一般化極値分布、3 母数ワイブル分布の最良かもしれない推定法 KLDMWR-GE
Takuya KAWANISHI
Kanazawa University

September 7, 2022

https://t-kawanishi.w3.kanazawa-u.ac.jp/ Takuya Kawanishi

1Introduction∙ For the three-parameter Weibull (TPW) and the generalized
extreme value (GEV) distributions, the maximum likelihood(ML) method fails for some range of parameters.

∙ We ﬁnd that our KLDMWR-GE, which is equivalent to Jiang’s
modiﬁed maximum product of spacings (JMMPS) method, isapplicable to these distributions for all ranges of parameters,
and in many cases has less bias than the original MPS.

∙ We know there are many alternatives to ML for TPW andGEV, but, in this presentation we only investigate the relative
performance of KLDMWR-CE (ML), SE (MPS) and GE(JMMPS).

https://t-kawanishi.w3.kanazawa-u.ac.jp/ Takuya Kawanishi

2Introduction 2

TPW[ ()𝜉 ]𝑥−𝜈𝐹 (𝑥; 𝜈, 𝜂, 𝜉) = 1 − exp −𝜂

(1)

For ML,∙ when 𝜉 < 1, MLE does not exist or inconsistent;∙ when 1 ≥ 𝜉 < 2, MLE may exist but not asymptotically normal;∙ when 2 ≤ 𝜉 , MLE exists and asymptotically normal.
In contrast, for KLDMWR-SE (MPS) and GE (JMMPS),∙ when 𝜉 < 2, the SE and GE exist but not asymptoticallynormal.

https://t-kawanishi.w3.kanazawa-u.ac.jp/ Takuya Kawanishi

3Introduction 3GEV

[ {()}−1∕𝜉 ]𝑥−𝜈
.
𝐺(𝑥; 𝜈, 𝜂, 𝜉) = exp − 1 + 𝜉𝜂(2)For ML,∙ when 𝜉 < −1, MLE does not exist or inconsistent;∙ when −1 ≥ 𝜉 < 0.5, MLE may exist but not asymptoticallynormal;∙ when −0, 5 ≤ 𝜉 , MLE exists and asymptotically normal.
In contrast, for KLDMWR-SE (MPS) and GE (JMMPS),∙ when 𝜉 < −0.5, the SE and GE exist but not asymptoticallynormal.
It is known that the TPW is a reparameterization of GEV, twodistributions are closely related.
https://t-kawanishi.w3.kanazawa-u.ac.jp/ Takuya Kawanishi

4Simulation∙ We generated samples using the probability integral
transformations.

∙ For TPW, 𝜈0 = 1, 𝜂0 = 1, 𝜉0 = 0.5: inconsistent ML, 𝜉0 = 1.5:
non-asymptotically normal ML, 𝜉0 = 2.5: regular ML.
∙ For GEV, 𝜈0 = 1, 𝜂0 = 1, 𝜉0 = −2: inconsistent ML, 𝜉0 = −0.75:
non-asymptotically normal ML, 𝜉0 = 0.75: regular ML.
∙ Sample size 𝑛 ∈ {20, 50, 100, 200, 500}.
∙ Number of samples: 10000.

https://t-kawanishi.w3.kanazawa-u.ac.jp/ Takuya Kawanishi

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MSE, TPW

10 1

100
10 1MSE

10 4

10 1010 2

10 2

10 7

10 3101102
n103

101102n

10 4103

101

102
n103

Figure: TPW, MSE vs sample size; circle: GE, triangle: SE, cross: MLE;
solid line: 𝜉0 = 0.75, Dotted line: 𝜉0 = 1.5, dot-dash line: 𝜉0 = 2.5.

https://t-kawanishi.w3.kanazawa-u.ac.jp/ Takuya Kawanishi

6Bias, TPWnbias

ν (ξ0 = 1.5)
0.05

20.00

0nbiasν (ξ0 = 1.5)
205 ν (ξ0 = 2.5)nbias

η (ξ0 = 1.5)

0.0
2.5
5.0
5

005

5102

n0

ξ (ξ0 = 1.5)

2

η (ξ0 = 1.5)5 ξ (ξ0 = 1.5)05

η (ξ0 = 2.5)

ξ (ξ0 = 2.5)
010

102

n102

nFigure: TPW, 𝑛bias vs sample size; circle: GE, triangle: SE, cross: MLE;
solid line: 𝜉0 = 0.5, Dotted line: 𝜉0 = 1.5, dot-dash line: 𝜉0 = 2.5.

https://t-kawanishi.w3.kanazawa-u.ac.jp/ Takuya Kawanishi

7Bias of return levels at 𝑝 = 0.01, TPW
nbias

0.10 z01(ξ0 = 0.5)
0.05

nbias

nbias

0.00
2z01(ξ0 = 1.5)

0
2.5
0.0
2.5z01(ξ0 = 2.5)

102

nFigure: TPW, 𝑛bias vs sample size; circle: GE, triangle: SE, cross: MLE;
solid line: 𝜉0 = 0.5, Dotted line: 𝜉0 = 1.5, dot-dash line: 𝜉0 = 2.5.

