Inference for dependence competing risks from bivariate exponential model under generalized progressive hybrid censoring with partially observed failure causes

doi:10.21629/JSEE.2019.01.19

Journal of Systems Engineering and Electronics ›› 2019, Vol. 30 ›› Issue (1): 201-208.doi: 10.21629/JSEE.2019.01.19

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Inference for dependence competing risks from bivariate exponential model under generalized progressive hybrid censoring with partially observed failure causes

Liang WANG*(), Huanyu LI(), Jin'ge MA()

School of Mathematics and Statistics, Xidian University, Xi'an 710071, China

Received:2017-11-20 Online:2019-02-27 Published:2019-02-27
Contact: Liang WANG E-mail:wangl@xidian.edu.cn;1363153391@qq.com;807369960@qq.com
About author:WANG Liang was born in 1983. He received his B.S. degree in applied mathematics from Northwest University in 2006, and M.S. and Ph.D. degrees in applied mathematics from Northwestern Polytechnical University in 2009 and 2012. Since 2012, he has worked as an associate professor at the School of Mathematics and Statistics in Xidian University. His research interests include Bayesian analysis and reliability theory. E-mail:wangl@xidian.edu.cn|LI Huanyu was born in 1992. She received her B.S. degree in statistics from Xinyan Normal University in 2017. Now, she is a master candidate at the School of Mathematics and Statistics in Xidian University. Her research interests include reliability theory and incomplete data analysis. E-mail:1363153391@qq.com|MA Jin'ge was born in 1995. She received her B.S. degree in statistics from Baoji Universuty of Arts and Sciences in 2018. Now, she is a master candidate at the School of Mathematics and Statistics in Xidian University. Her research interest is system reliability analysis. E-mail:807369960@qq.com
Supported by:
the National Natural Science Foundation of China(11501433);the Fundamental Research Funds for the Central Universities(JB180711);This work was supported by the National Natural Science Foundation of China (11501433) and the Fundamental Research Funds for the Central Universities (JB180711)

Abstract

Abstract:

Inference are considered for the dependence competing risks model by using the Marshal-Olkin bivariate exponential distribution. Under generalized progressively hybrid censoring with partially observed failure causes, the maximum likelihood estimators are established, and the approximate confidence intervals are also constructed via the observed Fisher information matrix. Moreover, Bayes estimates and highest probability density credible intervals are presented and the importance sampling technique is used to compute corresponding results. Finally, the numerical analysis is proposed for illustration.

Key words: dependence competing risk, generalized progressive hybrid censoring, bivariate exponential distribution, Bayesian inference

Liang WANG, Huanyu LI, Jin'ge MA. 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.

Figures/Tables 5

Table 1

Table 2

Table 3

Table 4

Table 5

References 13

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$p = 0.15$		MLE			Bayes
$(n,m)$	$k$	$\lambda _0 $	$\lambda _1 $	$\mu $	$\lambda _0 $	$\lambda _1 $	$\mu $
	12	0.873 2 [0.815 9]	0.812 8 [0.711 9]	0.901 2 [0.881 0]	0.845 2 [0.782 4]	0.793 8 [0.679 3]	0.884 2 [0.817 1]
(40,30)	18	0.791 6 [0.687 3]	0.751 1 [0.615 2]	0.847 9 [0.773 3]	0.764 8 [0.638 2]	0.734 5 [0.580 2]	0.831 5 [0.737 9]
	12	0.554 5 [0.339 8]	0.521 3 [0.322 4]	0.613 5 [0.434 0]	0.531 5 [0.312 6]	0.502 1 [0.289 1]	0.597 0 [0.382 7]
(60,50)	30	0.459 1 [0.231 6]	0.431 4 [0.216 9]	0.502 0 [0.298 8]	0.416 9 [0.208 7]	0.412 4 [0.205 9]	0.480 6 [0.268 9]

