Statistical inference for dependence competing risks model under middle censoring

doi:10.21629/JSEE.2019.01.20

Journal of Systems Engineering and Electronics ›› 2019, Vol. 30 ›› Issue (1): 209-222.doi: 10.21629/JSEE.2019.01.20

• Reliability • Previous Articles

Statistical inference for dependence competing risks model under middle censoring

Yan WANG^1,²(), Yimin SHI^1,*(), Min WU³()

¹ Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an 710072, China
² School of Science, Xi'an Polytechnic University, Xi'an 710048, China
³ School of Economics & Management, Shanghai Maritime University, Shanghai 201306, China

Received:2017-08-08 Online:2019-02-27 Published:2019-02-27
Contact: Yimin SHI E-mail:wywzyf@126.com;ymshi@nwpu.edu.cn;wm6543@126.com
About author:WANG Yan was born in 1987. She received her B.S. degree in mathematics and applied mathematics from Henan institute of Science and Technology in 2009, her M.S. degree in applied mathematics from Xi'an Polytechnic University in 2012, and now she is a Ph.D. candidate at Northwestern Polytechnical University. Her research interests include reliability theory and application, and applied probability and statistics. E-mail:wywzyf@126.com|SHI Yimin was born in 1952. He is a professor at Northwestern Polytechnical University. His research interests include applied probability and statistics, reliability theory and application, financial mathematics and information fusion theory. E-mail:ymshi@nwpu.edu.cn|WU Min was born in 1986. She received her B.S. degree in statistics and M.S. degree in applied mathematics from Xi'an Polytechnic University in 2009 and 2012, respectively. And she received her Ph.D. degree in applied mathematics from Northwestern Polytechnical University in 2017. Her research interests include reliability theory and application, and applied probability and statistics. E-mail:wm6543@126.com
Supported by:
the National Natural Science Foundation of China(71571144);the National Natural Science Foundation of China(71401134);the Program of International Cooperation and Exchanges in Science and Technology Funded by Shaanxi Province(2016KW-033);This work was supported by the National Natural Science Foundation of China (71571144; 71401134), the Program of International Cooperation and Exchanges in Science and Technology Funded by Shaanxi Province (2016KW-033)

Abstract

Abstract:

Middle censoring is an important censoring scheme, in which the actual failure data of an observation becomes unobservable if it falls into a random interval. This paper considers the statistical analysis of the dependent competing risks model by using the Marshall-Olkin bivariate Weibull (MOBW) distribution. The maximum likelihood estimations (MLEs), midpoint approximation (MPA) estimations and approximate confidence intervals (ACIs) of the unknown parameters are obtained. In addition, the Bayes approach is also considered based on the Gamma-Dirichlet prior of the scale parameters, with the given shape parameter. The acceptance-rejection sampling method is used to obtain the Bayes estimations and construct credible intervals (CIs). Finally, two numerical examples are used to show the performance of the proposed methods.

Key words: middle censoring, dependent competing risks model, Marshall-Olkin bivariate Weibull (MOBW) distribution, acceptancerejection sampling

Yan WANG, Yimin SHI, Min WU. Statistical inference for dependence competing risks model under middle censoring[J]. Journal of Systems Engineering and Electronics, 2019, 30(1): 209-222.

Figures/Tables 9

Table 1

Table 2

Table 3

Table 4

MSE of estimations for parameters when $\mathit{\boldsymbol{\alpha = 1.539, \lambda _0 = 0.38, \lambda _1 = 0.369, \lambda _2 = 0.458}}$"

