Weibull形状参数加权极大似然估计的偏差修正

doi:10.13229/j.cnki.jdxbgxb.20221589

吉林大学学报(工学版) ›› 2024, Vol. 54 ›› Issue (10): 2764-2770.doi: 10.13229/j.cnki.jdxbgxb.20221589

• 车辆工程·机械工程 • 上一篇

Weibull形状参数加权极大似然估计的偏差修正

杨小玉^1,²(),谢里阳^1,²(),杨奕凤^1,²,赵丙峰^1,²,李元³

^1.东北大学机械工程与自动化学院，沈阳 110819
^2.东北大学航空动力装备振动及控制教育部重点实验室，沈阳 110819
^3.西北工业大学航空学院，西安 710072

收稿日期:2022-10-12 出版日期:2024-10-01 发布日期:2024-11-22
通讯作者: 谢里阳 E-mail:yxy18210532358@126.com;lyxieneu@163.com
作者简介:杨小玉（1993-），女，博士研究生.研究方向：可靠性工程.E-mail：yxy18210532358@126.com
基金资助:
国家科技重大专项项目(J2019-V-0009-0103);国家自然科学基金项目(52005087)

Bias correction for weighted maximum likelihood estimation of Weibull shape parameters

Xiao-yu YANG^1,²(),Li-yang XIE^1,²(),Yi-feng YANG^1,²,Bing-feng ZHAO^1,²,Yuan LI³

^1.School of Mechanical Engineering and Automation，Northeastern University，Shenyang 110819，China
^2.Key Laboratory of Vibration and Control of Aero-Propulsion Systems，Ministry of Education，Northeastern University，Shenyang 110819，China
^3.School of Aeronautics，Northwestern Polytechnical University，Xi′an 710072，China

Received:2022-10-12 Online:2024-10-01 Published:2024-11-22
Contact: Li-yang XIE E-mail:yxy18210532358@126.com;lyxieneu@163.com

摘要/Abstract

摘要：

针对在小样本下加权最大似然方法（WMLE）估计形状参数存在偏差的问题，本文研究了在小样本情形下（样本量为3~15）加权极大似然估计形状参数（ $β > 1$ ）的偏差修正方法，提出了关于样本量的两个偏差修正函数，易于使用。通过蒙特卡洛（Monte Carlo）仿真，在不同的威布尔模型、不同小样本量下，验证了本文偏差修正方法估计的形状参数的准确性和有效性。最后通过实例分析，表明本文偏差修正方法具有可行性。

关键词: 应用统计数学, 三参数威布尔分布, 形状参数, 加权极大似然估计, 偏差修正

Abstract:

There is bias in estimating shape parameters（ $β > 1$ ） using the weighted maximum likelihood method（WMLE） under small samples. Hence， we study the bias correction method for weighted maximum likelihood estimation of shape parameters in the small sample case （sample sizes of 3 to 15）， and two bias correction functions about the sample size are proposed， which are easy to use. By Monte Carlo simulation， the accuracy and validity of the shape parameters estimated by this bias correction method are verified under different Weibull models and different small sample sizes. The example shows that the bias correction method is feasible.

Key words: applied statistical mathematics, three-parameter Weibull distribution, shape parameter, weighted maximum likelihood estimation, bias correction

中图分类号:

TB114.3

杨小玉,谢里阳,杨奕凤,赵丙峰,李元. Weibull形状参数加权极大似然估计的偏差修正[J]. 吉林大学学报(工学版), 2024, 54(10): 2764-2770.

Xiao-yu YANG,Li-yang XIE,Yi-feng YANG,Bing-feng ZHAO,Yuan LI. Bias correction for weighted maximum likelihood estimation of Weibull shape parameters[J]. Journal of Jilin University(Engineering and Technology Edition), 2024, 54(10): 2764-2770.

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参考文献 14

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特征	n=3	n=4	n=5	n=6	n=7	n=8	n=9	n=10	n=11	n=12	n=13	n=14	n=15
mean	1.4924	1.6632	1.7516	1.8106	1.8392	1.8566	1.8808	1.8942	1.9038	1.9176	1.9208	1.9282	1.9294
bias	-0.5076	-0.3368	-0.2484	-0.1894	-0.1608	-0.1434	-0.1192	-0.1058	-0.0962	-0.0824	-0.0792	-0.0718	-0.0706

特征	n=3	n=4	n=5	n=6	n=7	n=8	n=9	n=10	n=11	n=12	n=13	n=14	n=15
mean	2.2140	2.4711	2.5929	2.6748	2.7114	2.7579	2.7909	2.8146	2.8305	2.8404	2.8464	2.8623	2.8674
bias	-0.7860	-0.5289	-0.4071	-0.3252	-0.2886	-0.2421	-0.2091	-0.1854	-0.1695	-0.1596	-0.1536	-0.1377	-0.1326

