系统可靠性评估中的信息融合方法及应用
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摘要
本文综合运用Bayes、经验Bayes及多层Bayes理论,在产品具有多源验前信息的情况下,充分利用验前信息,并结合少量现场试验样本对系统可靠性进行了评估。接着研究了Bayes统计分析中利用验前信息的稳健性问题。同时通过数值仿真来说明应用过程和方法的正确性、合理性。
主要工作如下:
首先,在系统验前信息多源性的情况下,由不同的验前信息得到不同形式的验前分布,利用可信度加权法、相关函数法、极大似然法来融合系统可靠性验前信息,合理地确定了各验前分布在融合综合验前分布中的权重。
其次,讨论了多源验前信息情况下如何对产品的失效率进行融合估计的问题,利用经验Bayes及多层Bayes方法来融合系统的多源验前信息,得到了产品失效率的验前分布及后验分布,并分别在平方损失及Linex损失函数下得到产品失效率的经验Baves估计。
再次,讨论了经验Bayes和多层Bayes信息融合方法在k/n(G)系统可靠性评估中的应用,在多个验前信息源的情况下,得到了系统可靠性指标的Bayes估计。
最后,以平方损失下的Γ-后验期望损失为判别准则,讨论了指数寿命型产品失效率的最优Bayes稳健区间估计,导出了指数寿命型产品失效率的最优Bayes稳健点估计。
In this paper, when there has few samples, we discuss the estimation of the system reliability in multi-sources of prior information with Bayes, EB (Empirical Bayes) and HB (Hierarchical Bayes) theories. Then we study the robustness of prior distribution in Bayes statistical analysis. The results from simulation show that the method proposed in this paper is effective and reasonable.The main workis:Firstly, When prior information comes from different sources.we develop some methods to realize the fusion of the system information based on the correlation function、 the credibility and the ideology of the maximum like hood. And confirm weights of every prior distribution logically when fusing.Secondly, methods for pooling failure rate data obtained from different sources is discussed, introduce how to use the EB and HB theories to realize the fusion information from multi-sources, and find the prior distribution and posterior distribution of the failure rate. Then the EB estimate of the failure rate is obtained using the squared error loss and the Linex loss functions.Thirdly, When prior information comes from different sources, the EB and HB fusion methods of prior information and their applications in reliability analysis of k/n (G) system are discussed, and then give the Bayes estimation.Finally, the optimal Bayes robust credible set and the optimal Bayes robust point estimator of the failure rate X are discussed using the r- posterior expected loss under the squared error loss as the criterion.
主要工作如下:
首先,在系统验前信息多源性的情况下,由不同的验前信息得到不同形式的验前分布,利用可信度加权法、相关函数法、极大似然法来融合系统可靠性验前信息,合理地确定了各验前分布在融合综合验前分布中的权重。
其次,讨论了多源验前信息情况下如何对产品的失效率进行融合估计的问题,利用经验Bayes及多层Bayes方法来融合系统的多源验前信息,得到了产品失效率的验前分布及后验分布,并分别在平方损失及Linex损失函数下得到产品失效率的经验Baves估计。
再次,讨论了经验Bayes和多层Bayes信息融合方法在k/n(G)系统可靠性评估中的应用,在多个验前信息源的情况下,得到了系统可靠性指标的Bayes估计。
最后,以平方损失下的Γ-后验期望损失为判别准则,讨论了指数寿命型产品失效率的最优Bayes稳健区间估计,导出了指数寿命型产品失效率的最优Bayes稳健点估计。
In this paper, when there has few samples, we discuss the estimation of the system reliability in multi-sources of prior information with Bayes, EB (Empirical Bayes) and HB (Hierarchical Bayes) theories. Then we study the robustness of prior distribution in Bayes statistical analysis. The results from simulation show that the method proposed in this paper is effective and reasonable.The main workis:Firstly, When prior information comes from different sources.we develop some methods to realize the fusion of the system information based on the correlation function、 the credibility and the ideology of the maximum like hood. And confirm weights of every prior distribution logically when fusing.Secondly, methods for pooling failure rate data obtained from different sources is discussed, introduce how to use the EB and HB theories to realize the fusion information from multi-sources, and find the prior distribution and posterior distribution of the failure rate. Then the EB estimate of the failure rate is obtained using the squared error loss and the Linex loss functions.Thirdly, When prior information comes from different sources, the EB and HB fusion methods of prior information and their applications in reliability analysis of k/n (G) system are discussed, and then give the Bayes estimation.Finally, the optimal Bayes robust credible set and the optimal Bayes robust point estimator of the failure rate X are discussed using the r- posterior expected loss under the squared error loss as the criterion.