$n$	$m$	Estimator	$(\eta _1 , \eta _2) = (0.3, 0.5)$			$(\eta _1 , \eta _2 ) = (0.3, 1)$			$(\eta _1 , \eta _2) = (0.3, 1.5)$
$n$	$m$	Estimator	$\lambda _0 $	$\lambda _1 $	$\lambda _2 $	$\lambda _0 $	$\lambda _1 $	$\lambda _2 $	$\lambda _0 $	$\lambda _1 $	$\lambda _2 $
		MLE	0.517 1	0.502 1	0.514 4	0.527 7	0.524 4	0.533 2	0.551 5	0.561 9	0.573 3
	2	MPA	0.523 4	0.536 7	0.541 8	0.542 1	0.561 1	0.561 9	0.574 1	0.584 0	0.581 2
		Bayes	0.462 6	0.465 8	0.336 4	0.487 4	0.491 8	0.366 1	0.495 1	0.503 8	0.394 6
		MLE	0.552 9	0.531 1	0.563 9	0.578 9	0.566 9	0.588 4	0.605 8	0.597 4	0.606 6
30	4	MPA	0.562 5	0.550 4	0.570 3	0.591 4	0.576 8	0.597 7	0.615 2	0.612 7	0.622 9
		Bayes	0.471 6	0.497 8	0.403 1	0.508 2	0.509 2	0.446 1	0.519 1	0.516 9	0.454 4
		MLE	0.602 9	0.596 8	0.608 1	0.643 1	0.636 3	0.642 2	0.666 1	0.655 8	0.660 2
	6	MPA	0.658 7	0.677 1	0.674 6	0.687 2	0.704 6	0.700 3	0.701 0	0.712 4	0.714 2
		Bayes	0.577 4	0.513 9	0.462 6	0.607 3	0.527 9	0.557 4	0.619 4	0.542 4	0.563 0
		MLE	0.477 0	0.471 3	0.505 3	0.507 4	0.481 0	0.516 7	0.517 7	0.505 1	0.526 8
	2	MPA	0.502 6	0.494 6	0.510 3	0.518 9	0.500 6	0.525 3	0.523 4	0.525 6	0.534 2
		Bayes	0.420 8	0.417 8	0.316 5	0.434 9	0.431 6	0.338 2	0.436 8	0.458 2	0.360 6
		MLE	0.510 3	0.519 7	0.521 1	0.526 9	0.537 3	0.534 5	0.532 9	0.542 9	0.554 4
40	4	MPA	0.523 1	0.540 3	0.551 2	0.564 3	0.553 0	0.567 2	0.604 6	0.606 5	0.616 5
		Bayes	0.435 7	0.457 9	0.390 8	0.458 2	0.468 7	0.415 1	0.476 4	0.495 6	0.434 9
		MLE	0.523 6	0.521 0	0.559 8	0.551 7	0.541 5	0.572 7	0.591 4	0.582 9	0.594 6
	6	MPA	0.636 5	0.610 3	0.651 2	0.642 5	0.662 0	0.677 2	0.697 2	0.681 3	0.687 5
		Bayes	0.476 1	0.488 5	0.407 7	0.504 7	0.494 6	0.463 7	0.518 7	0.507 4	0.514 0
		MLE	0.427 6	0.455 4	0.458 2	0.442 9	0.464 3	0.478 4	0.469 3	0.477 3	0.483 8
	2	MPA	0.445 5	0.463 4	0.470 2	0.497 3	0.487 7	0.498 1	0.501 7	0.505 2	0.508 2
		Bayes	0.368 4	0.352 8	0.308 3	0.378 0	0.379 7	0.315 6	0.388 8	0.397 7	0.322 5
		MLE	0.434 7	0.469 6	0.465 1	0.454 6	0.488 8	0.483 9	0.464 2	0.494 1	0.499 2
50	4	MPA	0.489 6	0.479 3	0.501 3	0.505 3	0.494 5	0.509 3	0.519 6	0.522 7	0.529 1
		Bayes	0.385 1	0.380 1	0.325 7	0.399 1	0.393 3	0.358 3	0.409 1	0.404 9	0.415 3
		MLE	0.481 4	0.497 7	0.515 3	0.505 9	0.516 8	0.532 2	0.516 4	0.519 2	0.547 5
	6	MPA	0.507 4	0.501 5	0.530 5	0.531 5	0.536 9	0.543 9	0.552 3	0.559 2	0.559 7
		Bayes	0.409 9	0.410 5	0.351 1	0.427 3	0.444 5	0.374 0	0.456 3	0.461 9	0.386 7

Table 4

Table 5

MSE of estimations for parameters when $\mathit{\boldsymbol{\alpha = 1.539, \lambda _0 = 0.52, \lambda _1 = 0.369, \lambda _2 = 0.458}}$"

