Weibull distributions when the shape parameter is defined

Computational Statistics & Data Analysis 36 (2001) 299–310 www.elsevier.com/locate/csda

Weibull distributions when the shape parameter is de%ned  K.O. Bowmana;∗ , L.R. Shentonb a

Computer Science and Mathematics Division, Building 6012, Oak Ridge National Laboratory, P.O.Box 2008, MS-6367, Oak Ridge, TN 37831-6367, USA b Department of Statistics, University of Georgia, Athens, GA 30602, USA Received 1 April 2000; received in revised form 1 October 2000

Abstract The Weibull distribution, depending on parameters of location, scale, and shape, is often useful as a model for fracture data sets. If the location parameter is to be estimated then we have shown that maximum likelihood methods are not recommended. In the data set considered here the shape parameter is known to lie between 2 and 3 or so. We therefore studied the two-parameter model for which the shape parameter is known, or has a probability structure. Simple moment estimators are used and some c 2001 Elsevier Science B.V. All rights moments of these are studied and veri%ed by simulation.
reserved. Keywords: Envelope distribution; Moment estimator; Moments of sample moments; Pad8e sequences; Taylor series; Unbiased estimators

1. Introduction The Weibull model depends on three parameters, location a, scale b, and shape c. Bowman and Shenton (1998, 1999, 2000) have recently shown that in maximum likelihood estimation, moments of the estimators only exist for the shape parameter 

This manuscript has been authored by a contractor of the US Government under Contract No. DE-AC05-00OR22725. Accordingly, the US Government retains a nonexclusive, royalty-free license to publish or reproduce the published form of this contribution, or allow others to do so, for US Government purposes. ∗ Corresponding author. Tel.: +1-423-574-3126; fax: +1-423-574-0680. E-mail address: [email protected] (K.O. Bowman). c 2001 Elsevier Science B.V. All rights reserved. 0167-9473/01/$ - see front matter
PII: S 0 1 6 7 - 9 4 7 3 ( 0 0 ) 0 0 0 4 8 - 7

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in a certain range. For example, asymptotic covariances only exist if c ¿ 2, skewness only exists if c ¿ 3, and the standardized fourth moment only exists if c ¿ 4. If for example, c = √4:1 then the asymptotic √ skewness of the maximum√likelihood estimator aˆ is −1:07= N , that for bˆ is 1:4= N , and that for cˆ is 5:24= N . Thus, to reduce the asymptotic skewness of cˆ to 0.1, a sample size of N ¿ 5000 or so would be needed; to reduce it to 0.5, N would have to exceed 100 or so. To what extent this defect aIects the estimation procedures is not quite clear. As a general rule, we think samples of N = 250 or so would be a prudent restriction to keep in mind. The singularity that causes the problem arises from the logarithmic derivatives of the density with respect to the location parameter. It will arise whenever the Weibull model includes the location parameter which is to be estimated; it will not occur in the traditional two-parameter case for which the location parameter is zero or constant. The paper studies the two-parameter cases, giving new information on the moments of the parameter estimates, using extended Taylor series. Thus, for a density
of f(x; m1 ; m2 ; m3 ; m4 ), where ms = xjs =N for a random sample (x1 ; x2 ; : : : ; xN ), a s-dimensional Taylor series is set up in the moments j = mj − j ; j = 1; 2; : : : ; s, where E(j ) = 0. There is an algorithm (Shenton et al., 1971; Shenton and Bowman, 1975) for evaluating E[(m1 − 1 )r1 (m2 − 2 )r2 (m3 − 3 )r3 (m4 − 4 )r4 ]: Series for E[f(x; m)]s follow in terms of powers of 1=N , N being the sample size; the coeLcient of N −s includes contributions from several sources. We consider in detail an application of the two-parameter model using moment estimators. It involves the estimation of the location parameter, and properties of this estimator are a prime concern. In this connection, we introduce a Weibull model in which the shape parameter may be a random variable with a given distribution. It should be pointed out that although much study has been given to the actual estimators of the Weibull density, little attention has been focussed on the properties of the estimators themselves (see Johnson et al., 1994). By contrast, we consider estimators based on sample moments, along with a powerful algorithm to set up their moments. The paper involves statistical methodology, applied mathematics focusing on the analysis of extended Taylor series, and lastly procedures based on simulation studies. 2. The two-parameter Weibull density when the shape parameter c is known 2.1. Background setting of the data A case has arisen (Bowman and Williams, 2000) on “Technical Basis for Statistical Models of Extended Kic and Kia Fracture Toughness Databases for RPV Steels”. There are 27 distinct data sets, the largest sample size being 44. The sets are temperature-dependent. The report (ORNL=NRC=LTR-99 =27) has, for example, some 38 references on fracture analysis, and many more on statistical analysis including