引文
[1] 梅启智,廖炯生,孙惠中.系统可靠性工程基础[M].北京:科学出版社.1992.
[2] J. Berger, Statistical Decision Theory and Bayesian Analysis [M]. Springer Verlag. 2nd Edi., 1985.
[3] T. F. Li, Empirical Bayes Approach to Reliability Estimation for the Exponential Distribution [J]. IEEE Transactions on Reliability. 1984, 33(3), 233-236.
[4] J. Berger, L. M. Berliner, Robust Bayes empirical Bayes analysis with ε-contaminated priors [J]. Ann.Statist, 1986(14), 461-486.
[5] H. SCHABE. Bayes Estimates under Asymmetric Loss[J]. IEEE Transactions on Reliability,1991,40(1),63-67.
[6] A.P. Basu, Bayesian approach to life testing and reliability estimation using asymmetric loss function, Journal of Statistical Planning and Inference, 1991, 29, 21-31
[7] Jan M. van Noortwijk, Rommert Dekker, Roger M.Cooke,T.A.M. Expert judgment in Maintenance Optimization[J]. IEEE Transactions on Reliability. 1992, 41(3), 427-432.
[8] Yuanzhang Li, K. M, Lal Saxena. Optimal robust Bayes estimation[J].Journal of Statistical Planning and Inference, 1995(46) 365-380.
[9] Consonni, G., Veronese, P., A Bayes method for combining results from several binomial experiments [J]. Amer. Statist. Assoc. 1995(90), 935-944.
[10] Agata Boratyriska. On Bayes robustness with the ε-contamination class of priors[J]. Statistics&Probability Letters, 1996(26), 323-328.
[11] H. SCHABE. Combining failure rate data from various sources [J]. Microelectron. Reliab. 1996, 36(1), 47-54.
[12] Erto P., Grorgio M., Modified practical Bayes estimator, IEEE Transactions on Reliability, 1996,VoL 45,No. 1,132-137
[13] Huang S.Y, Liang T.C. Empirical Bayes estimation of the truncation parameter with Linex loss [J].Statistica Sinica, 1997:755-769
[14] Coit D. W., System reliability confidence intervals for complex-systems with estimated component reliability, IEEE Transactions on Reliability, 1997, VoL 46, No.4, 487-493
[15] Craig J. W., Series-system reliability-estimation using small binomial samples. IEEE Transactions on Reliability, 1997, VoL 46, No2, 296-302
[16] Tyoskin O. I., Sonkina T., Parametric reliability-prediction based on small Sample, IEEE Transactions on Reliability, 1997, VoL 46, No3, 394-399
[17] Lieber D., A fast Monte-Carlo method for evaluation reliability indexes, IEEE Transactions on Reliability, 1999, VoL 48, No3, 256-262
[18] Lee S. M. S., Young G. A., The effect of Monte-Carlo approximation on converge error of double-Bootstrap confidence intervals, Journal of Royal Statistical society, 1999, VoL61, No.2,
[19] Doganaksoy N., Practical aspects of corrected likelihood ratio confidence intervals: a discussion of Jeng-Meeker and Wong-Wu, Technometrics, 2000,VoL 42, No.2, 156
[20] Yimin Shi, Huayong Xiao, Empirical Bayes test for the parameter of the truncated type distribution families, Southeast Asian Bulletin of Mathematics, 2000, VOl.24, NO.3
[21] Ahmed A. Soliman, Comparison of Linex and quadratic Bayes estimators for the Rayleigh distribution, Communication in Statistics Theory and Methods, 2000,29(1), 95-107
[22] Chaudhuri G., A new approach to system reliability, IEEE Transactions on Reliability, 2001, 50(1), 75-84
[23] Levitin G., Reliability evaluation for linear consecutively-connected systems with multistate elements and retransmission delays, Quality and Reliability Engineering International 2001, 17: 373-378
[24] Yimin Shi, Yong Xu, Huiguang Kang, A strong limit theorem on random selection, Southeast Asian Bulletin of Mathematics, 2001, VOl.25, NO.3
[25] Malay Ghosh, Dal Ho Kim. Robust Bayes analysis with partially exchangeable priors[J].. Journal of Statistical Planning and Inference, 2002(102), 99-107
[26] Huiguang Kang, Yimin Shi, Xiaoshan Zhao, Empirical Bayes estimation for the parameter of two dimension one-side truncated distribution families with Linex loss, Chinese Quarterly Journal of Mathematics, 2001, VOl.16, NO.3.