$n$	$m$	Estimator	$(\eta _1 , \eta _2) = (0.3, 0.5)$			$(\eta _1 , \eta _2 ) = (0.3, 1)$			$(\eta _1 , \eta _2) = (0.3, 1.5)$
$n$	$m$	Estimator	$\lambda _0 $	$\lambda _1 $	$\lambda _2 $	$\lambda _0 $	$\lambda _1 $	$\lambda _2 $	$\lambda _0 $	$\lambda _1 $	$\lambda _2 $
		MLE	0.412 6	0.420 6	0.426 7	0.434 6	0.438 7	0.455 6	0.461 8	0.451 3	0.478 7
	2	MPA	0.468 2	0.468 3	0.473 0	0.483 7	0.471 8	0.497 2	0.492 9	0.485 0	0.505 6
		Bayes	0.307 2	0.350 1	0.274 4	0.324 1	0.371 8	0.298 6	0.388 6	0.412 8	0.309 0
		MLE	0.428 2	0.462 2	0.476 8	0.469 2	0.480 8	0.489 6	0.474 4	0.495 7	0.499 6
30	4	MPA	0.494 7	0.506 6	0.524 6	0.512 6	0.520 9	0.567 1	0.532 3	0.589 8	0.585 3
		Bayes	0.312 5	0.409 3	0.288 6	0.320 6	0.423 6	0.308 3	0.334 6	0.435 8	0.310 2
		MLE	0.458 7	0.503 4	0.525 3	0.547 4	0.526 7	0.504 6	0.490 7	0.540 1	0.529 5
	6	MPA	0.539 9	0.524 6	0.551 1	0.560 8	0.538 9	0.589 2	0.576 8	0.559 7	0.592 7
		Bayes	0.325 2	0.416 4	0.302 1	0.346 8	0.438 2	0.320 4	0.373 6	0.458 6	0.340 6
		MLE	0.401 9	0.387 1	0.356 5	0.403 1	0.390 9	0.412 8	0.434 7	0.402 8	0.431 1
	2	MPA	0.427 7	0.414 2	0.416 0	0.437 5	0.438 4	0.434 4	0.467 8	0.441 0	0.458 6
		Bayes	0.222 1	0.291 9	0.215 5	0.254 1	0.334 9	0.249 8	0.264 7	0.355 8	0.261 1
		MLE	0.428 9	0.392 5	0.403 7	0.435 9	0.414 8	0.427 4	0.463 4	0.427 8	0.458 9
40	4	MPA	0.447 6	0.456 8	0.445 7	0.469 9	0.477 5	0.480 9	0.483 4	0.501 6	0.510 6
		Bayes	0.230 1	0.331 1	0.239 8	0.270 3	0.341 6	0.265 8	0.306 7	0.351 3	0.291 5
		MLE	0.447 0	0.417 0	0.438 8	0.456 5	0.437 1	0.447 2	0.487 4	0.442 5	0.474 9
	6	MPA	0.453 2	0.476 7	0.467 9	0.475 5	0.494 4	0.488 2	0.496 9	0.515 9	0.520 1
		Bayes	0.302 1	0.378 7	0.296 2	0.293 1	0.381 8	0.299 8	0.323 0	0.397 9	0.303 9
		MLE	0.370 1	0.333 7	0.302 2	0.382 1	0.349 6	0.310 9	0.393 1	0.367 5	0.321 0
	2	MPA	0.416 0	0.382 8	0.382 2	0.424 8	0.383 9	0.402 4	0.430 6	0.427 8	0.410 3
		Bayes	0.152 0	0.249 7	0.190 3	0.163 1	0.272 9	0.202 9	0.177 1	0.283 1	0.223 1
		MLE	0.383 9	0.357 7	0.323 2	0.405 1	0.375 6	0.333 2	0.417 9	0.390 7	0.348 5
50	4	MPA	0.424 9	0.414 5	0.418 7	0.433 9	0.426 0	0.430 2	0.455 8	0.443 7	0.453 8
		Bayes	0.163 4	0.301 9	0.236 9	0.173 6	0.311 0	0.259 8	0.177 4	0.311 5	0.266 1
		MLE	0.391 3	0.360 7	0.343 2	0.412 0	0.398 4	0.349 4	0.423 3	0.402 1	0.352 6
	6	MPA	0.431 9	0.424 7	0.428 0	0.456 8	0.444 1	0.463 2	0.462 4	0.502 5	0.508 4
		Bayes	0.177 3	0.332 1	0.259 3	0.180 3	0.342 0	0.275 7	0.183 1	0.358 2	0.278 5

Table 5

Table 6

MSE of estimations for parameters when $\mathit{\boldsymbol{\alpha = 1.539, \lambda _0 = 0.75, \lambda _1 = 0.369, \lambda _2 = 0.458}}$"

