双应力恒加寿命试验下相依竞争失效模型的统计分析

doi:10.3969/j.issn.1001-506X.2020.11.29

摘要/Abstract

摘要：

针对传统加速寿命试验仅考虑单一应力且假定产品各失效机理之间独立的问题,首先结合自适应逐步Ⅱ型混合截尾(adaptive type-Ⅱ progressively hybrid censored scheme, AT-Ⅱ PHCS)寿命试验,建立AT-Ⅱ PHCS双应力恒加寿命试验。其次,利用Marshall-Olkin二维Gompertz (Marshall-Olkin bivariate Gompertz, MOBG)分布构建产品失效机理间相依关系。然后,基于AT-Ⅱ PHCS双应力加速寿命试验下相依竞争失效数据,利用极大似然理论,计算了参数的极大似然估计(maximum likelihood estimation, MLE),采用渐近似然理论和Bootstrap抽样方法构建参数的置信区间(confidence interval, CI)。最后,结合一个实例对构建的相依竞争失效模型进行了算例验证,结果表明其具有很好的统计推断性能。

关键词: 双应力恒加寿命试验, 相依竞争失效模型, Marshall-Olkin二维Gompertz分布, 极大似然估计, Bootstrap方法

Abstract:

To solve problem that traditional accelerated life test only considers a single stress and assumes the competing risks indenpendce, firstly, the double constant-stress accelerated life test under adaptive Type-Ⅱ progressively hybrid censored scheme (AT-Ⅱ PHCS) is established. Secondly, the Marshall-Olkin bivariate Gompertz (MOBG) distribution is used to construct the dependence of the competing risks. Based on the dependent competing data under AT-Ⅱ PHCS double-stress accelerated life test, the maximum likelihood estimation (MLE) of the unknown parameters is obtained. The asymptotic likelihood theory and the Bootstrap sampling method are used to construct the confidence interval (CI) of the unknown parameters. Finally, an example is used to verify the constructed dependent competing risks model, and the results show that it has good statistical inference performance.

Key words: double constant-stress accelerated life test, dependent competing risks model, Marshall-Olkin bivariate Gompertz (MOBG) distribution, maximum likelihood estimation (MLE), Bootstrap method

中图分类号:

O213.2

王燕, 师义民. 双应力恒加寿命试验下相依竞争失效模型的统计分析[J]. 系统工程与电子技术, 2020, 42(11): 2644-2653.

Yan WANG, Yimin SHI. Statistical analysis of the dependent competing risks model under the double constant-stress accelerated life test[J]. Systems Engineering and Electronics, 2020, 42(11): 2644-2653.

