A statistical inference for generalized Rayleigh model under Type-Ⅱ progressive censoring with binomial removals

doi:10.21629/JSEE.2020.01.20

Abstract

Abstract:

This paper considers the parameters and reliability characteristics estimation problem of the generalized Rayleigh distribution under progressively Type-Ⅱ censoring with random removals, that is, the number of units removed at each failure time follows the binomial distribution. The maximum likelihood estimation and the Bayesian estimation are derived. In the meanwhile, through a great quantity of Monte Carlo simulation experiments we have studied different hyperparameters as well as symmetric and asymmetric loss functions in the Bayesian estimation procedure. A real industrial case is presented to justify and illustrate the proposed methods. We also investigate the expected experimentation time and discuss the influence of the parameters on the termination point to complete the censoring test.

Key words: Type-Ⅱ progressive censoring with random removals, generalized Rayleigh distribution, reliability characteristic, maximum likelihood estimation, Markov chain Monte Carlo method, expected experimentation time

Junru REN, Wenhao GUI. A statistical inference for generalized Rayleigh model under Type-Ⅱ progressive censoring with binomial removals[J]. Journal of Systems Engineering and Electronics, 2020, 31(1): 206-223.

Figures/Tables 22

Table 1

Fig 1

Fig 2

Table 2

Table 3

Table 4

Table 5

Bayesian estimation of $\alpha $ for different hyperparameters and different LFs"