$p = 0$		MLE			Bayes
$(n,m)$	$k$	$\lambda _0 $	$\lambda _1 $	$\mu $	$\lambda _0 $	$\lambda _1 $	$\mu $
	12	0.693 6 [0.559 7]	0.660 9 [0.482 5]	0.732 4 [0.615 7]	0.679 3 [0.486 6]	0.653 4 [0.468 6]	0.711 8 [0.540 5]
(30,20)	18	0.635 8 [0.455 1]	0.597 4 [0.396 2]	0.701 3 [0.560 2]	0.604 8 [0.422 2]	0.578 0 [0.368 3]	0.581 3 [0.381 1]
	12	0.479 3 [0.256 8]	0.453 5 [0.236 5]	0.535 5 [0.322 4]	0.461 2 [0.216 5]	0.431 9 [0.208 0]	0.516 4 [0.298 3]
(60,45)	30	0.402 1 [0.201 4]	0.396 8 [0.192 4]	0.474 1 [0.265 9]	0.381 9 [0.176 1]	0.365 8 [0.155 7]	0.443 5 [0.226 1]

$p = 0.15$		MLE			Bayes
$(n,m)$	$k$	$\lambda _0 $	$\lambda _1 $	$\mu $	$\lambda _0 $	$\lambda _1 $	$\mu $
	12	0.783 6 [0.664 5]	0.651 9 [0.468 2]	0.842 7 [0.746 4]	0.759 0 [0.615 8]	0.634 2 [0.427 1]	0.824 7 [0.728 7]
(30,20)	18	0.741 3 [0.610 8]	0.600 3 [0.415 2]	0.786 2 [0.659 7]	0.713 2 [0.560 2]	0.583 1 [0.368 3]	0.756 9 [0.606 3]
	12	0.512 9 [0.311 2]	0.463 7 [0.246 5]	0.562 3 [0.357 8]	0.472 8 [0.249 7]	0.445 4 [0.216 3]	0.532 8 [0.311 1]
(60,45)	30	0.433 2 [0.236 1]	0.401 8 [0.192 1]	0.472 1 [0.270 1]	0.407 1 [0.188 9]	0.382 3 [0.165 2]	0.419 0 [0.189 7]

$p = 0$		MLE			Bayes
$(n,m)$	$k$	$\lambda _0 $	$\lambda _1 $	$\mu $	$\lambda _0 $	$\lambda _1 $	$\mu $
	12	0.640 1 [0.456 4]	0.599 8 [0.388 3]	0.673 2 [0.497 3]	0.631 0 [0.454 7]	0.581 1 [0.364 9]	0.660 1 [0.488 1]
(30,20)	18	0.576 4 [0.369 1]	0.524 3 [0.316 2]	0.621 4 [0.445 2]	0.559 4 [0.343 7]	0.500 9 [0.276 7]	0.599 2 [0.384 2]
	12	0.447 1 [0.229 5]	0.431 9 [0.221 9]	0.498 2 [0.278 9]	0.405 2 [0.190 0]	0.413 8 [0.189 5]	0.473 6 [0.246 2]
(60,45)	30	0.380 9 [0.173 8]	0.374 2 [0.166 1]	0.413 8 [0.204 3]	0.312 3 [0.112 8]	0.345 2 [0.134 3]	0.389 5 [0.167 8]

$p = 0.15$		MLE			Bayes
$(n,m)$	$k$	$\lambda _0 $	$\lambda _1 $	$\mu $	$\lambda _0 $	$\lambda _1 $	$\mu $
	12	0.923 1 [1.652 0]	0.922 4 [1.623 5]	—	0.924 3 [1.595 5]	0.923 0 [1.460 3]	0.932 1 [1.647 4]
(30,20)	18	0.925 1 [1.445 1]	0.924 9 [1.437 3]	—	0.926 5 [1.461 0]	0.935 3 [1.393 2]	0.934 4 [1.554 0]
	12	0.935 8 [1.072 6]	0.933 6 [1.008 0]	—	0.937 9 [1.038 1]	0.934 7 [0.991 5]	0.939 0 [1.135 4]
(60,45)	30	0.939 6 [0.822 7]	0.937 8 [0.804 8]	—	0.940 1 [0.785 4]	0.938 5 [0.780 8]	0.941 3 [0.891 9]