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301

references to the Weibull model. The statistical focus centered on the estimation of the location parameter (the smallest toughness to be expected for a particular temperature). Samples as small as one occur. 2.2. Statistical background The proven model for structure failure is the Weibull density; it has been intensely studied and the chapter in Johnson et al. (1994) has hundreds of references. However, few applications are discussed in detail; in addition, as mentioned in our introduction, maximum likelihood methods are of doubtful value unless large samples (N ¿ 250 or so) are available, particularly when location, scale, and shape are to be estimated. Attention is therefore focussed on the two-parameter Weibull models, with the location parameter included. The choice is therefore estimating a and b or a and c. Now, the skewness, measured by the standardized third central sample moment (m3 =m23=2 ), does not involve scale and location, and for the toughness data it was surmised that 1:3 ≤ c ≤ 3 was a reasonable assumption. This suggested studying the two-parameter Weibull model where a and b are to be estimated, given that the shape parameter is known or de%ned. 3. Properties of the two-parameter Weibull density when c is known 3.1. Basic formulas The density is c c w(x; a; b|c) = yc−1 e−y b



x−a ; x ¿ a; b; c ¿ 0 y= b



with distribution function W (x; a; b|c) =



x

t=a

c

w(t; a; b|c) dt = 1 − e−y ;

and c is known. The moments are mean: 1 (x; a; b|c) = a + b(1 + 1=c) (c ¿ 0); variance: 2 (x; a; b|c) = b2 [(1 + 2=c) − 2 (1 + 1=c)]; with moment estimators √  b∗ = m2 = [(1 + 2=c) − 2 (1 + 1=c)]b

or

b∗ =b =

√

√ m2 = 2

and a∗ = m1 − b∗ (1 + 1=c): Series for the mean, variance, third and fourth central moments for a∗ and b∗ are given in Table 1.

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Table 1 Series for the moments of a∗ and b∗ when c = 2:1 N −s

a

1

2

3

4

0:67891756D + 00 0:12065197D + 00 0:22596801D + 00 −0:94143717D − 01 0:96779137D + 00 −0:10428443D + 02 0:66616458D + 02 −0:82602889D + 03 0:10768022D + 05 −0:15348588D + 06 0:26402856D + 07 −0:49216457D + 08

0:39190262D + 00 −0:12936388D + 00 0:39926399D + 00 −0:57533240D + 00 0:23255354D + 01 −0:24242021D + 02 0:16238149D + 03 −0:18977491D + 04 0:24561289D + 05 −0:34850034D + 06 0:59300523D + 07 −0:11011743D + 09

0:00000000D + 00 −0:62858268D − 01 0:65259996D + 00 −0:12345449D + 01 0:39983611D + 01 −0:38215069D + 02 0:26774912D + 03 −0:30275244D + 04 0:39176514D + 05 −0:55694455D + 06 0:94425785D + 07 −0:17548365D + 09

0:00000000D + 00 0:46076300D + 00 0:87633222D − 01 −0:45578535D + 00 0:34831723D + 01 −0:40062234D + 02 0:28350636D + 03 −0:34750509D + 04 0:46256651D + 05 −0:67302490D + 06 0:11655858D + 08 −0:21976221D + 09

1 −0:76653772D + 00 0:53307543D + 00 2 −0:13622315D + 00 −0:31513378D + 00 3 −0:25513113D + 00 0:30142190D + 00 4 0:10629377D + 00 −0:62227956D + 00 5 −0:10926932D + 01 0:22788332D + 01 6 0:11774324D + 02 −0:25259961D + 02 7 −0:75213885D + 02 0:16823523D + 03 8 0:93263503D + 03 −0:19779396D + 04 9 −0:12157728D + 05 0:25731005D + 05 10 0:17329456D + 06 −0:36501582D + 06 11 −0:29810372D + 07 0:62249534D + 07 12 0:55568265D + 08 −0:11566618D + 09 a Calculation based on a = 0; b = 1.