var	$n$	$m$	SELF			GELF
			$\text{AB}$	$\text{MSE }$	$\text{PR}$	$q=0.5$			$q=-0.5$
			$\text{AB}$	$\text{MSE }$	$\text{PR}$	$\text{AB}$	$\text{MSE }$	$\text{PR}$	$\text{AB}$	$\text{MSE }$	$\text{PR}$
		5	0.127 2	0.105 2	0.082 1	0.061 9	0.132 7	0.010 4	0.125 0	0.061 4	0.010 1
	10	7	0.055 8	0.097 2	0.078 9	-0.012 4	0.107 9	0.010 0	0.000 6	0.086 5	0.009 6
		9	0.148 7	0.166 7	0.078 8	-0.064 1	0.101 4	0.009 9	0.022 0	0.097 1	0.009 4
		8	0.061 8	0.061 2	0.069 9	0.115 9	0.080 2	0.008 6	0.100 8	0.101 9	0.008 3
	16	10	0.267 4	0.204 7	0.068 0	0.067 2	0.053 1	0.008 4	0.141 4	0.181 9	0.008 5
		14	0.016 7	0.143 7	0.065 8	0.119 0	0.179 2	0.008 3	0.197 2	0.121 8	0.007 9
		10	0.134 9	0.153 3	0.065 5	0.045 4	0.076 6	0.008 3	0.362 8	0.297 7	0.008 2
0.5	20	15	0.258 9	0.125 5	0.061 3	0.065 2	0.100 9	0.007 4	-0.209 4	0.112 5	0.007 3
		18	0.093 8	0.110 3	0.060 9	0.253 7	0.192 6	0.007 4	0.112 3	0.090 1	0.007 2
		10	0.243 0	0.200 2	0.059 7	-0.079 1	0.062 8	0.007 7	0.197 7	0.117 4	0.007 4
	25	15	0.142 6	0.171 9	0.057 5	0.146 6	0.126 7	0.007 2	0.043 9	0.119 0	0.007 1
		20	-0.019 1	0.081 0	0.053 1	0.109 3	0.113 5	0.006 8	0.193 4	0.135 7	0.006 8
		15	0.045 0	0.166 8	0.053 0	-0.051 4	0.163 3	0.006 9	0.268 2	0.135 3	0.006 7
	30	20	0.016 2	0.088 6	0.048 4	0.163 6	0.169 5	0.006 3	0.247 3	0.148 8	0.006 4
		25	0.481 9	0.316 9	0.050 9	0.110 1	0.136 4	0.005 9	0.146 0	0.080 4	0.005 9
		5	0.043 9	0.097 2	0.126 3	0.356 3	0.260 1	0.016 0	0.235 0	0.232 4	0.015 0
	10	7	-0.097 9	0.084 7	0.116 1	0.151 2	0.213 7	0.015 5	0.159 7	0.188 5	0.014 7
		9	0.045 3	0.299 2	0.114 2	0.200 8	0.261 4	0.014 5	0.108 3	0.351 3	0.014 0
		8	0.221 9	0.377 4	0.097 5	0.164 2	0.230 3	0.012 4	0.265 1	0.136 0	0.012 8
	16	10	0.101 5	0.111 1	0.095 4	0.184 4	0.190 4	0.012 3	0.158 3	0.198 8	0.011 8
		14	0.366 0	0.497 4	0.088 5	-0.047 1	0.148 4	0.010 7	0.056 2	0.314 4	0.011 1
		10	0.079 9	0.263 2	0.088 6	0.161 7	0.271 4	0.011 0	0.131 2	0.308 1	0.010 7
1	20	15	0.111 8	0.105 3	0.082 1	0.406 6	0.376 4	0.011 2	0.385 2	0.428 1	0.010 2
		18	0.111 5	0.209 3	0.074 9	-0.200 5	0.099 9	0.009 1	0.144 9	0.249 5	0.009 4
		10	0.136 5	0.195 1	0.081 3	0.268 9	0.201 7	0.010 2	0.083 5	0.204 0	0.010 1
	25	15	0.107 9	0.068 5	0.072 8	0.001 2	0.122 0	0.009 4	0.238 5	0.242 2	0.009 0
		20	0.117 9	0.160 5	0.070 0	-0.047 4	0.145 6	0.008 5	0.274 7	0.248 5	0.008 2
		15	0.154 2	0.192 2	0.070 1	0.163 7	0.324 7	0.008 6	0.144 4	0.260 1	0.008 5
	30	20	0.245 4	0.250 9	0.066 3	0.262 3	0.320 4	0.008 2	-0.005 1	0.209 9	0.007 8
		25	0.168 1	0.136 5	0.058 9	0.273 2	0.204 2	0.007 6	0.077 1	0.125 8	0.007 2
		5	0.824 1	1.553 7	0.225 9	0.496 1	1.362 7	0.026 0	0.942 5	1.413 9	0.026 8
	10	7	0.542 7	1.130 9	0.197 2	0.561 2	0.679 5	0.025 2	1.277 5	3.009 3	0.025 7
		9	0.086 3	0.644 4	0.171 1	0.452 6	0.508 9	0.023 5	0.677 4	2.119 2	0.023 7
		8	0.086 1	0.274 8	0.144 2	0.410 9	1.022 1	0.017 8	1.071 9	1.952 5	0.020 0
	16	10	0.581 7	1.259 1	0.139 7	1.158 4	2.892 1	0.019 5	1.064 4	2.583 8	0.018 1
		14	0.438 0	0.636 9	0.127 4	0.067 6	0.307 8	0.015 4	0.339 3	0.403 7	0.015 7
		10	0.389 7	0.819 1	0.122 7	0.414 6	0.839 1	0.015 6	1.369 0	3.252 6	0.017 2
4	20	15	0.509 8	1.055 4	0.114 9	0.651 6	1.493 7	0.014 1	0.500 3	1.260 5	0.014 1
		18	0.416 3	0.517 7	0.104 4	0.329 5	0.705 8	0.013 5	0.552 0	0.691 9	0.013 1
		10	0.095 4	0.312 4	0.112 8	-0.013 2	0.345 2	0.014 2	0.603 2	1.002 2	0.015 3
	25	15	0.023 8	0.165 9	0.092 9	0.047 9	0.405 5	0.011 8	-0.069 9	0.089 3	0.011 6
		20	0.123 0	0.414 2	0.086 2	0.036 8	0.203 9	0.011 1	0.518 9	0.637 7	0.011 2
		15	0.598 7	1.113 4	0.091 6	0.109 8	0.426 6	0.011 1	0.130 7	0.622 4	0.010 6
	30	20	0.451 7	0.784 0	0.084 4	0.673 8	2.460 7	0.010 5	0.575 9	0.864 8	0.010 4
		25	-0.053 8	0.156 7	0.070 2	0.552 6	0.572 0	0.009 4	0.547 8	0.765 6	0.009 8