图/表 9

表1

表2

表3

表4

表5

表6

表7

图1

图2

参考文献 30

1	LI P C , TING W , KWONG D L . Time-dependent dielectric breakdown of chemical-vapour-depositedSiO2 gate dielectrics[J]. Electronics Letters, 1989, 25 (10): 665- 666. doi: 10.1049/el:19890450
2	鲍志晖, 李玲. Weibull分布下双应力恒加试验的一种区间估计方法[J]. 池州学院报, 2019, 33 (3): 9- 12.
	BAO Z H , LI L . The method of inverse on double alternative step-stress accelerated life test for Weibull distribution[J]. Journal of Chizhou University, 2019, 33 (3): 9- 12.
3	孙天宇, 师义民, 谢奇. Weibull分布下双应力交叉步加试验的可靠性分析[J]. 机械强度, 2013, 35 (3): 253- 257.
	SUN T Y , SHI Y M , XIE Q . Analysis ofreliability performances on two alternative step-stress accelerated life test for Weibull distribution[J]. Journal of Mechanical Strength, 2013, 35 (3): 253- 257.
4	寇海霞, 安宗文, 刘波. 双应力同步步降加速命试验方法[J]. 电子科报, 2016, 45 (2): 316- 320.
	KOU H X , AN Z W , LIU B . Double synchronous-step-down-stress accelerated life test method[J]. Journal of University of Electronic Science and Technology of China, 2016, 45 (2): 316- 320.
5	ZHAO Z C, SONG B W, WANG X P. Maximum likelihood estimation of double alternative step-stressaccelerated life test for electronics[C]//Proc.of the 3rd International Conference on Artificial Intelligence and Computational Intelligenc, 2011: 27-34.
6	CHEN J , LI J , WANG D Y , et al. Double-crossed step-down-stress accelerated life testing for pneumatic cylinder based on cumulative damage model[J]. Advanced Materials Research, 2014, 87 (1): 56- 63.
7	王德义, 陈娟. 双应力交叉步降加速寿命试验研究与应用[J]. 质量与可靠性, 2019, 202 (4): 35- 45.
	WANG D Y , CHEN J . Research and application on double-crossed step-stress accelerated life testing[J]. Quality and Reliability, 2019, 202 (4): 35- 45.
8	张亚涛, 邱明, 周大威, 等. 基于双应力加速寿命试验的关节轴承寿命预测与可靠性分析[J]. 润滑与密封, 2020, 45 (3): 51- 56.
	ZHAN Y T , QIU M , ZHOU D W , et al. Life prediction and reliability analysis of spherical plain bearings based on double stress accelerated life test[J]. Lubrication Engineering, 2020, 45 (3): 51- 56.
9	AHMADI K , REZAEI M , YOUSEFZADEH F . Statistical analysis of middle censored competing risks data with exponential distribution[J]. Journal of Statistical Computation and Simulation, 2017, 87 (16): 3082- 3110. doi: 10.1080/00949655.2017.1357721
10	BAI X C , SHI Y M , LIU Y M , et al. Statistical analysis of competing risks model from Marshall-Olkin extended Chen distribution under adaptive progressively interval censoring with random removals[J]. Communications in Statistics-Theory and Methods, 2019, 48 (15): 3683- 3702. doi: 10.1080/03610926.2018.1481973
11	ZHANG C F , SHI Y M , WU M . Statistical inference for competing risks model in step-stress partially accelerated life tests with progressively type-Ⅰ hybrid censored Weibull life data[J]. Journal of Computational and Applied Mathematics, 2016, 297 (1): 65- 74.
12	黄文平, 周经伦, 宁菊红, 等. 基于竞争失效数据的Lindley分布参数估计[J]. 系统工程与电子技术, 2016, 38 (2): 464- 469.
	HANG W P , ZHOU J L , NING J H , et al. Parameters estimation of Lindley distribution with competing risks data[J]. Systems Engineering and Electronics, 2016, 38 (2): 464- 469.
13	AZIZI F , HAGHIGHI F , TABIBI G N . Statistical inference for competing risks model under progressive interval censored Weibull data[J]. Communications in Statistics-Simulation and Computation, 2018, 47 (3): 1- 14.
14	WANG L . Inference of progressively censored competing risks data from Kumaraswamy distributions[J]. Journal of Computational and Applied Mathematics, 2018, 343 (9): 719- 736.
15	BAGHESTANI A R , HOSSEINE-BAHARANCHI F S . An improper form of Weibull distribution for competing risks analysis with Bayesian approach[J]. Journal of Applied Statistics, 2019, 46 (13): 2409- 2417. doi: 10.1080/02664763.2019.1597027
16	LIN D Y , SUN W , YING Z . Nonparametric estimation of the gap time distributions for serial events with censored data[J]. Biometrika, 1999, 86 (1): 59- 70.
17	WANG L , LI H Y , MA J G . Inference for dependence competing risks from bivariate exponential model under generalized progressive hybrid censoring with partially observed failure causes[J]. Journal of Systems Engineering and Electronics, 2019, 30 (1): 201- 208.
18	SHEN Y , XU A C . On the dependent competing risks using Marshall-Olkin bivariate Weibull model: parameter estimation with different methods[J]. Communications in Statistics-Theory and Methods, 2018, 47 (22): 5558- 5572. doi: 10.1080/03610926.2017.1397170
19	BAI X C , SHI Y M , ZHANG C F . Statistical inference of constant stress accelerated life testwith dependent competing risks[J]. Quality and Reliability Engineering International, 2020, 36 (2): 511- 528.
20	WU M , SHI Y M , ZHANG C F . Statistical analysis of dependent competing risks model in accelerated life testing under progressively hybrid censoring using copula function[J]. Communication in Statistics-Simulation and Computation, 2017, 46 (5): 4004- 4017.
21	SRIVASTAVA P W , GUPTA T . Copula based constant-stress PALT using tampered failure rate model with dependent competing risks[J]. International Journal of Quality & Reliability Management, 2019. doi: 10.1108/IJQRM-08-2017-0170
22	NG H K T , KUNDUD , CHAN P S . Statistical analysis of exponential lifetimes under an adaptive type-Ⅱ progressive censoring scheme[J]. Naval Research Logistics, 2009, 56 (8): 687- 698. doi: 10.1002/nav.20371
23	BLOCK H W , BASU A P . A continuous bivariate exponential extension[J]. Journal of the American Statistical Association, 1974, 69 (384): 1031- 1037.
24	KUNDU D . On hybrid censored Weibull distribution[J]. Journal of Statistical Planning and Inference, 2007, 137 (7): 2127- 2142. doi: 10.1016/j.jspi.2006.06.043
25	MEEKER W Q , ESCOBARL A . Statistical methods for reliability data[M]. NewYork: Wiley, 1998.
26	EFRON B . The Jack knife, the bootstrap and other resampling plans[M]. Philadelphia: SIAM, 1982.
27	PHILLIPS M J . Statistical Methods for Reliability Data Analysis[M]. London: Springer, 2003.
28	BALAKRISHNAN N , SANDHURA . A simple simulation algorithm for generating progressive type-Ⅱ censored samples[J]. The American Statistician, 1995, 49 (2): 229- 230.
29	EBRAHIM I N . Analysis of bivariate accelerated life test data for the bivariate exponential distribution[J]. American Journal of Mathematical and Management Sciences, 1987, 7 (1/2): 175- 190.
30	KUNDN D , GUPTA A K . On bivariate Weibull-Geometric distribution[J]. Journal of Multivariate Analysis, 2014, 123 (1): 19- 29.