$n$	$m$	Estimator	$(\eta _1 , \eta _2) = (0.3, 0.5)$			$(\eta _1 , \eta _2 ) = (0.3, 1)$			$(\eta _1 , \eta _2) = (0.3, 1.5)$
$n$	$m$	Estimator	$\lambda _0 $	$\lambda _1 $	$\lambda _2 $	$\lambda _0 $	$\lambda _1 $	$\lambda _2 $	$\lambda _0 $	$\lambda _1 $	$\lambda _2 $
		MLE	0.273 7	0.289 7	0.266 6	0.323 5	0.310 2	0.270 2	0.339 9	0.323 5	0.308 7
	2	MPA	0.293 0	0.299 7	0.278 5	0.334 5	0.332 6	0.280 9	0.356 4	0.378 0	0.282 8
		Bayes	0.030 8	0.217 2	0.165 3	0.032 6	0.223 3	0.166 6	0.032 7	0.227 8	0.167 1
		MLE	0.316 9	0.318 1	0.277 3	0.324 2	0.329 6	0.290 1	0.379 4	0.336 6	0.318 4
30	4	MPA	0.319 3	0.331 3	0.336 2	0.347 3	0.345 2	0.341 6	0.364 0	0.396 9	0.341 6
		Bayes	0.034 1	0.226 9	0.192 3	0.063 7	0.235 4	0.202 2	0.086 9	0.246 9	0.205 3
		MLE	0.327 9	0.336 8	0.284 5	0.346 7	0.370 4	0.300 3	0.374 6	0.342 2	0.322 8
	6	MPA	0.321 6	0.356 7	0.359 6	0.350 9	0.365 0	0.360 9	0.378 6	0.376 5	0.368 9
		Bayes	0.076 4	0.350 2	0.293 9	0.081 0	0.362 4	0.295 3	0.103 3	0.373 1	0.303 1
		MLE	0.133 8	0.207 6	0.193 2	0.138 6	0.242 8	0.201 0	0.149 6	0.243 1	0.210 4
	2	MPA	0.138 2	0.258 0	0.258 4	0.166 0	0.260 5	0.271 4	0.162 4	0.274 3	0.293 1
		Bayes	0.021 5	0.152 6	0.126 9	0.024 1	0.173 5	0.130 3	0.025 4	0.174 6	0.147 8
		MLE	0.148 4	0.230 5	0.202 2	0.159 1	0.254 9	0.205 6	0.165 1	0.258 3	0.220 1
40	4	MPA	0.154 6	0.267 9	0.274 1	0.170 1	0.283 4	0.298 1	0.187 4	0.319 4	0.320 9
		Bayes	0.031 0	0.207 8	0.162 3	0.041 6	0.249 9	0.176 9	0.091 6	0.251 0	0.181 5
		MLE	0.151 2	0.242 6	0.220 4	0.164 6	0.267 5	0.220 8	0.174 6	0.279 8	0.224 6
	6	MPA	0.169 7	0.280 1	0.277 6	0.170 4	0.293 3	0.304 2	0.191 4	0.329 2	0.322 8
		Bayes	0.054 0	0.252 3	0.203 9	0.078 3	0.258 6	0.205 2	0.099 4	0.269 7	0.209 1
		MLE	0.098 1	0.153 9	0.130 9	0.102 1	0.175 2	0.159 3	0.107 8	0.186 5	0.175 4
	2	MPA	0.124 6	0.166 0	0.228 2	0.132 9	0.178 7	0.233 5	0.143 2	0.194 5	0.241 0
		Bayes	0.017 4	0.126 0	0.109 0	0.017 5	0.136 3	0.113 8	0.018 1	0.156 8	0.116 9
		MLE	0.106 4	0.183 4	0.157 9	0.127 6	0.194 5	0.160 9	0.137 7	0.215 0	0.182 4
50	4	MPA	0.134 6	0.243 9	0.250 1	0.054 3	0.255 8	0.261 1	0.054 4	0.269 0	0.279 0
		Bayes	0.018 0	0.172 1	0.154 6	0.018 1	0.175 3	0.140 8	0.018 6	0.178 1	0.175 8
		MLE	0.127 8	0.200 8	0.177 1	0.139 6	0.225 1	0.190 6	0.140 6	0.225 7	0.233 0
	6	MPA	0.147 1	0.252 4	0.262 5	0.158 2	0.283 8	0.283 7	0.178 3	0.395 6	0.295 4
		Bayes	0.022 7	0.205 2	0.176 0	0.023 3	0.207 3	0.176 4	0.024 1	0.207 5	0.175 3

Table 6

Table 7

Coverage percentages (%) of parameters when $\mathit{\boldsymbol{\alpha = 1.539, \lambda _0 = 0.38, \lambda _1 = 0.369, \lambda _2 = 0.458}}$"