Table 5

Table 6

Bayesian estimation of $\lambda $ for different hyperparameters and different LFs"

var	$n$	$m$	SELF			GELF
			$\text{AB}$	$\text{MSE }$	$\text{PR}$	$q=0.5$			$q=-0.5$
			$\text{AB}$	$\text{MSE }$	$\text{PR}$	$\text{AB}$	$\text{MSE }$	$\text{PR}$	$\text{AB}$	$\text{MSE }$	$\text{PR}$
		5	0.011 2	0.016 3	0.028 7	0.043 3	0.014 9	0.003 5	0.092 1	0.030 6	0.003 4
	10	7	0.111 5	0.078 8	0.022 7	0.069 4	0.015 9	0.002 8	0.092 1	0.018 7	0.002 7
		9	0.034 9	0.008 1	0.017 7	0.031 1	0.031 8	0.002 4	0.034 8	0.012 1	0.002 3
		8	0.010 4	0.011 3	0.020 5	0.043 0	0.009 5	0.002 5	0.132 7	0.033 9	0.002 5
	16	10	0.074 8	0.011 9	0.016 0	0.050 6	0.017 7	0.002 2	0.118 4	0.025 7	0.002 1
		14	0.030 5	0.009 0	0.014 0	0.013 4	0.004 2	0.001 7	0.096 9	0.013 9	0.001 6
		10	0.025 1	0.010 6	0.017 6	0.114 3	0.026 8	0.002 2	0.163 4	0.043 7	0.002 0
0.5	20	15	0.052 7	0.012 1	0.012 1	-0.003 6	0.003 8	0.001 6	0.027 7	0.002 2	0.001 8
		18	0.041 2	0.009 8	0.011 2	0.132 5	0.021 5	0.001 3	0.060 3	0.009 9	0.001 3
		10	-0.005 9	0.011 8	0.016 9	0.028 1	0.034 8	0.002 5	0.071 4	0.015 7	0.002 2
	25	15	0.039 6	0.005 0	0.013 3	0.000 3	0.005 2	0.001 5	0.018 4	0.002 7	0.001 7
		20	-0.023 0	0.006 7	0.010 9	0.008 7	0.007 5	0.001 3	0.054 5	0.009 4	0.001 3
		15	0.006 0	0.009 6	0.013 7	0.026 3	0.010 2	0.001 8	0.046 6	0.016 6	0.001 6
	30	20	0.038 0	0.007 9	0.010 1	0.018 0	0.007 5	0.001 2	0.066 0	0.017 7	0.001 3
		25	0.050 7	0.010 1	0.008 1	0.033 7	0.007 4	0.001 1	0.058 8	0.005 3	0.001 1
		5	0.037 6	0.026 1	0.030 7	0.160 7	0.058 5	0.003 8	0.149 3	0.056 4	0.003 8
	10	7	0.004 5	0.008 8	0.026 5	0.072 6	0.014 7	0.003 1	0.101 0	0.021 3	0.003 1
		9	0.088 0	0.032 7	0.021 6	0.062 7	0.035 8	0.002 4	0.063 1	0.023 7	0.002 7
		8	0.104 0	0.026 1	0.023 3	0.063 3	0.034 2	0.003 0	0.071 7	0.016 4	0.002 8
	16	10	0.100 6	0.017 8	0.020 1	0.117 3	0.029 9	0.002 4	0.038 0	0.017 0	0.002 5
		14	0.030 9	0.008 0	0.014 3	0.046 3	0.015 2	0.001 9	0.029 8	0.019 4	0.001 9