方案	n₁₁=n₂₁=n₂₂=20	方案	n₁₁=n₂₁=n₂₂=30	方案	n₁₁=n₂₁=n₂₂=40
方案	(m₁₁, m₂₁, m₂₂, τ₁₁, τ₂₁, τ₂₂)	方案	(m₁₁, m₂₁, m₂₂, τ₁₁, τ₂₁, τ₂₂)	方案	(m₁₁, m₂₁, m₂₂, τ₁₁, τ₂₁, τ₂₂)
Ⅰ	(6, 8, 10, 0.8, 0.6, 0.4)	Ⅰ	(8, 10, 12, 0.8, 0.6, 0.4)	Ⅰ	(12, 14, 18, 0.8, 0.6, 0.4)
Ⅱ	(8, 10, 12, 0.8, 0.6, 0.4)	Ⅱ	(12, 14, 18, 0.8, 0.6, 0.4)	Ⅱ	(18, 22, 26, 0.8, 0.6, 0.4)
Ⅲ	(6, 8, 10, 1, 0.8, 0.6)	Ⅲ	(8, 10, 12, 1, 0.8, 0.6)	Ⅲ	(12, 14, 18, 1, 0.8, 0.6)
Ⅳ	(8, 10, 12, 1, 0.8, 0.6)	Ⅳ	(12, 14, 18, 1, 0.8, 0.6)	Ⅳ	(18, 22, 26, 1, 0.8, 0.6)

