失效概率的E-Bayes估计和E-MSE及其应用

Abstract

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

CLC Number:

O213.2

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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Figures/Tables 14

References 28

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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

E-Bayesian Estimation and E-MSE of Failure Probability and Its Applications

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Figures/Tables 14

References 28

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$ i $	$ t_{i} $	$ n_{i} $	$ r_{i} $	$ s_{i} $
1	450	10	0	70
2	650	10	0	60
3	850	10	0	50
4	1050	10	0	40
5	1250	10	1	30
6	1450	10	1	20
7	1650	10	2	10

估计方法(不同损失函数)	$ \widehat{m} $	$ \widehat{\eta} $
定理1的(ⅰ)	3.3174965	2958.5512
定理1的(ⅱ)	3.2787667	3015.4834
以上两个估计值之差	0.0387298	56.9322

$ t $	100	300	500	800	1000	1200	1500	1700
$ \widehat{R}_{EB1}(t) $	0.9999	0.9995	0.9971	0.9870	0.9730	0.9511	0.9003	0.8529
$ \widehat{R}_{EB2}(t) $	0.9999	0.9995	0.9972	0.9872	0.9735	0.9524	0.9037	0.8584

$ i $	$ t_{i} $	mean	sd	MC-error	val2.5pc	val97.5pc	start	sample
1	450	0.0069	0.0075	9.710e-5	1.1374e-5	0.0245	1001	9000
2	650	0.0079	0.0085	1.208e-4	4.9915e-5	0.0287	1001	9000
3	850	0.0096	0.0104	1.563e-4	7.0336e-5	0.0366	1001	9000
4	1050	0.0118	0.0136	1.856e-4	9.0227e-5	0.0469	1001	9000
5	1250	0.0456	0.0370	3.923e-4	0.00252	0.1386	1001	9000
6	1450	0.0675	0.0535	5.319e-4	0.00409	0.2015	1001	9000
7	1650	0.2021	0.1145	0.001368	0.04232	0.4748	1001	9000

估计方法	$ \widehat{m} $	$ \widehat{\eta} $
定理1的(ⅰ)	3.3174965	2958.5512
定理1的(ⅱ)	3.2787667	3015.4834
MCMC方法	3.2942417	2993.9772
以上三个估计值之间的最大差值	0.0387298	56.9322

[1]	Ming Han. E-Bayesian Estimation and Its E-MSE of Poisson Distribution Parameter Under Different Loss Functions [J]. Acta mathematica scientia,Series A, 2019, 39(3): 664-673.
[2]	Han Ming. The E-Bayesian Estimation of the Parameter for Poisson Distribution and Its Properties [J]. Acta mathematica scientia,Series A, 2016, 36(5): 1010-1016.
[3]	HAN Ming. Properties of E-Bayesian Estimation for the Reliability Derived form Binomial Distribution [J]. Acta mathematica scientia,Series A, 2013, 33(1): 62-70.
[4]	HAN Ming. E-Bayesian Estimation of Failure Probability in the Case of Data Only One Failure [J]. Acta mathematica scientia,Series A, 2011, 31(2): 577-583.
[5]	Han Ming. Expected Bayesian Estimation of Failure Probability and Its Character [J]. Acta mathematica scientia,Series A, 2007, 27(3): 488-495.