		10	0.002 7	0.010 6	0.020 9	0.075 3	0.010 3	0.002 5	0.018 7	0.017 9	0.002 5
1	20	15	0.076 1	0.015 8	0.014 2	0.046 1	0.009 5	0.001 8	0.066 0	0.016 9	0.001 8
		18	0.037 7	0.012 2	0.012 3	0.010 0	0.013 1	0.001 7	0.029 9	0.013 9	0.001 6
		10	0.099 2	0.026 4	0.020 3	-0.015 7	0.003 2	0.002 5	0.055 8	0.017 5	0.002 6
	25	15	0.046 5	0.012 2	0.014 3	-0.016 6	0.013 1	0.001 9	0.028 8	0.006 3	0.001 7
		20	-0.017 4	0.007 8	0.011 8	-0.026 6	0.008 0	0.001 5	0.028 5	0.010 4	0.001 3
		15	0.018 6	0.020 9	0.015 3	0.074 4	0.015 1	0.001 9	-0.017 1	0.007 6	0.001 9
	30	20	0.055 9	0.009 8	0.011 6	0.032 7	0.022 3	0.001 5	0.010 2	0.009 4	0.001 5
		25	0.068 2	0.009 4	0.009 4	0.084 1	0.014 9	0.001 2	0.057 1	0.006 3	0.001 2
		5	0.194 0	0.078 1	0.035 3	0.142 5	0.040 9	0.004 6	0.296 9	0.121 0	0.004 3
	10	7	0.142 5	0.079 5	0.029 2	0.109 2	0.052 0	0.003 4	0.125 3	0.061 3	0.003 2
		9	0.037 6	0.028 9	0.026 2	0.095 7	0.023 4	0.003 1	0.109 1	0.029 7	0.003 1
		8	0.044 1	0.047 2	0.028 6	0.103 5	0.039 6	0.003 3	0.156 3	0.044 1	0.003 0
	16	10	0.053 8	0.032 1	0.021 2	0.176 1	0.070 2	0.002 6	0.170 7	0.053 5	0.002 5
		14	0.084 2	0.024 4	0.017 2	0.035 1	0.014 6	0.002 2	0.122 1	0.038 9	0.002 2
		10	0.091 0	0.047 7	0.022 8	0.067 2	0.034 2	0.002 8	0.076 2	0.017 5	0.002 5
4	20	15	0.087 8	0.018 5	0.016 3	0.081 1	0.018 2	0.001 9	0.148 9	0.048 8	0.002 1
		18	0.062 8	0.024 8	0.013 6	0.027 0	0.019 0	0.001 8	0.079 2	0.014 4	0.001 7
		10	0.024 5	0.012 4	0.024 8	0.115 0	0.042 2	0.003 2	0.192 4	0.103 6	0.002 9
	25	15	-0.022 6	0.006 1	0.017 3	0.034 1	0.018 3	0.002 1	0.010 7	0.001 6	0.002 2
		20	0.038 7	0.017 0	0.013 1	0.014 6	0.005 3	0.001 7	0.068 1	0.029 0	0.001 5
		15	0.091 8	0.044 3	0.014 8	0.028 2	0.007 6	0.002 1	0.028 2	0.038 0	0.002 1
	30	20	0.035 3	0.016 8	0.012 2	0.070 6	0.025 9	0.001 5	0.095 1	0.019 0	0.001 5
		25	0.062 5	0.012 4	0.011 2	0.058 6	0.006 2	0.001 2	0.056 8	0.015 3	0.001 3