$n$	$m$	Estimator	$(\eta _1 , \eta _2) = (0.3, 0.5)$			$(\eta _1 , \eta _2 ) = (0.3, 1)$			$(\eta _1 , \eta _2) = (0.3, 1.5)$
$n$	$m$	Estimator	$\lambda _0 $	$\lambda _1 $	$\lambda _2 $	$\lambda _0 $	$\lambda _1 $	$\lambda _2 $	$\lambda _0 $	$\lambda _1 $	$\lambda _2 $
30	2	ACI	0.605 0	0.609 0	0.675 0	0.585 0	0.600 0	0.628 0	0.577 0	0.594 0	0.626 0
		AL	2.222 5	2.334 0	2.216 2	2.456 6	2.519 9	2.758 8	2.881 7	3.120 6	2.975 3
		HPD	0.679 0	0.716 0	0.854 0	0.664 0	0.711 0	0.850 0	0.640 0	0.709 0	0.842 0
		HL	1.914 5	1.972 6	1.960 3	2.197 1	2.248 7	2.304 5	2.462 3	2.514 5	2.631 5
	4	ACI	0.590 0	0.594 0	0.660 0	0.581 0	0.590 0	0.623 0	0.573 0	0.585 0	0.614 0
		AL	2.647 4	2.597 7	2.495 9	2.873 3	2.797 8	2.971 3	3.149 3	3.329 9	3.527 8
		HPD	0.663 0	0.705 0	0.851 0	0.651 0	0.698 0	0.849 0	0.649 0	0.690 0	0.840 0
		HL	2.462 1	2.339 1	2.361 3	2.784 0	2.895 9	2.876 1	2.978 0	3.055 1	2.996 8
	6	ACI	0.581 0	0.588 0	0.639 0	0.578 0	0.584 0	0.621 0	0.561 0	0.579 0	0.610 0
		AL	2.848 6	3.132 1	3.151 9	2.943 1	3.751 6	4.200 6	3.547 7	4.121 7	4.798 9
		HPD	0.649 0	0.698 0	0.841 0	0.629 0	0.668 0	0.834 0	0.627 0	0.659 0	0.829 0
		HL	2.693 7	2.616 9	2.501 9	2.810 3	2.959 1	2.991 1	3.187 9	3.392 2	3.534 2
40	2	ACI	0.672 0	0.684 0	0.729 0	0.670 0	0.681 0	0.726 0	0.667 0	0.671 0	0.721 0
		AL	1.917 3	2.004 1	2.015 3	2.033 3	2.275 3	2.235 5	2.270 8	2.413 5	2.620 2
		HPD	0.772 0	0.771 0	0.857 0	0.763 0	0.768 0	0.847 0	0.757 0	0.764 0	0.841 0
		HL	1.702 8	1.719 2	1.949 1	1.912 3	1.916 6	2.164 5	2.151 2	2.100 4	2.344 4
	4	ACI	0.666 0	0.679 0	0.719 0	0.656 0	0.675 0	0.714 0	0.647 0	0.672 0	0.708 0
		AL	2.124 6	2.261 9	2.250 1	2.476 3	2.699 6	2.880 6	2.916 7	2.976 7	3.160 9
		HPD	0.769 0	0.765 0	0.848 0	0.757 0	0.758 0	0.844 0	0.754 0	0.752 0	0.838 0
		HL	1.964 4	2.058 2	2.189 2	2.156 0	2.336 5	2.453 3	2.395 8	2.530 8	2.750 4
	6	ACI	0.656 0	0.672 0	0.684 0	0.651 0	0.670 0	0.683 0	0.648 0	0.667 0	0.672 0
		AL	2.450 1	2.940 3	2.919 5	2.896 4	3.545 5	3.553 3	3.339 5	3.730 6	4.184 7
		HPD	0.766 0	0.761 0	0.843 0	0.754 0	0.752 0	0.840 0	0.750 0	0.739 0	0.835 0
		HL	2.115 5	2.342 6	2.459 5	2.319 5	2.590 1	2.682 6	2.816 3	3.013 0	3.161 0
50	2	ACI	0.691 0	0.731 0	0.772 0	0.686 0	0.728 0	0.768 0	0.683 0	0.725 0	0.766 0
		AL	1.654 2	1.797 7	1.677 4	1.912 1	2.055 7	2.015 0	2.120 2	2.292 8	2.311 6
		HPD	0.842 0	0.866 0	0.915 0	0.837 0	0.862 0	0.909 0	0.835 0	0.854 0	0.903 0
		HL	1.611 1	1.549 1	1.603 0	1.692 2	1.697 6	1.732 3	1.751 6	1.894 2	1.900 4
	4	ACI	0.674 0	0.729 0	0.773 0	0.673 0	0.727 0	0.764 0	0.669 0	0.724 0	0.763 0
		AL	1.865 5	1.984 4	1.891 0	2.009 8	2.223 6	2.498 9	2.313 7	2.861 7	3.021 4
		HPD	0.819 0	0.853 0	0.910 0	0.815 0	0.849 0	0.905 0	0.811 0	0.828 0	0.894 0
		HL	1.637 8	1.599 0	1.680 3	1.763 1	1.714 8	1.817 4	1.839 3	1.913 8	1.997 9
	6	ACI	0.655 0	0.729 0	0.771 0	0.647 0	0.723 0	0.762 0	0.621 0	0.724 0	0.759 0
		AL	1.901 2	2.091 2	2.146 8	2.223 6	2.539 4	2.588 9	2.993 2	3.674 2	3.282 1
		HPD	0.810 0	0.809 0	0.875 0	0.797 0	0.794 0	0.867 0	0.784 0	0.791 0	0.864 0
		HL	1.735 5	1.620 1	1.731 4	1.811 1	1.998 2	1.981 2	1.922 6	2.051 6	2.196 2

