失效概率的E-Bayes估计和E-MSE及其应用

摘要/Abstract

摘要：

为了度量估计误差, 该文在E-Bayes估计(expected Bayesian estimation)的基础上引入了E-MSE(expected mean square error)的定义, 并推导不同损失函数(包括平方损失函数和LINEX损失函数)下失效概率的E-Bayes估计及其E-MSE的表达式.通过MonteCarlo模拟比较了所提出估计方法的性能(比较基于E-MSE).最后, 分别采用E-Bayes方法和MCMC方法, 结合发动机可靠性问题进行了计算和分析.在考虑评价不同损失函数下参数的E-Bayes估计时, 该文提出用E-MSE作为评价标准.

关键词: E-Bayes估计, E-MSE, 失效概率, Monte Carlo模拟, MCMC方法

Abstract:

In order to measure the estimated error, this paper based on the E-Bayesian estimation (expected Bayesian estimation) introduced the definition of E-MSE (expected mean square error), and derive the expressions of E-Bayesian estimation of failure probability and their the E-MSE under different loss functions (including: squared error loss function and LINEX loss function). By Monte Carlo simulations compared with the performances of the proposed the estimation method (the comparison of the results is based on the E-MSE). Finally, combined with the engine reliability problem, used respectively E-Bayesian estimation method and the MCMC method the calculation and analysis are performed. When considering evaluating the E-Bayesian estimations under different loss functions, this paper proposed the E-posterior risk as an evaluation standard.

Key words: E-Bayesian estimation, E-MSE, Failure probability, Monte Carlo simulation, MCMC method

中图分类号:

O213.2

韩明. 失效概率的E-Bayes估计和E-MSE及其应用[J]. 数学物理学报, 2022, 42(6): 1790-1801.

Ming Han. E-Bayesian Estimation and E-MSE of Failure Probability and Its Applications[J]. Acta mathematica scientia,Series A, 2022, 42(6): 1790-1801.

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$ s_{i} $	$ r_{i} $	$ \widehat p_{iEB1} $	$ \widehat p_{iEB2} $	$ \widehat p_{iEB1}-\widehat p_{iEB2} $
10	0	0.0364	0.0350	0.0014
	1	0.1171	0.1123	0.0048
	5	0.4586	0.4533	0.0053
	10	0.8806	0.8732	0.0074
20	0	0.0193	0.0189	4.0000e-004
	5	0.2500	0.2416	0.0084
	10	0.4761	0.4670	0.0091
	20	0.9298	0.9213	0.0085
50	0	0.0111	0.0110	1.0000e-004
	10	0.2026	0.1991	0.0035
	30	0.5865	0.5804	0.0061
	50	0.9711	0.9706	5.0000e-004
100	0	0.0047	0.0046	1.0000e-004
	20	0.2015	0.1989	0.0026
	50	0.4952	0.4901	0.0051
	100	0.9849	0.9830	0.0019

s_i	r_i	E-MSE$\left(\widehat{p}_{i E B 1}\right) $	E-MSE$ \left(\widehat{p}_{i E B 2}\right)$	E-MSE$\left(\widehat{p}_{i E B 2}\right)- $E-MSE$ \left(\widehat{p}_{i E B 1}\right)$
10	0	0.0025	0.0027	2.0000e-004
	1	0.0079	0.0081	2.0000e-004
	5	0.0190	0.0194	4.0000e-004
	10	0.0081	0.0086	5.0000e-004
20	0	8.1042e-004	8.1169e-004	1.2700e-006
	5	0.0081	0.0084	3.0000e-004
	10	0.0030	0.0034	4.0000e-004
	20	0.0028	0.0033	5.0000e-004
50	0	2.0616e-004	2.1619e-004	1.0030e-005
	10	0.2026	0.2033	7.0000e-004
	30	0.0046	0.0051	5.0000e-004
	50	5.2910e-004	5.3048e-004	1.3800e-006
100	0	4.4859e-005	4.5969e-005	1.1100e-006
	20	0.0016	0.0018	2.0000e-004
	50	0.0024	0.0028	4.0000e-004
	100	1.4368e-004	1.4859e-004	4.9100e-006

$ i $	$ t_{i} $	$ \widehat p_{iEB1} $	$ \widehat p_{iEB2} $	$ \widehat p_{iEB1}-\widehat p_{iEB2} $
1	450	0.0070	0.0069	1.0000e-004
2	650	0.0081	0.0080	1.0000e-004
3	850	0.0097	0.0096	1.0000e-004
4	1050	0.0120	0.0118	2.0000e-004
5	1250	0.0475	0.0471	4.0000e-004
6	1450	0.0694	0.0688	6.0000e-004
7	1650	0.2160	0.2151	9.0000e-004

i	t_i	E-MSE$\left(\widehat{p}_{i E B 1}\right) $	E-MSE$\left(\widehat{p}_{i E B 2}\right) $	E-MSE$\left(\widehat{p}_{i E B 2}\right)- $E-MSE$\left(\widehat{p}_{i E B 1}\right) $
1	450	9.4992e-005	1.5319e-004	5.8198e-005
2	650	1.2779e-004	2.0635e-004	7.8560e-005
3	850	1.8106e-004	2.9287e-004	1.1181e-004
4	1050	2.7615e-004	3.8963e-004	1.1348e-004
5	1250	0.0014	0.0033	0.0019
6	1450	0.0029	0.0063	0.0034
7	1650	0.0047	0.0158	0.0111

i	t_i	$\widehat{p}_{i E B 1} $	$\widehat{p}_{i E B 2} $	$\widehat{p}_{i E B 1}-\widehat{p}_{i E B 2} $
1	450	0.0069	0.0068	1.0000e-004
2	650	0.0080	0.0079	1.0000e-004
3	850	0.0096	0.0095	1.0000e-004
4	1050	0.0119	0.0117	2.0000e-004
5	1250	0.0468	0.0464	4.0000e-004
6	1450	0.0680	0.0672	8.0000e-004
7	1650	0.2080	0.2071	9.0000e-004