0:00000000D + 00 0:42414305D + 00 −0:54175408D + 00 0:10815229D + 01 −0:34612660D + 01 0:35523787D + 02 −0:25469337D + 03 0:28792702D + 04 −0:37781724D + 05 0:53803912D + 06 −0:91590334D + 07 0:17061368D + 09

0:00000000D + 00 0:85250825D + 00 −0:54631884D + 00 0:42079527D − 01 0:14272741D + 01 −0:28133440D + 02 0:21130414D + 03 −0:27350607D + 04 0:38011792D + 05 −0:56144208D + 06 0:98972354D + 07 −0:18861713D + 09

a

∗

1 2 3 4 5 6 7 8 9 10 11 12 b∗

Then if a = a0 and b = b0 , 1 (a∗ |a0 ; b0 ) = a0 + b0 1 (a∗ |0; 1) and for the variance 2 (a∗ |a0 ; b0 ) = b20 2 (a∗ |0; 1): Similarly, 1 (b∗ |a0 ; b0 ) = b0 1 (b∗ |0; 1) and 2 (b∗ |a0 ; b0 ) = b20 2 (b∗ |0; 1): • Note that for the Weibull density the skewness is zero when c = 3:6 approximately. In the vicinity of 3.6 Weibull density is similar in shape to normal density (Johnson et al., 1994, p. 635).

K.O. Bowman, L.R. Shenton / Computational Statistics & Data Analysis 36 (2001) 299–310

303

Table 2a Pad8e approximation of moments and percentage points of a∗ when c = 2:1, b = 1:0, a = 0a N

1





1

2

5%

25%

50%

75%

95%

10

Pad8e MC

0.069 0.070

0.196 0.196

−0:011 −0:000

3.185 3.205

−0:25

−0:06

0.07

0.20

0.39

11

Pad8e MC

0.063 0.063

0.187 0.186

−0:016 −0:030

3.174 3.139

−0:24

−0:06

0.06

0.19

0.37

12

Pad8e MC

0.058 0.058

0.179 0.179

−0:019 −0:017

3.164 3.144

−0:24

−0:06

0.06

0.18

0.35

13

Pad8e MC

0.053 0.054

0.172 0.172

−0:022 −0:019

3.155 3.138

−0:23

−0:06

0.05

0.17

0.33

14

Pad8e MC

0.049 0.050

0.166 0.166

−0:024 −0:023

3.146 3.106

−0:22

−0:06

0.05

0.16

0.32

15

Pad8e MC

0.046 0.046

0.160 0.161

−0:026 −0:022

3.139 3.122

−0:22

−0:06

0.05

0.15

0.31

20

Pad8e MC

0.034 0.035

0.139 0.139

−0:031 −0:028

3.110 3.099

−0:20

−0:06

0.03

0.13

0.26

30

Pad8e MC

0.023 0.024

0.114 0.114

−0:032 −0:029

3.077 3.104

−0:17

−0:05

0.02

0.10

0.21

40

Pad8e MC

0.017 0.018

0.099 0.099

−0:031 −0:041

3.059 3.057

−0:15

−0:05

0.02

0.08

0.18

50

Pad8e 0.014 0.088 −0:029 3.048 −0:13 −0:05 0.01 0.07 0.16 MC 0.014 0.089 −0:029 3.076 a Percentage points are based on the Pearson 4 moment approximating distributions. (Bowman and Shenton, 1979a, b).