特征	n=3	n=4	n=5	n=6	n=7	n=8	n=9	n=10	n=11	n=12	n=13	n=14	n=15
mean	1.4875	1.6643	1.7527	1.8118	1.8405	1.8642	1.8810	1.8963	1.9023	1.9158	1.9257	1.9280	1.9301
bias	-0.5125	-0.3357	-0.2473	-0.1882	-0.1595	-0.1358	-0.1190	-0.1037	-0.0977	-0.0842	-0.0743	-0.0720	-0.0699

特征	n=3	n=4	n=5	n=6	n=7	n=8	n=9	n=10	n=11	n=12	n=13	n=14	n=15
mean	2.2033	2.4784	2.5961	2.6798	2.7248	2.7611	2.7984	2.8155	2.8333	2.8364	2.8456	2.8652	2.8638
bias	-0.7967	-0.5216	-0.4039	-0.3202	-0.2752	-0.2389	-0.2016	-0.1845	-0.1667	-0.1636	-0.1544	-0.1348	-0.1362

β_true	β_mean/β_true
β_true	n=3	n=4	n=5	n=6	n=7	n=8	n=9	n=10	n=11	n=12	n=13	n=14	n=15
β=1.5	0.7611	0.8534	0.8969	0.9224	0.9389	0.9492	0.9543	0.9588	0.9657	0.9691	0.9715	0.9719	0.9738
β=1.8	0.7507	0.8379	0.8787	0.9093	0.9253	0.9372	0.9440	0.9508	0.9573	0.9590	0.9654	0.9669	0.9706
β=2	0.7462	0.8316	0.8758	0.9053	0.9196	0.9283	0.9404	0.9471	0.9519	0.9588	0.9604	0.9641	0.9647
β=2.5	0.7380	0.8227	0.8692	0.8947	0.9130	0.9247	0.9326	0.9392	0.9450	0.9514	0.9531	0.9586	0.9605
β=3	0.7380	0.8237	0.8643	0.8916	0.9038	0.9193	0.9303	0.9382	0.9435	0.9468	0.9488	0.9541	0.9558
β=3.5	0.7351	0.8211	0.8576	0.8907	0.9073	0.9204	0.9269	0.9338	0.9397	0.9434	0.9520	0.9536	0.9573
β=4	0.7325	0.8219	0.8609	0.8877	0.9066	0.9192	0.9295	0.9339	0.9396	0.9453	0.9493	0.9553	0.9557
β=4.5	0.7361	0.8181	0.8669	0.8897	0.9095	0.9199	0.9297	0.9350	0.9391	0.9481	0.9483	0.9553	0.9558
β=5	0.7368	0.8237	0.8636	0.8883	0.9077	0.9173	0.9254	0.9334	0.9415	0.9492	0.9498	0.9563	0.9593
β=6	0.7359	0.8269	0.8712	0.8957	0.9084	0.9232	0.9320	0.9396	0.9464	0.9497	0.9541	0.9556	0.9568
β=7	0.7434	0.8311	0.8706	0.8939	0.9103	0.9230	0.9342	0.9387	0.9441	0.9452	0.9549	0.9591	0.9655
β=8	0.7504	0.8315	0.8746	0.8993	0.9186	0.9198	0.9294	0.9332	0.9416	0.9421	0.9506	0.9540	0.9541
mean	0.7420	0.8286	0.8709	0.8974	0.9141	0.9251	0.9341	0.9401	0.9463	0.9507	0.9549	0.9587	0.9608

Weibull形状参数加权极大似然估计的偏差修正

Bias correction for weighted maximum likelihood estimation of Weibull shape parameters

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No.	β_WMLE	β_MR	β_MH
1	2.0901	2.5215	2.5220
2	2.8982	3.4963	3.4970
3	2.0759	2.5043	2.5048
4	2.2605	2.7269	2.7275
5	2.9762	3.5904	3.5912
6	2.0379	2.4585	2.4590
7	2.4504	2.9561	2.9567
8	2.4835	2.9960	2.9966
9	2.0966	2.5293	2.5298
10	2.4077	2.9046	2.9052

特征	β_WMLE	β_MR	β_MH
mean	2.4569	2.9646	2.9640
bias	-0.5431	-0.0354	-0.0360
RMSE	0.5954	0.2967	0.2967

No.	β_WMLE	β_MR	β_MH
1	3.1042	3.5629	3.5624
2	3.1874	3.6584	3.6578
3	4.0166	4.6102	4.6095
4	3.9846	4.5735	4.5728
5	2.9308	3.3639	3.3634
6	3.0927	3.5498	3.5492
7	3.2942	3.7811	3.7805
8	3.5857	4.1156	4.1150
9	3.5555	4.0810	4.0803
10	3.5897	4.1202	4.1195

特征	β_WMLE	β_MR	β_MH
mean	3.4443	3.9527	3.9533
bias	-0.5557	-0.0473	-0.0467
RMSE	0.6211	0.3218	0.3218

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形状参数	数值
β_WMLE	1.8296
β_MR	1.9441
β_MH	1.9440