Table 7

Table 8

Coverage percentages (%) of parameters when $\mathit{\boldsymbol{\alpha = 1.539, \lambda _0 = 0.52, \lambda _1 = 0.369, \lambda _2 = 0.458}}$"

$n$	$m$	Estimator	$(\eta _1 , \eta _2) = (0.3, 0.5)$			$(\eta _1 , \eta _2 ) = (0.3, 1)$			$(\eta _1 , \eta _2) = (0.3, 1.5)$
$n$	$m$	Estimator	$\lambda _0 $	$\lambda _1 $	$\lambda _2 $	$\lambda _0 $	$\lambda _1 $	$\lambda _2 $	$\lambda _0 $	$\lambda _1 $	$\lambda _2 $
30	2	ACI	0.795 0	0.845 0	0.835 0	0.785 0	0.835 0	0.828 0	0.784 0	0.829 0	0.824 0
		AL	1.832 5	1.856 1	1.905 3	1.909 4	2.014 6	2.035 9	2.013 5	2.114 3	2.215 2
		HPD	0.845 0	0.859 0	0.865 0	0.833 0	0.851 0	0.853 0	0.821 0	0.842 0	0.845 0
		HL	1.704 4	1.795 9	1.707 5	1.802 1	1.965 7	1.878 8	1.973 4	1.980 4	1.946 0
	4	ACI	0.781 0	0.831 0	0.828 0	0.776 0	0.823 0	0.819 0	0.770 0	0.812 0	0.813 0
		AL	1.994 9	2.050 8	2.163 5	2.175 8	2.349 4	2.457 5	2.152 3	2.527 1	2.649 0
		HPD	0.826 0	0.833 0	0.837 0	0.822 0	0.826 0	0.835 0	0.818 0	0.830 0	0.825 0
		HL	1.902 6	1.853 0	1.857 5	2.083 4	2.036 6	2.147 6	2.156 0	2.281 4	2.336 1
	6	ACI	0.772 0	0.811 0	0.820 0	0.761 0	0.805 0	0.806 0	0.754 0	0.794 0	0.795 0
		AL	2.333 0	2.413 4	2.661 7	2.411 7	2.510 4	2.882 1	2.537 4	2.796 5	2.951 8
		HPD	0.821 0	0.822 0	0.826 0	0.814 0	0.819 0	0.816 0	0.813 0	0.816 0	0.813 0
		HL	2.117 4	2.392 5	2.357 3	2.398 0	2.471 7	2.496 0	2.589 1	2.610 0	2.890 5
40	2	ACI	0.849 0	0.858 0	0.908 0	0.835 0	0.844 0	0.897 0	0.834 0	0.835 0	0.893 0
		AL	1.733 5	1.871 7	1.811 9	1.954 6	1.951 4	1.925 9	2.053 8	2.107 7	2.138 8
		HPD	0.872 0	0.867 0	0.913 0	0.869 0	0.861 0	0.910 0	0.863 0	0.856 0	0.904 0
		HL	1.597 6	1.647 5	1.660 4	1.641 0	1.791 8	1.844 5	1.726 7	1.926 5	1.943 1
	4	ACI	0.831 0	0.832 0	0.879 0	0.825 0	0.823 0	0.868 0	0.819 0	0.812 0	0.860 0
		AL	1.940 5	1.909 9	2.144 8	2.124 0	2.062 7	2.312 8	2.202 0	2.591 3	2.203'6
		HPD	0.864 0	0.856 0	0.904 0	0.856 0	0.851 0	0.898 0	0.853 0	0.848 0	0.890 0
		HL	1.947 8	2.053 1	1.923 7	2.049 5	2.150 5	2.151 5	2.135 7	2.294 8	2.205 6
	6	ACI	0.823 0	0.828 0	0.865 0	0.819 0	0.819 0	0.839 0	0.815 0	0.804 0	0.832 0
		AL	2.191 6	2.389 2	2.358 0	2.478 4	2.492 4	2.492 4	2.574 5	2.627 7	2.891 3
		HPD	0.863 0	0.846 0	0.865 0	0.859 0	0.841 0	0.882 0	0.852 0	0.838 0	0.876 0
		HL	2.136 6	2.253 0	2.335 2	2.301 2	2.392 7	2.421 1	2.462 2	2.574 0	2.766 3
50	2	ACI	0.881 0	0.864 0	0.915 0	0.873 0	0.852 0	0.904 0	0.853 0	0.851 0	0.900 0
		AL	1.752 0	1.750 9	1.725 3	1.831 8	1.877 7	1.915 4	1.932 3	1.928 5	2.045 5
		HPD	0.920 0	0.885 0	0.931 0	0.918 0	0.882 0	0.929 0	0.917 0	0.872 0	0.926 0
		HL	1.583 6	1.507 7	1.589 9	1.628 1	1.618 2	1.764 8	1.810 5	1.811 2	1.903 7
	4	ACI	0.865 0	0.854 0	0.893 0	0.857 0	0.841 0	0.880 0	0.846 0	0.847 0	0.873 0
		AL	1.932 1	1.805 7	1.955 6	1.919 1	2.015 0	2.052 5	2.121 6	2.328 5	2.149 5
		HPD	0.916 0	0.866 0	0.920 0	0.913 0	0.865 0	0.918 0	0.911 0	0.864 0	0.917 0
		HL	1.585 1	1.611 8	1.612 9	1.642 5	1.767 0	1.808 0	1.819 7	1.855 2	1.929 5
	6	ACI	0.846 0	0.842 0	0.876 0	0.846 0	0.824 0	0.871 0	0.834 0	0.811 0	0.868 0
		AL	2.101 9	2.116 1	2.130 3	2.286 5	2.377 7	2.725 4	2.535 3	2.333 6	2.289 2
		HPD	0.914 0	0.856 0	0.922 0	0.911 0	0.852 0	0.915 0	0.905 0	0.857 0	0.909 0
		HL	1.616 9	1.645 9	1.624 7	1.713 1	1.806 8	1.876 6	1.922 5	1.989 6	1.931 4