https://t-kawanishi.w3.kanazawa-u.ac.jp/ Takuya Kawanishi

8MSEMSE, GEV

10110110110210210210310310310110n210310110n210310110n2103Figure: GEV, MSE vs sample size; circle: GE, triangle: SE, cross: MLE;
solid line: 𝜉0 = −2.0, Dotted line: 𝜉0 = −0.75, dot-dash line: 𝜉0 = 0.75.

https://t-kawanishi.w3.kanazawa-u.ac.jp/ Takuya Kawanishi

9Bias, GEVnbias0η (ξ0 = - 2.0)
512nbiasν (ξ0 = - 2.0)0ξ (ξ0 = - 2.0)100ν (ξ0 = - 0.75)

0η (ξ0 = - 0.75)

22.500.02

ξ (ξ0 = - 0.75)

4nbiasν (ξ0 = 0.75)
15.0

η (ξ0 = 0.75)

2.5

10n2

ξ (ξ0 = 0.75)

10.0

0

2010

n210

2n

Figure: GEV, 𝑛bias vs sample size; circle: GE, triangle: SE, cross: MLE;
solid line: 𝜉0 = −2.0, Dotted line: 𝜉0 = −0.75, dot-dash line: 𝜉0 = 0.75.

https://t-kawanishi.w3.kanazawa-u.ac.jp/ Takuya Kawanishi

10Bias of return levels at 𝑝 = 0.01, GEV

nbias

nbias

nbias

0.00 z01(ξ0 = - 2.0)
0.02

0z01(ξ0 = - 0.75)

22000z01(ξ0 = 0.75)

1000
0102

nFigure: GEV, 𝑛bias vs sample size; circle: GE, triangle: SE, cross: MLE;
solid line: 𝜉0 = −2.0, Dotted line: 𝜉0 = −0.75, dot-dash line: 𝜉0 = 0.75.

https://t-kawanishi.w3.kanazawa-u.ac.jp/ Takuya Kawanishi

11Proxy∙ Log likelihood 𝑙L (𝜃̂L ), and 𝑙L 𝜃̂G ), and 𝑙L (𝜃̂S )∙ Left panel: GEV, Regular case (MLE exists and asymptoticallynormal), data from Coles (2001), center panel: TPW, MLE
exsits but not asymptotically normal, data from Cheng andAmin (1983), right panel: TPW, MLE not existing, data fromCheng and Amin (1983).

lL

4

17

194

16195

3196

15
0.2

0.0

0.2

1.0

1.5

2.0

0.5

1.0

1.5

Figure: Proﬁle likelihood, comparison of 𝑙L (𝜃̂L ), 𝑙L (𝜃̂G ) and 𝑙L (𝜃̂S ), circle:GE, triangle: SE, cross: MLE.

https://t-kawanishi.w3.kanazawa-u.ac.jp/ Takuya Kawanishi

12Discussion
Parameter estimation of TPW and GEV∙ MSE of the KLDMWR-GE, SE and CE are nearly identical.
∙ For the bias, KLDMWR-GE outperforms SE and CE (MLE) inmost cases.
∙ Exceptions are: location parameter in non-ML-existing case for
TPW, location parameter in regular case for GEV.

∙ However, for the bias of the return levels, the superiority of GE
to SE is questionable.
∙ We will investigate the effect of bias correction.
KLDMWR-GE as a proxy for MLE√∙ For regular cases, while ||𝜃̂L − 𝜃̂S || = 𝑜𝑝 (1∕ 𝑛) (Anatolyev and||
Kosenok, 2005), we have for GE,||̂|𝜃L − 𝜃̂G | = 𝑂(1∕𝑛),||∙ Need more investigation for non-regular cases
https://t-kawanishi.w3.kanazawa-u.ac.jp/ Takuya Kawanishi

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References
S. Anatolyev and G. Kosenok, Econ. Theory 21, (2005).
R. C. H. Cheng and N. a. K. Amin, Journal of the Royal StatisticalSociety: Series B (Methodological) 45, 394 (1983).
S. Coles, An Introduction to Statistical Modeling of Extreme Values(Springer, London; New York, 2001).
R. Jiang, A modiﬁed MPS method for ﬁtting the 3-parameter
Weibull distribution. Presented at the QR2MSE 2013 Proceedings of 2013 International Conference on Quality,Reliability, Risk, Maintenance, and Safety Engineering (2013)
T. Kawanishi, Comp. Appl. Math. 39, (2020).
B. Ranneby, Scandinavian Journal of Statistics 11, 93 (1984).
R. L. SMITH, Biometrika 72, 67 (1985).
T. S. T. Wong and W. K. Li, in Institute of Mathematical Statistics
Lecture Notes - Monograph Series (Institute of Mathematical
Statistics, Beachwood, Ohio, USA, 2006), pp. 272 ‒ 283.
https://t-kawanishi.w3.kanazawa-u.ac.jp/ Takuya Kawanishi

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