, the power

calculations pertain to a wider class of alternatives than is immediately apparent.

Two-Parameter Case – Let X be distributed as GO(x) and let Y = b0 X

a

0 . Then log Yi =

a,, log Xi + log b0. On the log scale, Rw e is a correlation and hence it has the same

value whether calculated in terms of the log Yi or the log Xi.

Further,

by the properties in Equation (A.4). Consequently, the estimated uniform variables are

the same on both scales, so A2 and D do not depend on the scale.

Thus, the power of each of the three tests remains the same for any lognormal

distribution. Also, the powers for the uniform (0, 1) alternative hold for the uniform

(0, > 0 and, more generally, for

25
The basic relations between the maximum

be shown directly. Let Y = Then

likelihood estimators on the two scales can

Consequently, the maximum likelihood estimates based on x and y (or equivalently on

ln y) satisfy

Because of this relation the estimated uniform variables are also equal.

Three-Parameter Case-Although there is a wider invariance class, we note that the

power for the three-parameter case is the same for any scale change. Let X be

distributed as G0(x) and let Y = b0 X + c0. On the original scale, Rw, is a correlation

and so its value is unchanged. The scores involving â are the same on both scales.

More generally, the maximum likelihood estimates are related as follows:

Consequently, if maximize L(y | a, b, c), then

maximize L(x | a, b, c). In this sense, the maximum likelihood estimators and

are equivariant.

26
According to Equation (A.9)

By Equation (A.2), the estimated uniform variables are equal, and we conclude that

power is the same under both G0(x) and any location scale change

27
Invariance When the Underlying Distribution Is Weibull

Two-Parameter Case – On the log scale

is distributed free of the location parameter log b and scale parameter a

– 1 (David and

Johnson 1948; Lawless 1982, p. 147). From Equation (A.l) the distribution of

estimated uniform observations does not depend on a and b, and hence the A

2

and D

statistics do not depend on a and b.

The result in Equation (A.3) depends on a property of the maximum likelihood

estimator (see Lawless (1982), p. 147) that

That is, if X is transformed to b0 X

a0 , the maximum likelihood estimates of location

and scale for the extreme value distribution obey Equation (A.4).

The statistic, on the log scale, has a distribution that is free of the parameters since

is a function of the standard exponential variables Zi = (X( i)/b)

a

and is the

correlation of these variables with the scores log mw,i where the 1/ factor can be

ignored in the calculation of correlation.

Three-Parameter Case-For the three-parameter case, Lemon (1975) gives the pivotal

functions

where the equality in distribution is for the values of (a,b,c) indicated in the subscript.

In other words, these quantities depend only on the underlying population shape

parameter. Because

24
where the distribution of each term depends only on a, the estimated uniform variables

in Equation (A.2) and hence the distributions of A2 and D depend only on a, not the

other parameters.

In the case of the R w statistic, on the original scale,

which is a function of the standard Weibull with parameters a10. Since Rw is the

correlation of these variables with the scores mw , i

 and the scores only depend on , and

hence a, the distribution of Rw depends only on a.

For the log scale,

Similar to Equation (A.6),

has a distribution that depends only on a. Since Rw is the correlation of the variables

(A.7) with the scores ln mw ,i, its distribution also depends only on a.

Invariance and Power

Because of the properties (Eq. (A.4)) of the maximum likelihood estimators, the power

calculations pertain to a wider class of alternatives than is immediately apparent.

Two-Parameter Case – Let X be distributed as GO(x) and let Y = b0 X

a

0 . Then log Yi =

a,, log Xi + log b0. On the log scale, Rw e is a correlation and hence it has the same

value whether calculated in terms of the log Yi or the log Xi.

Further,

by the properties in Equation (A.4). Consequently, the estimated uniform variables are

the same on both scales, so A2 and D do not depend on the scale.

Thus, the power of each of the three tests remains the same for any lognormal

distribution. Also, the powers for the uniform (0, 1) alternative hold for the uniform

(0, > 0 and, more generally, for

25
The basic relations between the maximum

be shown directly. Let Y = Then

likelihood estimators on the two scales can

Consequently, the maximum likelihood estimates based on x and y (or equivalently on

ln y) satisfy

Because of this relation the estimated uniform variables are also equal.

Three-Parameter Case-Although there is a wider invariance class, we note that the

power for the three-parameter case is the same for any scale change. Let X be

distributed as G0(x) and let Y = b0 X + c0. On the original scale, Rw, is a correlation

and so its value is unchanged. The scores involving â are the same on both scales.

More generally, the maximum likelihood estimates are related as follows:

Consequently, if maximize L(y | a, b, c), then

maximize L(x | a, b, c). In this sense, the maximum likelihood estimators and

are equivariant.

26
According to Equation (A.9)

By Equation (A.2), the estimated uniform variables are equal, and we conclude that

power is the same under both G0(x) and any location scale change

27