Table 6

Table 7

Fig 3

Table 8

Table 9

Table 10

Table 11

Fig 4

Table 12

Table 13

Table 14

Table 15

Table 16

Fig 5

Fig 6

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Failure times	Number of failure units	Random removal	Number of remaining units
1	1	$R_1 \sim B(n - m, p)$	$n- 1 - R_1 $
2	1	$R_2 \sim B(n - m - R_1, p)$	$n - 2 - R_1 - R_2 $
$\vdots$	$\vdots$	$\vdots$	$\vdots$
$m - 1$	1	$R_{m - 1} \sim B(n - m - \sum\limits_{i=1}^{m - 2} {R_i }, p)$	$n - (m - 1) - \sum\limits _{i=1}^{m - 1} R_i $
$m$	1	$R_m =n -m - \sum\limits _{i=1}^{m- 1} R_i $	0

$n$	$m$	$\widehat{p}$_MLE	AB	MSE	$CI_L $	$CI_U $	Cover
	5	0.543 0	0.043 0	0.033 1	0.254 1	0.831 9	84.2
10	7	0.576 8	0.076 8	0.055 3	0.248 1	0.905 5	82.6
	9	0.698 4	0.198 4	0.142 1	0.420 5	0.976 3	45.6
	8	0.530 4	0.030 4	0.018 5	0.291 5	0.769 3	92.0
16	10	0.542 5	0.042 5	0.024 9	0.271 4	0.813 7	91.3
	14	0.614 9	0.114 9	0.080 3	0.274 4	0.955 4	71.4
	10	0.523 2	0.023 2	0.014 4	0.308 0	0.738 4	92.7
20	15	0.550 1	0.050 1	0.031 1	0.260 4	0.839 8	86.6
	18	0.610 6	0.110 6	0.080 2	0.271 6	0.949 6	71.0
	10	0.517 0	0.017 0	0.009 0	0.339 3	0.694 6	93.8
25	15	0.524 7	0.024 7	0.014 1	0.309 0	0.740 3	93.3
	20	0.550 4	0.050 4	0.030 9	0.260 4	0.840 3	86.4
	15	0.517 4	0.017 4	0.009 1	0.339 8	0.695 0	94.2
30	20	0.522 9	0.022 9	0.013 9	0.307 5	0.738 3	93.4
	25	0.550 0	0.050 0	0.030 6	0.259 3	0.840 7	86.5

$n$	$m$	$\widehat{\alpha} _{\text{MLE }} $	$\text{AB}$	$\text{MSE }$	$CI_L $	$CI_U $	Cover
	5	6.839 1	4.839 1	136.165 8	0	22.786 6	97.7
10	7	8.408 2	6.408 2	1 335.444 0	0	31.100 1	96.4
	9	3.238 4	1.238 4	0.828 1	0	6.931 6	96.8
	8	3.446 1	1.446 1	11.727 3	0	7.606 0	96.8
16	10	2.872 2	0.872 2	0.559 0	0.143 9	5.600 6	96.9
	14	2.585 3	0.585 3	0.201 9	0.446 1	4.724 5	96.4
	10	2.955 6	0.955 6	6.841 8	0	5.912 2	96.3
20	15	2.474 6	0.474 6	0.159 8	0.598 9	4.350 4	96.2
	18	2.524 3	0.524 3	0.155 8	0.690 4	4.358 2	97.5
	10	2.532 7	0.532 7	0.189 1	0.506 5	4.558 9	95.6
25	15	2.437 8	0.437 8	0.142 7	0.701 2	4.174 4	96.5
	20	2.398 6	0.398 6	0.111 2	0.820 1	3.977 0	96.2
	15	2.345 3	0.345 3	0.098 5	0.777 1	3.913 5	96.6
30	20	2.363 4	0.363 4	0.094 0	0.887 2	3.839 6	97.1
	25	2.257 9	0.257 9	0.063 4	0.938 1	3.577 7	95.7