• Note that the series, as far as our calculations go, suggest convergence for N ¿ 20 or so. • For smaller values of c, the series will require larger N for convergence, and this is related to the fact that the Weibull density tends to reverse J shape in this case. Moments and moment ratios for a∗ (Tables 2a and 2b) for c = 2:1 (known) shows the distribution to be nearly normal especially for 20 ≤ N ≤ 40. As a check simulation results (cycles 50,000) are included, and the comparison with Pad8e methods is most satisfactory. The 50,000 cycles of simulation were carried out in batches of 10,000 and studied for consistency. Also there was a check for the generated moments against those of the Weibull density itself. The program used was from the Nag Library (www.nag.com). 3.2. Some useful formulas and algorithms (c known)   1 E(a∗ ) =  1 + [b − E(b∗ )]; c

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Table 2b Pad8e approximation of moments and percentage points of b∗ when c = 2:1, b = 1:0, a = 0 N

1





1

2

5%

25%

50%

75%

95%

10

Pad8e MC

0.922 0.921

0.225 0.224

0.335 0.327

3.141 3.122

0.58

0.76

0.91

1.07

1.31

11

Pad8e MC

0.929 0.929

0.215 0.215

0.320 0.330

3.130 3.113

0.60

0.78

0.92

1.07

1.30

12

Pad8e MC

0.935 0.934

0.206 0.206

0.307 0.302

3.121 3.085

0.62

0.79

0.92

1.07

1.29

13

Pad8e MC

0.940 0.939

0.198 0.199

0.295 0.296

3.112 3.118

0.63

0.80

0.93

1.07

1.28

14

Pad8e MC

0.944 0.944

0.191 0.192

0.285 0.285

3.105 3.091

0.65

0.81

0.94

1.07

1.27

15

Pad8e MC

0.948 0.948

0.185 0.186

0.275 0.285

3.099 3.124

0.66

0.82

0.94

1.07

1.27

20

Pad8e MC

0.961 0.960

0.161 0.161

0.240 0.236

3.076 3.068

0.71

0.85

0.95

1.07

1.24

30

Pad8e MC

0.974 0.973

0.132 0.132

0.197 0.193

3.052 3.061

0.76

0.88

0.97

1.06

1.20

40

Pad8e MC

0.981 0.980

0.115 0.115

0.171 0.194

3.039 3.070

0.80

0.90

0.98

1.06

1.17

50

Pad8e MC

0.985 0.984

0.103 0.103

0.153 0.157

3.032 3.057

0.82

0.91

0.98

1.05

1.16

    1 1  2 (1 + 1=c)(2 − 1) Var(a ) ∼ 2 −  1 + 2 1 + N c 4 ∗



b∗ E b





=E

∼

(N → ∞);

m2 2

1 1−2 4 =16−1522 =128 + 32 =64−31 =8−31=128 1− 1+ + N 8N N2



(N → ∞); 

b∗ Var b





= Var 

1 ∼ 1− N

2 ,



m2 2







1 =1− − E N

m2 2

2

2 − 1 4 =8 − 722 =32 + 2 =16 − 31 =4 − 15=32 − 4N N2

(N → ∞) 1 , 2 , and 4 = 6 =23 refer to the Weibull Density.



K.O. Bowman, L.R. Shenton / Computational Statistics & Data Analysis 36 (2001) 299–310

Almost unbiased estimators are b∗ b∗∗ = ; 1 + A1 =N + A2 =N 2 where 0:806668 1:423234 A1 = −0:827043 + − ; c c2

305

(1)

3:990936 5:659460 + : c c2 Using the Pad8e approximants for 1:1 ≤ c ≤ 4:0 and 10 ≤ N ≤ 100, the maximum absolute error is less than 0.1% except at c = 1:1 and N = 10 where the error is 0.175%. Similarly, an almost unbiased estimator for a is A2 = 0:436183 −

a∗∗ = m1 − b∗∗ (1 + 1=c):

(2)



We %nd (b∗ =b) = 1 − 1=N − [E(b∗ =b)]2 , and this can be set up from E(b∗ =b) ∗ or the Pad8e form for  Var(b =b). ∗ The formula for 1 (b =b) is



1

b∗ b



=



1 1 (b∗ ) ∼ √ N





B1 +

B2 ; N

where B1 = −0:600783 +

1:964495 3:330215 ; + c c2

13:437573 25:060284 ; − c c2 for 1:1 ¡ c ≤ 3:0 and 10  ≤ N ≤ 100, the maximum error is less than 5%. For c ¿ 3:0, 10 ≤ N ≤ 100, 1 ¡ 0:15. The formula for 2 (b∗ =b) is  ∗   b C1 C2 ∗ 2 = 2 (b ) ∼ 3 + + 2 ; b N N where 102:464711 125:090738 + C1 = 21:924414 − ; c c2 B2 = −0:847900 +