[27] Yong Xu, The Applications of Statistical Principles to Information Fusion, The Fifth International Conference on Information Fusion, 7-11 July 2002.
[28] Zehua Chen, Component reliability analysis of k-out-of-n system with censored data [J]. Journal of Statistical Planning and Inference. 2002,12,1-11.
[29] Ahmed A. Soliman, Reliability estimation in a generalized life model with application to the Burr-ⅩⅡ, IEEE Transactions on Reliability, 2002, 51 (3) 337-343
[30] Zehua chen, Component reliability analysis of k-out of n system with censored data. J. of statistical planning and inference. 2003, (1), 1-11
[31] Yimin Shi, Limei Ren, Xuyan Wang, The extension of fixed point theorems for set valued mapping, Journal of Applied Mathematics and Computing, 2003, 14(1-2).
[32] Yimin Shi, Yong Xu, Xiaolin Shi, Convergence rate of empirical Bayes test with truncation parameters under the Linex loss, Southeast Asian Bulletin of Mathematics, 2003,27: 705-717
[33] Yimin Shi, Xiaolin Shi, Shesheng Gao, Empirical Bayes estimation of the truncation parameter with asymmetric loss function using NA samples, Journal of Applied Mathematics and Computing, 2004, 14(1-2),
[34] 赵林城.一类离散分布参数的经验Baye估计的收敛速度[J].数学研究与评论,1981,1:59～69
[35] 成平.经验贝叶斯估计[J].数理统计与应用概率,1985,3(2),156-163.
[36] 唐雪梅.复杂系统可靠性鉴定方法[J].航天控制,1994,4,61-66.
[37] 师义民,黄希利,指数分布下恒定应力加速寿命试验的贝叶斯统计分析,工程数学学报,1995,vol.12,NO.2.
[38] 师义民,宋乾坤,一类单边截断型分布族参数的EB检验,纯粹数学与应用数,1996,vol..12,NO.1.
[39] 施军,邓克绪,段文颖.K/n(G)系统可靠性评定的熵近似限[J].航空学报,1998,19(1),30-34.
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[41] 师义民,加速寿命试验中产品可靠度的贝叶斯估计,西北大学学报,1999,vol.26
[42] 师义民,刘小冬,截断型分布族参数的EB检验,西北大学学报,1999,vol.25
[43] 韩明.无失效数据失效率的综合多层Bayes估计[J].运筹与管理,,1999,8(1),1-5.
[44] 张金槐.Bayes试验分析中验前分布的表示[J].国防科技大学学报.1999,21(6):109-113.
[45] 宋笔锋,李为吉,吉国明,大型结构可靠性优化设计的大系统方法,力学进展,30(1),2000
[46] 韦来生,刻度指数族参数的经验Bayes检验问题:NA样本情形,应用数学学报,2000,23(3),403-412
[47] 师义民,双边截断型分布族参数的经验Bayes估计,高校应用数学学报2000,VOl.15.NO.4
[48] 张士峰,蔡洪.Bayes分析中的多源信息融合问题[J].系统仿真学报,2000,1(12):54-57.