$n$	$m$	$\widehat{\lambda }_{\text{MLE }} $	$\text{AB}$	$\text{MSE }$	$CI_L $	$CI_U $	Cover
	5	1.189 0	0.189 0	0.015 8	0.644 4	1.733 6	89.7
10	7	1.136 3	0.136 3	0.009 6	0.691 7	1.580 9	90.4
	9	1.101 3	0.101 3	0.006 0	0.719 1	1.483 6	91.4
	8	1.111 3	0.111 3	0.006 7	0.700 7	1.522 0	92.2
16	10	1.088 8	0.088 8	0.004 9	0.728 5	1.449 2	93.4
	14	1.051 1	0.051 1	0.002 6	0.754 1	1.348 0	94.6
	10	1.081 6	0.081 6	0.004 8	0.723 2	1.440 1	92.8
20	15	1.052 2	0.052 2	0.002 7	0.764 7	1.339 7	93.7
	18	1.050 4	0.050 4	0.002 3	0.788 5	1.312 2	92.7
	10	1.071 7	0.071 7	0.004 5	0.714 7	1.428 6	93.4
25	15	1.050 8	0.050 8	0.002 7	0.764 7	1.336 9	93.1
	20	1.044 0	0.044 0	0.001 9	0.796 6	1.291 5	93.3
	15	1.046 7	0.046 7	0.002 5	0.762 4	1.330 9	94.4
30	20	1.047 3	0.047 3	0.002 0	0.800 8	1.293 9	93.8
	25	1.027 0	0.027 0	0.001 4	0.807 4	1.246 7	93.9

$n$	$m$	$\alpha $		$\lambda $
$n$	$m$	$(\alpha _L^T, \alpha _U^T)$	$(\alpha _L^H, \alpha _U^H)$	$(\lambda _L^T, \lambda _U^T)$	$(\lambda _L^H, \lambda _U^H)$
	5	(0.892 6, 2.316 5)	(0.901 2, 2.305 5)	(0.896 1, 2.568 1)	(0.904 8, 2.382 1)
10	7	(0.953 1, 2.718 1)	(1.023 3, 2.678 9)	(0.961 8, 2.710 4)	(0.991 8, 2.103 9)
	9	(0.800 9, 2.180 3)	(0.953 2, 2.085 2)	(0.800 9, 2.180 3)	(0.833 7, 2.032 3)
	8	(0.948 2, 2.244 6)	(0.956 7, 2.143 5)	(1.005 1, 2.252 7)	(1.014 2, 1.997 6)
16	10	(0.937 5, 2.312 6)	(0.958 9, 2.248 8)	(0.937 5, 2.312 6)	(1.092 5, 2.297 3)
	14	(0.983 5, 2.314 3)	(1.010 0, 2.167 8)	(0.983 5, 2.314 3)	(0.997 6, 2.145 9)
	10	(0.918 1, 2.252 5)	(0.953 4, 2.148 6)	(0.849 2, 2.238 2)	(0.903 7, 2.134 6)
20	15	(0.858 5, 2.376 0)	(0.907 8, 2.267 3)	(0.855 9, 2.360 4)	(0.897 6, 2.287 7)
	18	(1.019 5, 2.231 5)	(1.101 0, 2.134 2)	(0.990 4, 2.292 1)	(1.124 3, 2.136 2)
	10	(0.898 1, 2.269 2)	(0.935 2, 2.187 3)	(0.897 7, 2.294 1)	(0.957 0, 2.190 8)
25	15	(0.897 6, 2.182 6)	(0.943 2, 2.040 8)	(0.854 6, 2.179 8)	(0.904 5, 2.090 3)
	20	(0.818 4, 1.929 9)	(0.858 1, 1.918 4)	(0.818 4, 1.929 9)	(0.836 2, 1.910 0)
	15	(0.838 2, 2.067 2)	(0.865 8, 2.078 3)	(0.838 2, 2.067 2)	(0.873 6, 1.998 7)
30	20	(0.961 1, 2.012 2)	(0.987 7, 2.135 2)	(0.931 1, 1.936 1)	(0.971 1, 1.942 1)
	25	(0.989 5, 2.863 6)	(1.032 4, 2.659 1)	(0.917 3, 2.755 3)	(0.932 8, 2.487 3)