995:457126 1025:128798 − ; c c2 for values 1:1 ≤ c ≤ 3:2 and 12 ¡ N ¡ 100 with maximum error less than 2%. For c ¿ 3:2 the kurtosis 2 is approximately 3. With the four moments of b∗∗ given by these algorithms we can set up con%dence intervals for b and a, using the Pearson four moment approximating distribution (Bowman and Shenton, 1979a, b). For a data set studied in the sequel properties of the two-parameter Weibull density with c known are required. C2 = −242:161370 +

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4. Fracture toughness data set In the report ORNL=NRC=LTR-93=15, “Preliminary Review of the Bases for the KIc Curve in the ASME Code”, sets of fracture toughness data for given temperatures are given and the distribution is taken to be a Weibull model. The largest sample size for a given temperature is 26 and the ordered data points are Ordered data 29.40 35.50 44.00

31.10 37.10 44.90

31.30 37.10 47.30

31.30 37.20 50.40

31.40 38.10 50.90

32.60 39.30 54.00

33.00 39.40

33.20 39.40

34.70 41.20

35.00 43.90

The statistics of these data are Number of observations 26 Mean 38.5654 Variance 44.1976 Skewness 0.72 Kurtosis 2.57 From a metallurgical point of view it is known that the location parameter a is ¿ 18:2 and ¡ 29:4, the smallest data point. There are many temperature levels with various sample sizes, and %nding the values of the location parameters is most important. A preliminary study showed that 2 ≤ c ≤ 3, so we decided to use the two-parameter Weibull density with c known. We %t the data with 27 values of c (1.1(0.1)3.7), excluding a solution if the location a∗ is greater than the smallest sample value. There are 19 acceptable %ts judged by the $2 test, and for 1:4 ≤ c ≤ 3:2 the best case arises with c = 1:5 (Table 3). We %nd a∗ = 28:77;

b∗ = 10:85:

Using (1) and (2), the almost unbiased estimators are a∗∗ = 28:42;

b∗∗ = 11:24:

The distribution of b∗ has 22 −31 −6=0 using the values given in Table 2, so that a gamma distribution is suggested for the scale parameter. Note that (a∗ ) ∼ 1:5. Using the grouped data we found Data Theo

26:3 ≤ x ¡ 32:5,

32:5 ≤ x ¡ 38:6,

38:6 ≤ x ¡ 44:8,

44:8 ≤ x ¡ 57:1.

5 5.07

10 9.99

6 6.45

5 4.49

For the goodness of %t criterion $2 =1:2 and the degrees of freedom %=4−2−1=1, the last two groups being combined. Now, Pr($2 ¿ 1:2; % = 1) is 0.2, so from this point of view the %t is acceptable.

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307

Table 3 Estimated values of a∗ , b∗ , a∗∗ , and b∗∗ c

a∗

a∗∗

b∗

b∗∗

$2

N1 (5)

N2 (10)

N3 (6)

N4+5 (5)

1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2

29.38 28.77 28.18 27.59 27.00 26.42 25.85 25.28 24.71 24.15 23.59 23.03 22.47 21.92 21.37 20.82 20.27 19.73 19.18

29.03 28.42 27.82 27.22 26.62 26.04 25.45 24.87 24.29 23.72 23.14 22.57 22.00 21.44 20.87 20.31 19.75 19.19 18.63

10.08 10.85 11.59 12.31 13.00 13.69 14.35 15.00 15.64 16.28 16.90 17.51 18.12 18.72 19.31 19.90 20.48 21.06 21.64

10.46 11.24 11.99 12.72 13.43 14.12 14.80 15.46 16.12 16.76 17.40 18.02 18.65 19.26 19.87 20.47 21.07 21.67 22.26

0.1016 0.0894 0.1082 0.1443 0.1904 0.2422 0.2970 0.3532 0.4096 0.4656 0.5207 0.5747 0.6274 0.6787 0.7285 0.7768 0.8236 0.8690 0.9130

4.95 5.07 5.14 5.19 5.22 5.24 5.24 5.25 5.24 5.23 5.22 5.21 5.20 5.18 5.17 5.16 5.14 5.13 5.11

