Bayesian estimation of a power law process with incomplete data

doi:10.23919/JSEE.2021.000021

Journal of Systems Engineering and Electronics ›› 2021, Vol. 32 ›› Issue (1): 243-251.doi: 10.23919/JSEE.2021.000021

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Bayesian estimation of a power law process with incomplete data

Junming HU^1,^2,³(), Hongzhong HUANG^1,^3,*(), Yanfeng LI^1,³()

¹ School of Mechanical and Electrical Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China
² School of Transportation and Automotive Engineering, Xihua University, Chengdu 610039, China
³ Center for System Reliability and Safety, University of Electronic Science and Technology of China, Chengdu 611731, China

Received:2020-06-20 Online:2021-02-25 Published:2021-02-25
Contact: Hongzhong HUANG E-mail:hujunming@std.uestc.edu.cn;hzhuang@uestc.edu.cn;yanfengli@uestc.edu.cn
About author:|HU Junming was born in 1987. He is currently a lecturer at Xihua University and also a Ph.D. candidate at the University of Electronic Science and Technology of China. He received his M.S. degree in weapon system and utilization engineering from Academy of Engineering Physics in 2012. His research interests include system reliability analysis, reliability growth, maintenance modeling, and uncertainty quantification. E-mail: hujunming@std.uestc.edu.cn||HUANG Hongzhong was born in 1963. He received his Ph.D. degree in reliability engineering from Shanghai Jiao Tong University in 1999. He is a full professor at the School of Mechanical and Electrical Engineering, University of Electronic Science and Technology of China. He has held visiting appointments at several universities in the USA, Canada, and several Asian countries. He is an ISEAM fellow, a technical committee member of ESRA, a co-editor-in-chief of the International Journal of Reliability and Applications, and editorial board members of several international journals. His research interests include reliability engineering, optimization design, fuzzy sets story, and product development. E-mail: hzhuang@uestc.edu.cn||LI Yanfeng was born in 1981. He received his Ph.D. degree in mechatronics engineering from the University of Electronic Science and Technology of China in 2013. He is an associate professor in the School of Mechanical and Electrical Engineering, University of Electronic Science and Technology of China. His research interests include reliability modeling and analysis of complex systems, dynamic fault tree analysis, Bayesian network modeling, and probabilistic inference. E-mail: yanfengli@uestc.edu.cn
Supported by:
This work was supported by the National Natural Science Foundation of China (51775090)

Abstract

Abstract:

Due to the simplicity and flexibility of the power law process, it is widely used to model the failures of repairable systems. Although statistical inference on the parameters of the power law process has been well developed, numerous studies largely depend on complete failure data. A few methods on incomplete data are reported to process such data, but they are limited to their specific cases, especially to that where missing data occur at the early stage of the failures. No framework to handle generic scenarios is available. To overcome this problem, from the point of view of order statistics, the statistical inference of the power law process with incomplete data is established in this paper. The theoretical derivation is carried out and the case studies demonstrate and verify the proposed method. Order statistics offer an alternative to the statistical inference of the power law process with incomplete data as they can reformulate current studies on the left censored failure data and interval censored data in a unified framework. The results show that the proposed method has more flexibility and more applicability.

Key words: incomplete data, power law process, Bayesian inference, order statistics, repairable system

Junming HU, Hongzhong HUANG, Yanfeng LI. Bayesian estimation of a power law process with incomplete data[J]. Journal of Systems Engineering and Electronics, 2021, 32(1): 243-251.

Figures/Tables 5

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Table 1

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References 38

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Step	Content
1	Set the parameters $\alpha $ , $\beta $ and the truncated testing time ${t_c}$
2	Generate the failure number $n$ from the Poisson distribution with mean parameter $\lambda = {({{{t_c}} / \alpha })^\beta }$
3	Generate $n$ random numbers ${u_i}$ from the standard uniform distribution
4	Generate the failure times ${{T}}$ , where ${t_i}$ is equal to ${t_i} = {t_c} \cdot {({r_i})^{\frac{1}{\beta } } }$
5	Sort ${t_i}$ in an ascending order

Data type	Mean estimation		Credible interval
Data type	$\alpha $	$\beta $	$\alpha $	$\beta $
Complete data	40.61	0.591 5	[21.17, 59.05]	[0.505 1, 0.675 9]
Left censored	42.14	0.600 7	[21.55, 59.24]	[0.514 9, 0.685 2]
Interval censored	41.91	0.599 8	[21.57, 59.20]	[0.511 7, 0.683 0]

$r$	$\hat \gamma $		$\hat \beta $		95% credible interval of $\gamma $		95% confidence interval of $\gamma $		95% confidence interval of $\gamma $
$r$	Mean	MLE	Mean	MLE	MCMC	Width	CCI	Width	GCI	Width
1	0.131 5	0.107 2	0.551 2	0.569 0	[0.060 6, 0.246 9]	0.186 3	[0.039 4, 2.104 5]	2.065 1	[0.008 1, 1.424 4]	1.416 4
2	0.146 6	0.1238	0.536 0	0.551 9	[0.073 8, 0.257 2]	0.183 4	[0.046 3, 2.681 1]	2.634 8	[0.010 3, 1.614 0]	1.603 7
3	0.150 6	0.1676	0.530 3	0.515 9	[0.075 4, 0.262 3]	0.186 9	[0.059 0, 5.265 2]	5.206 2	[0.013 7, 2.074 1]	2.060 5
4	0.151 3	0.1835	0.529 3	0.505 2	[0.074 6, 0.265 1]	0.190 5	[0.065 6, 7.211 6]	7.146 0	[0.015 1, 2.343 7]	2.328 6

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Bayesian estimation of a power law process with incomplete data

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