Table 8

Table 9

Coverage percentages (%) of parameters when $\mathit{\boldsymbol{\alpha = 1.539, \lambda _0 = 0.75, \lambda _1 = 0.369, \lambda _2 = 0.458}}$"

$n$	$m$	Estimator	$(\eta _1 , \eta _2) = (0.3, 0.5)$			$(\eta _1 , \eta _2 ) = (0.3, 1)$			$(\eta _1 , \eta _2) = (0.3, 1.5)$
$n$	$m$	Estimator	$\lambda _0 $	$\lambda _1 $	$\lambda _2 $	$\lambda _0 $	$\lambda _1 $	$\lambda _2 $	$\lambda _0 $	$\lambda _1 $	$\lambda _2 $
30	2	ACI	0.932 0	0.868 0	0.933 0	0.929 0	0.865 0	0.931 0	0.926 0	0.862 0	0.928 0
		AL	1.841 7	1.851 5	1.855 3	1.912 7	1.932 3	2.023 9	2.041 7	2.178 6	2.145 8
		HPD	0.930 0	0.925 0	0.948 0	0.926 0	0.920 0	0.941 0	0.923 0	0.915 0	0.936 0
		HL	1.647 7	1.731 3	1.784 2	1.817 3	1.933 8	1.895 7	1.912 4	2.034 1	2.124 1
	4	ACI	0.923 0	0.861 0	0.913 0	0.917 0	0.857 0	0.911 0	0.909 0	0.847 0	0.908 0
		AL	1.811 7	1.933 5	1.996 8	1.955 8	2.053 5	2.157 3	2.131 7	2.103 5	2.333 0
		HPD	0.925 0	0.901 0	0.936 0	0.919 0	0.899 0	0.934 0	0.918 0	0.896 0	0.925 0
		HL	1.768 4	1.956 9	1.849 1	1.873 3	2.094 5	1.947 8	2.132 2	2.165 4	2.184 5
	6	ACI	0.905 0	0.854 0	0.905 0	0.899 0	0.841 0	0.894 0	0.891 0	0.836 0	0.891 0
		AL	1.963 5	2.065 9	2.121 2	2.049 1	2.187 8	2.325 4	2.324 1	2.262 7	2.407 5
		HPD	0.921 0	0.890 0	0.925 0	0.913 0	0.889 0	0.923 0	0.912 0	0.887 0	0.918 0
		HL	1.825 4	2.088 3	1.937 9	1.978 9	2.297 1	2.161 2	2.284 5	2.419 8	2.392 5
40	2	ACI	0.949 0	0.891 0	0.938 0	0.938 0	0.882 0	0.930 0	0.933 0	0.868 0	0.925 0
		AL	1.625 5	1.638 3	1.532 3	1.725 6	1.812 7	1.759 8	1.921 7	1.923 5	2.025 3
		HPD	0.951 0	0.931 0	0.951 0	0.949 0	0.927 0	0.945 0	0.945 0	0.923 0	0.940 0
		HL	1.617 8	1.603 1	1.573 7	1.706 1	1.742 3	1.640 7	1.713 7	1.799 3	1.736 2
	4	ACI	0.935 0	0.875 0	0.920 0	0.930 0	0.871 0	0.918 0	0.925 0	0.859 0	0.916 0
		AL	1.655 6	1.784 2	1.595 6	1.796 4	1.999 4	1.849 3	2.076 9	2.149 8	2.178 5
		HPD	0.945 0	0.915 0	0.939 0	0.941 0	0.911 0	0.934 0	0.937 0	0.907 0	0.930 0
		HL	1.622 7	1.625 9	1.586 0	1.712 2	1.746 5	1.696 0	1.758 4	1.821 3	1.788 4
	6	ACI	0.933 0	0.866 0	0.910 0	0.928 0	0.859 0	0.906 0	0.927 0	0.854 0	0.900 0
		AL	1.750 9	1.954 7	1.690 4	1.804 8	2.047 6	1.907 9	2.172 8	2.198 6	2.275 9
		HPD	0.940 0	0.913 0	0.929 0	0.937 0	0.905 0	0.927 0	0.934 0	0.900 0	0.920 0
		HL	1.690 4	1.712 4	1.642 1	1.736 1	1.755 1	1.746 6	1.838 3	1.875 0	1.814 2
50	2	ACI	0.952 0	0.901 0	0.943 0	0.947 0	0.891 0	0.941 0	0.944 0	0.887 0	0.939 0
		AL	1.584 6	1.639 0	1.576 6	1.598 2	1.604 7	1.867 4	1.714 5	1.904 7	1.916 6
		HPD	0.974 0	0.937 0	0.977 0	0.962 0	0.928 0	0.974 0	0.961 0	0.919 0	0.965 0
		HL	1.297 7	1.331 3	1.184 2	1.317 3	1.333 8	1.275 7	1.404 3	1.445 2	1.326 5
	4	ACI	0.944 0	0.890 0	0.923 0	0.941 0	0.882 0	0.921 0	0.938 0	0.877 0	0.921 0
		AL	1.667 2	1.706 0	1.649 5	1.684 1	1.852 6	1.560 9	1.748 6	1.950 1	1.923 5
		HPD	0.958 0	0.929 0	0.971 0	0.956 0	0.917 0	0.969 0	0.953 0	0.920 0	0.965 0
		HL	1.368 4	1.556 9	1.249 1	1.449 6	1.637 8	1.356 3	1.504 3	1.654 1	1.615 7
	6	ACI	0.940 0	0.881 0	0.913 0	0.936 0	0.873 0	0.910 0	0.931 0	0.868 0	0.907 0
		AL	1.692 9	1.871 7	1.683 9	1.702 7	1.972 7	1.797 1	1.823 2	1.982 4	1.942 1
		HPD	0.951 0	0.915 0	0.960 0	0.949 0	0.913 0	0.949 0	0.947 0	0.904 0	0.946 0
		HL	1.501 8	1.609 0	1.446 9	1.503 7	1.638 8	1.553 1	1.621 9	1.719 9	1.627 0