10.33 9.99 9.71 9.48 9.29 9.13 8.99 8.86 8.75 8.65 8.57 8.49 8.42 8.35 8.29 8.24 8.19 8.14 8.10

6.30 6.45 6.59 6.71 6.83 6.94 7.05 7.14 7.23 7.32 7.40 7.47 7.55 7.61 7.68 7.74 7.80 7.86 7.91

4.42 4.49 4.56 4.61 4.65 4.69 4.72 4.75 4.78 4.80 4.81 4.83 4.84 4.85 4.86 4.86 4.87 4.87 4.88

5. The Weibull density when c is a random variable 5.1. Introduction We study the Weibull density with c taken to be a random variable. Recall that there is a modi%ed binomial distribution where the probability parameter p is taken to be a Beta variable (see Bowman et al., 1992). One might call this kind of distribution, an Envelope distribution. 5.2. The two-parameter Weibull envelope distribution The Weibull density is c c w(x; a; b; c) = yc−1 e−y b with distribution function W (x; a; b; c) =



x

−∞

(y = (x − a)=b; x ¿ a; b; c ¿ 0)

w(t; a; b; c) dt = 1 − e−y

c

(c ¿ 0):

Consider a distribution B function (t); A ¡ t ¡ B, with a %nite or in%nite set of points of increase and A d(t) = 1. For the Weibull, the envelope density is w(x S : a; b) = w(x; a; b; c) d(c) with WS (a; a:b) =

 A

B

c

(1 − e−y ) d(c):

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As a particular case let c be a Beta variable. Then the density of c is b(c; p; q; A; B) =

(c − A)p−1 (B − c)q−1 Bp; q (B − A)p+q−1

(A ≤ c ≤ B; B ¿ A ¿ 0; p ¿ 0; q ¿ 0);

where Bp; q = (p)(q)=(p + q). It is clear that we must work in terms of non-central moments because of linearity considerations. Thus, for the envelope distribution moments, S s = bs





B

b(c; p; q; A; B) 1 +

A

s c



dc:

Using the transformation (c − A)=(B − A) = t, this becomes S s

=b

s





s t p−1 (1 − t)q−1  1+ Bp; q A + (B − A)t

1

0



dt:

For moment estimators of a and b the corresponding envelope moments are 2

m2 = bS



t p−1 (1 − t)q−1 2  1+ Bp; q A + (B − A)t

1

t p−1 (1 − t)q−1 1  1+ Bp; q A + (B − A)t

0

−





0

and m1



1

= aS + bS



1

0





dt



t p−1 (1 − t)q−1 1  1+ Bp; q A + (B − A)t

dt

2   



dt

(B ¿ A ¿ 0):

In particular, for the uniform Beta, p = q = 1; A = 1; B = 2: m2 = bS

2

  

m1 = aS + bS

1

0

 0



2  1+ 1+t 1



 1+



1 1+t

dt − 



0

1



1  1+ 1+t



dt

2   

;

dt:

In Table 4, the envelope has p=q=1; A, and B. Clearly, the case A=1:42; B=1:44 is the best yielding aS = 29:2, and bS = 10:3. 6. Conclusion It is pointed out that • Moments of maximum likelihood estimators in the three- and two-parameter case with a unknown, only exist if the shape parameter c is large enough. Means only exist if c ¿ 1, covariances if c ¿ 2, skewness if c ¿ 3, and kurtosis for c ¿ 4. Note that this phenomenon is associated with the fact that the shape of the Weibull distribution changes from reverse J-shaped to the usual bell-shape as c increases from unity.

K.O. Bowman, L.R. Shenton / Computational Statistics & Data Analysis 36 (2001) 299–310