方案	参数	λ	α₁₁₀	α₁₁₁	α₁₁₂	α₂₁₀	α₂₁₁	α₂₁₂	α₂₂₀	α₂₂₁	α₂₂₂
I	EMLE	0.651 3	0.181	0.265 4	0.251 3	0.274 3	0.419 5	0.385 6	0.374 6	0.350 3	0.617 7
	MSE	0.121 3	0.084 3	0.092 7	0.106 7	0.105 8	0.107 7	0.121 4	0.114 2	0.116 3	0.125 4
	ACI(CP)	91.500 0	93.300 0	93.100 0	92.200 0	92.600 0	91.900 0	91.800 0	92.200 0	92.200 0	90.900 0
	BCI(CP)	93.200 0	94.900 0	94.800 0	93.900 0	94.200 0	93.500 0	93.600 0	93.800 0	9400 0	93.100 0
Ⅱ	EMLE	0.944 8	0.187 5	0.177 7	0.384 6	0.280 3	0.227 7	0.611 7	0.386 8	0.362 7	0.626 4
	MSE	0.112 6	0.073 8	0.084 6	0.096 4	0.096 9	0.096 6	0.115 1	0.103 3	0.104 2	0.118 7
	ACI(CP)	92.800 0	94.400 0	94.000 0	93.100 0	93.500 0	93.000 0	92.800 0	93.100 0	93.400 0	92.300 0
	BCI(CP)	94.200 0	95.800 0	95.600 0	94.600 0	95.000 0	94.500 0	94.400 0	94.700 0	95.000 0	94.000 0
Ⅲ	EMLE	0.667 4	0.186 8	0.175 4	0.261 7	0.281 6	0.223	0.394 5	0.483 5	0.361 1	0.624 9
	MSE	0.118 2	0.081 7	0.088 3	0.097 5	0.095 1	0.095 8	0.114 4	0.102 2	0.102 4	0.117 7
	ACI(CP)	92.500 0	94.200 0	93.900 0	92.900 0	93.500 0	93.100 0	93.000 0	93.200 0	93.500 0	92.500 0
	BCI(CP)	94.200 0	95.700 0	95.400 0	94.500 0	95.100 0	94.800 0	94.700 0	94.600 0	95.200 0	94.100 0
Ⅳ	EMLE	0.934 8	0.191	0.186 1	0.272 1	0.288 6	0.239 7	0.411 1	0.393 1	0.516 2	0.637 3
	MSE	0.101 2	0.068 3	0.079 1	0.088 4	0.871	0.092 6	0.101 1	0.092 5	0.091 6	0.105 1
	ACI(CP)	93.600 0	95.400 0	95.400 0	94.100 0	94.400 0	94.200 0	94.100 0	94.500 0	94.700 0	93.200 0
	BCI(CP)	95.200 0	96.700 0	96.700 0	95.700 0	95.900 0	95.700 0	95.600 0	96.000 0	96.100 0	94.800 0

方案	参数	λ	α₁₁₀	α₁₁₁	α₁₁₂	α₂₁₀	α₂₁₁	α₂₁₂	α₂₂₀	α₂₂₁	α₂₂₂
Ⅰ	EMLE	0.665 6	0.188 3	0.181 9	0.259 1	0.281 8	0.223 4	0.615 7	0.381 5	0.362 2	0.624 8
	MSE	0.116 6	0.075 2	0.081 3	0.094 3	0.092 2	0.097 7	0.107 8	0.102 5	0.102 5	0.112 4
	ACI(CP)	93.100 0	94.400 0	94.300 0	93.600 0	93.800 0	93.400 0	93.200 0	93.400 0	93.200 0	92.400 0
	BCI(CP)	94.000 0	95.300 0	95.200 0	94.500 0	94.600 0	94.300 0	94.100 0	94.200 0	94.500 0	93.900 0
Ⅱ	EMLE	0.678 3	0.197 2	0.191 3	0.271 4	0.289 4	0.235 9	0.408 3	0.394 5	0.508 6	0.634 6
	MSE	0.104 5	0.065 1	0.075 8	0.086 9	0.083 4	0.086 1	0.099 3	0.094 7	0.092 6	0.104 3
	ACI(CP)	94.100 0	95.400 0	95.400 0	94.400 0	94.800 0	94.400 0	94.400 0	94.100 0	94.300 0	93.400 0
	BCI(CP)	94.800 0	96.200 0	96.300 0	95.200 0	95.500 0	95.100 0	94.900 0	94.900 0	95.300 0	94.500 0
Ⅲ	EMLE	0.677 3	0.195 1	0.190 9	0.271 2	0.288 7	0.236 7	0.405 4	0.388 7	0.374 7	0.635 6
	MSE	0.105 3	0.067 7	0.076 9	0.087 6	0.084 1	0.085 6	0.101 1	0.099 5	0.093 1	0.105 6
	ACI(CP)	93.800 0	95.300 0	95.500 0	94.200 0	94.800 0	94.500 0	94.400 0	94.400 0	94.300 0	93.500 0
	BCI(CP)	94.500 0	95.900 0	96.100 0	94.900 0	95.600 0	95.300 0	95.300 0	95.200 0	95.100 0	94.300 0
Ⅳ	EMLE	0.685 7	0.207 8	0.199 3	0.280 7	0.292 2	0.247 8	0.418 5	0.399 1	0.392 3	0.644 5
	MSE	0.091 7	0.054 3	0.065 7	0.078 6	0.078 3	0.078 6	0.915	0.085 7	0.082 6	0.092 4
	ACI(CP)	95.000 0	96.600 0	96.200 0	95.500 0	95.800 0	95.400 0	95.400 0	95.500 0	95.400 0	94.800 0
	BCI(CP)	96.100 0	97.300 0	97.300 0	96.200 0	96.300 0	96.100 0	96.100 0	96.700 0	96.900 0	95.600 0