Table 9

References 22

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Estimation	$\lambda _0 $	$\lambda _1 $	$\lambda _2 $
MLE	0.105 2 (0.056 2, 1.606 6)	0.306 3 (0.113 5, 1.679 1)	0.303 9 (0.143 8, 1.824 1)
MPA	0.135 1	0.307 0	0.317 7
Bayes	0.097 9 (0.037 6, 1.484 3)	0.276 4 (0.142 9, 1.578 1)	0.276 1 (0.141 6, 1.786 9)

Statistical inference for dependence competing risks model under middle censoring

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$i$	$T$	$c$
1	266	1
2	91	2
3	154	2
4	285	0
5	583	1
6	547	2
7	79	1
8	622	0
9	707	2
10	469	2
11	93	1
12	1 313	2
13	805	1
14	344	1
15	790	2
16	125	2
17	777	2
18	306	1
19	415	1
20	307	2
21	637	2
22	577	2
23	178	1
24	517	2
25	272	0
26	1 137	0
27	1 484	1
28	315	1
29	287	2
30	1 252	1
31	717	2
32	642	1
33	141	2
34	407	1
35	356	1
36	1 653	0
37	427	2
38	699	1
39	36	2
40	667	1
41	588	2
42	471	0
43	126	1
44	350	2
45	350	1
46	663	0
47	567	2
48	966	0
49	203	0
50	84	1
51	392	1
52	1 140	2
53	901	1
54	1 247	0
55	448	2
56	904	2
57	276	1
58	520	1
59	485	2
60	248	2
61	503	1
62	423	2
63	285	2
64	315	2
65	727	2
66	210	2
67	409	2
68	584	1
69	355	1
70	1 302	1
71	227	2

Case	Data
	79	(81.771 9 194.242 9)			93	126	178	266	276	306	315	344	350
Case1	355	356	392	407	415	503	520	583	584	642	667	699	805
	(218.235 7 963.491 2)			1252	1 302	1 484
	36	91	125	141	154	210	227	248	285	287	307	315
Case2	(171.692 7 364.090 0)			409	423	427	448	469	485	517	547	567	577
	588	637	707	717	727	777	(193.762 1 823.838 3)			904	1 140	1 313
Case0	203	272	285	471	622	663	966	1 137	1 247	1 653