309

Table 4 Examples of aS and bS for the Weibull envelope distribution A

B

aS

bS

N1

N2

N3

N4+5

$2

1.00 1.00 1.00 1.00 1.20 1.30 1.39 1.39 1.41 1.42

4.00 3.00 2.00 1.50 1.80 1.70 1.41 1.45 1.45 1.44

25.9376 27.5543 29.4254 30.5323 28.9924 28.8690 29.3800 29.2602 29.1981 29.1973

14.0176 12.2152 9.9907 8.5658 10.5584 10.7209 10.0780 10.2311 10.3104 10.3115

4.21 4.32 4.22 3.85 4.56 4.64 4.54 4.58 4.60 4.60

9.86 10.24 11.02 11.90 10.56 10.44 10.73 10.65 10.61 10.61

8.49 7.74 6.83 6.29 6.68 6.64 6.46 6.50 6.51 6.51

3.44 3.70 3.93 3.96 4.20 4.28 4.27 4.27 4.28 4.28

1.5823 0.9604 0.6311 0.9405 0.2954 0.2302 0.2587 0.2395 0.2299 0.2295

5

10

6

5

Fracture toughness data

• When the location parameter a is to be estimated either by moment or maximum likelihood estimation there is always the possibility that the estimated value of the location parameter may be larger than the sample minimum value. The chance of “no solution” is large when c is near to unity and the sample is small, but becomes smaller as the sample size increases. This is related to the fact that large samples have a close aLnity to the basic distribution structure. • An algorithm for assessing moments of functions of sample moments was introduced by Shenton et al. in 1971 and extended in 1975. This algorithm is computer oriented and uses computer extended Taylor series for f(m1 ; m2 ; m3 ; m4 ), √ where ma is a non-central sample moment. A simple example is m2 , the sample standard deviation. In the present study, we have given four moments for  a∗ = m1 − b∗ (1 + 1=c), where b∗ =b = m2 =2 . Sometimes the series for E[f(m1 ; m2 ; m3 ; m4 )s ; s = 1; 2; 3; 4 when arrayed in descending powers of N , may show divergency tendencies. Rational fraction (Pad8e) sequences may be set up in several forms, and these generally decrease the divergency. • The paper includes an application to fracture toughness for steels studied by Nanstad et al. (1993), and Bowman and Williams (2000), there being a temperature dependency. Sample sizes were 1– 44. These sample sizes were too small to %t the three-parameter model successfully. Series and associated moments of the moment estimators of location and scale parameters are given including checks of these by simulation. The scale estimator is estimated by the standard deviation, and location parameter by a linear function of the mean and standard deviation. Acknowledgements This research was sponsored by the Applied Mathematical Sciences Research Program, Oak Ridge National Laboratory, is operated by UT-Battelle, LLC for the US Department of Energy, under Contract no. DE-AC05-00OR22725.

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References Bowman, K.O., Kastenbaum, M.A., Shenton, L.R., 1992. The negative hypergeometric distribution and estimation by moments. Comm. Statist. Simulation Comput. 21 (2), 301–332. Bowman, K.O., Shenton, L.R., 1979a. Approximate percentage points for Pearson distributions. Biometrika 66, 147–151. Bowman, K.O., Shenton, L.R., 1979b. Further approximate Pearson percentage points and Cornish-Fisher. Comm. Statist. Simulation Comput. B (8), 231–244. Bowman, K.O., Shenton, L.R., 1998. Asymptotic skewness and the distribution of maximum likelihood estimators. Comm. Statist. Theory Method 27 (11), 2743–2760. Bowman, K.O., Shenton, L.R., 1999. The asymptotic kurtosis for maximum likelihood estimators. Comm. Statist. Theory Method 28 (11), 2641–2654. Bowman, K.O., Shenton, L.R., 2000. Maximum likelihood and the Weibull distribution. Far. East J. Theory Statist., to appear. Bowman, K.O., Williams, P.T., 2000. Technical basis for statistical models of extended KIc and KIa fracture toughness database for RPV steels. ORNL=NRC=LTR-99=27. Johnson, N.L., Kotz, S., Balakrishnan, N., 1994. Continuous Univariate Distributions, Vol. 1, 2nd Edition. Wiley, New York. Nanstad, R.K., Keeney, J.A., McCabe, D.E., 1993. Preliminary review of the bases for the KIc curve in the ASME code. ORNL=NRC=LTR-93=15. Shenton, L.R., Bowman, K.O., 1975. The development of techniques for the evaluation of sampling moments. Internat. Statist. Rev. 43 (3), 317–334. Shenton, L.R., Bowman, K.O., Sheehan, D., 1971. Sampling moments of moments associated with univariate distributions. J. Roy. Statist. Soc. Ser. B 33, 444–457.