$p = 0$		MLE			Bayes
$(n,m)$	$k$	$\lambda _0 $	$\lambda _1 $	$\mu $	$\lambda _0 $	$\lambda _1 $	$\mu $
	12	0.926 9 [1.267 1]	0.924 6 [1.202 7]	—	0.927 5 [1.280 7]	0.926 1 [1.248 2]	0.934 2 [1.386 8]
(30,20)	18	0.929 0 [1.192 5]	0.927 3 [1.043 4]	—	0.930 1 [1.116 5]	0.928 2 [1.086 2]	0.937 1 [1.112 9]
	12	0.937 2 [0.920 2]	0.934 1 [0.880 2]	—	0.938 4 [0.880 2]	0.936 0 [0.821 5]	0.941 3 [1.021 4]
(60,45)	30	0.940 7 [0.811 0]	0.938 5 [0.741 9]	—	0.941 3 [0.745 4]	0.939 2 [0.705 2]	0.944 0 [0.872 3]

$p = 0.15$		ACI			HPD
$(n,m)$	$k$	$\lambda _0 $	$\lambda _1 $	$\mu $	$\lambda _0 $	$\lambda _1 $	$\mu $
	12	0.924 8 [1.518 0]	0.924 5 [1.245 2]	—	0.926 0 [1.453 2]	0.925 8 [1.223 1]	0.934 2 [1.526 3]
(30,20)	18	0.928 0 [1.418 4]	0.926 7 [1.169 9]	—	0.929 5 [1.380 9]	0.928 1 [1.014 6]	0.937 0 [1.434 2]
	12	0.937 9 [1.001 7]	0.935 1 [0.884 0]	—	0.939 8 [0.904 8]	0.936 9 [0.862 6]	0.940 8 [1.033 1]
(60,45)	30	0.941 2 [0.798 5]	0.938 2 [0.788 2]	—	0.942 3 [0.702 5]	0.940 1 [0.716 9]	0.943 1 [0.798 5]

$p = 0$		ACI			HPD
$(n,m)$	$k$	$\lambda _0 $	$\lambda _1 $	$\mu $	$\lambda _0 $	$\lambda _1 $	$\mu $
	12	0.928 5 [1.200 5]	0.926 0 [1.134 7]	—	0.929 6 [1.196 1]	0.928 0 [1.015 2]	0.936 9 [1.166 3]
(30,20)	18	0.931 7 [1.013 1]	0.928 3 [0.945 5]	—	0.932 8 [0.965 9]	0.931 4 [0.886 2]	0.939 4 [1.017 9]
	12	0.939 0 [0.868 0]	0.937 4 [0.837 5]	—	0.940 3 [0.792 8]	0.939 9 [0.789 2]	0.942 8 [0.885 2]
(60,45)	30	0.941 3 [0.730 8]	0.940 1 [0.702 2]	—	0.943 7 [0.602 5]	0.942 7 [0.654 7]	0.947 1 [0.714 8]

Estimate	$\lambda _0 $	$\lambda _1 $	$\lambda _2 $	$\mu $
MLE	0.013 8	0.005 5	0.015 1	2
Bayes	0.014 2	0.005 4	0.016 0	1.727 1
ACI	(0.005 2,0.022 3) [0.017 1]	(0.000 1,0.010 9) [0.010 8]	(0.006 2,0.024 1) [0.017 9]	—
HPD	(0.009 1,0.025 7) [0.016 6]	(0.002 1,0.008 5) [0.006 4]	(0.010 8,0.025 9) [0.015 1]	(1.173 0,1.874 6) [0.701 6]

Inference for dependence competing risks from bivariate exponential model under generalized progressive hybrid censoring with partially observed failure causes

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