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

Abstract

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

CLC Number:

O213.2

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.

Figures/Tables 9

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References 30

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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)

Statistical analysis of the dependent competing risks model under the double constant-stress accelerated life test

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分布	F(t; λ, α₁₁₀₁)	F(t; λ, α₁₁₀₂)	F(t; λ, α₁₁)	F(t; λ, α₂₁₀₁)	F(t; λ, α₂₁₀₂)	F(t; λ, α₂₁)	F(t; λ, α₂₂₀₁)	F(t; λ, α₂₂₀₂)	F(t; λ, α₂₂)
K-S距离	0.215 1	0.141 8	0.185 5	0.145 6	0.134 7	0.164 7	0.208 2	0.213 6	0.226 8
p值	0.563 6	0.941 8	0.738 5	0.909 0	0.947 4	0.818 4	0.513 0	0.480 7	0.406 7

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参数	λ	α₁₁₀	α₁₁₁	α₁₁₂	α₂₁₀	α₂₁₁	α₂₁₂	α₂₂₀	α₂₂₁	α₂₂₂
MLE	1.120 1	0.091 2	0.174 5	0.127 7	0.156 2	0.438 6	0.635 7	0.602 3	0.097 6	0.2048
ACI	2.312 7	0.201 6	0.358 4	0.251 2	0.312 4	0.901 2	1.236 9	1.198 3	0.235 1	0.415 4
BCI	2.378 2	0.211 4	0.362 9	0.262 3	0.327 8	0.915 6	1, 315 4	1.246 9	0.241 3	0.424 3