方案	参数	λ	α₁₁₀	α₁₁₁	α₁₁₂	α₂₁₀	α₂₁₁	α₂₁₂	α₂₂₀	α₂₂₁	α₂₂₂
Ⅰ	EMLE	0.688	0.191 4	0.189 8	0.264 6	0.292 7	0.237 5	0.404 5	0.390 6	0.513 5	0.634 9
	MSE	0.103 1	0.065 6	0.071 8	0.085 2	0.088 5	0.086 8	0.094 2	0.091 6	0.093 5	0.101 5
	ACI(CP)	94.100 0	95.300 0	95.100 0	94.300 0	94.800 0	94.300 0	9400 0	94.300 0	94.100 0	93.300 0
	BCI(CP)	94.600 0	95.800 0	95.700 0	95.000 0	95.400 0	94.900 0	94.700 0	94.900 0	95.200 0	94.500 0
Ⅱ	EMLE	0.706 4	0.202 6	0.197 1	0.279 9	0.300 7	0.245 8	0.411 1	0.403 1	0.386 2	0.644 4
	MSE	0.091 6	0.056 1	0.062 3	0.075 4	0.073 1	0.075 3	0.087 6	0.082 3	0.084 5	0.092 5
	ACI(CP)	95.500 0	96.700 0	96.800 0	96.100 0	96.100 0	95.800 0	95.400 0	95.800 0	94.600 0	94.800 0
	BCI(CP)	95.900 0	97.100 0	97.200 0	96.500 0	96.400 0	96.100 0	95.800 0	96.100 0	95.800 0	95.200 0
Ⅲ	EMLE	0.707 4	0.201 8	0.194 2	0.279 7	0.297 5	0.242 1	0.598 9	0.391 9	0.385 2	0.644 5
	MSE	0.090 3	0.059 7	0.061 3	0.074 8	0.077 4	0.076 5	0.090 5	0.086 3	0.085 8	0.096 8
	ACI(CP)	95.300 0	96.700 0	96.600 0	95.900 0	96.100 0	95.900 0	95.600 0	95.900 0	94.800 0	94.900 0
	BCI(CP)	95.700 0	97.300 0	97.300 0	96.500 0	96.700 0	96.600 0	96.300 0	96.700 0	95.600 0	95.700 0
Ⅳ	EMLE	0.714 7	0.210 3	0.203 3	0.287 3	0.309 3	0.253 2	0.427 5	0.411 7	0.406 2	0.657 2
	MSE	0.081 2	0.045 7	0.054 5	0.063 8	0.068 1	0.068 6	0.083 2	0.073 7	0.073 3	0.083 1
	ACI(CP)	96.800 0	97.900 0	97.500 0	97.100 0	97.300 0	96.900 0	96.800 0	97.200 0	96.100 0	96.700 0
	BCI(CP)	97.100 0	98.100 0	97.800 0	97.400 0	97.600 0	97.200 0	97.100 0	97.600 0	96.500 0	97.100 0

序号	应力水平组合(1, 1)	应力水平组合(2, 1)	应力水平组合(2, 2)
1	(0.04, 0, 0)	(0.02, 1, 0)	(0.07, 0, 1)
2	(0.10, 1, 0)	(0.06, 0, 1)	(0.13, 1, 0)
3	(0.16, 0, 0)	(0.27, 0, 0)	(0.20, 0, 0)
4	(0.17, 0, 1)	(0.30, 0, 1)	(0.29, 1, 0)
5	(0.42, 0, 1)	(0.30, 0, 1)	(0.30, 0, 1)
6	(0.46, 0, 0)	(0.34, 1, 0)	(0.30, 0, 1)
7	(0.47, 1, 0)	(0.36, 1, 0)	(0.30, 1, 0)
8	(0.47, 1, 0)	(0.70, 0, 1)	(0.33, 0, 1)
9	(1.05, 1, 0)	(0.80, 0, 1)	(0.33, 0, 1)
10	(1.50, 0, 1)	(0.80, 0, 0)	(0.40, 0, 0)
11	(1.80, 0, 0)	(0.92, 0, 1)	(0.40, 1, 0)
12	(2.23, 1, 0)	(1.06, 0, 1)	(0.63, 0, 0)
13	—	(1.30, 0, 0)	(0.82, 0, 0)
14	—	—	(0.84, 0, 1)