[49] 师义民,魏玲,肖华勇.冷储备系统可靠性指标的估计,西北工业大学学报,2001,NO.1,EIP0139661390
[50] 张士峰等,复杂系统的Bayes可靠性评估,航天控制,2000(2),72-79
[51] 张士峰等,多源验前信息的融合方法,飞行器控制学报,2000,19(1),26-30
[52] 张金槐.验前分布的稳健性[J].国防科技大学学报,2000,22(6),17-22.
[53] 张士峰,樊树江,王慧频.复杂系统的Bayes可靠性评估[J].航天控制,2000,2:72-78.
[54] 许勇,师义民,单边截断型分布族参数的EB检验:NA样本情形,应用数学,2001,NO.4
[55] 康会光,师义民,Linex损失下单边截断型分布族参数的EB估计,应用数学,2001,NO.3
[56] 孙有朝,基于信息理论的复杂产品的可靠性综合评估,应用科学学报,2001,19(2),113-116
[57] 凌能祥,杜雪樵.NA样本下单边截断型分布族位置参数的经验Bayes估计[J].2002,25(5),743-747.
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[2] J. Berger, Statistical Decision Theory and Bayesian Analysis [M]. Springer Verlag. 2nd Edi., 1985.
[3] T. F. Li, Empirical Bayes Approach to Reliability Estimation for the Exponential Distribution [J]. IEEE Transactions on Reliability. 1984, 33(3), 233-236.
[4] J. Berger, L. M. Berliner, Robust Bayes empirical Bayes analysis with ε-contaminated priors [J]. Ann.Statist, 1986(14), 461-486.
[5] H. SCHABE. Bayes Estimates under Asymmetric Loss[J]. IEEE Transactions on Reliability,1991,40(1),63-67.
[6] A.P. Basu, Bayesian approach to life testing and reliability estimation using asymmetric loss function, Journal of Statistical Planning and Inference, 1991, 29, 21-31
[7] Jan M. van Noortwijk, Rommert Dekker, Roger M.Cooke,T.A.M. Expert judgment in Maintenance Optimization[J]. IEEE Transactions on Reliability. 1992, 41(3), 427-432.
[8] Yuanzhang Li, K. M, Lal Saxena. Optimal robust Bayes estimation[J].Journal of Statistical Planning and Inference, 1995(46) 365-380.
[9] Consonni, G., Veronese, P., A Bayes method for combining results from several binomial experiments [J]. Amer. Statist. Assoc. 1995(90), 935-944.
[10] Agata Boratyriska. On Bayes robustness with the ε-contamination class of priors[J]. Statistics&Probability Letters, 1996(26), 323-328.
[11] H. SCHABE. Combining failure rate data from various sources [J]. Microelectron. Reliab. 1996, 36(1), 47-54.
[12] Erto P., Grorgio M., Modified practical Bayes estimator, IEEE Transactions on Reliability, 1996,VoL 45,No. 1,132-137
[13] Huang S.Y, Liang T.C. Empirical Bayes estimation of the truncation parameter with Linex loss [J].Statistica Sinica, 1997:755-769
[14] Coit D. W., System reliability confidence intervals for complex-systems with estimated component reliability, IEEE Transactions on Reliability, 1997, VoL 46, No.4, 487-493
[15] Craig J. W., Series-system reliability-estimation using small binomial samples. IEEE Transactions on Reliability, 1997, VoL 46, No2, 296-302
[16] Tyoskin O. I., Sonkina T., Parametric reliability-prediction based on small Sample, IEEE Transactions on Reliability, 1997, VoL 46, No3, 394-399
[17] Lieber D., A fast Monte-Carlo method for evaluation reliability indexes, IEEE Transactions on Reliability, 1999, VoL 48, No3, 256-262
[18] Lee S. M. S., Young G. A., The effect of Monte-Carlo approximation on converge error of double-Bootstrap confidence intervals, Journal of Royal Statistical society, 1999, VoL61, No.2,
[19] Doganaksoy N., Practical aspects of corrected likelihood ratio confidence intervals: a discussion of Jeng-Meeker and Wong-Wu, Technometrics, 2000,VoL 42, No.2, 156
[20] Yimin Shi, Huayong Xiao, Empirical Bayes test for the parameter of the truncated type distribution families, Southeast Asian Bulletin of Mathematics, 2000, VOl.24, NO.3
[21] Ahmed A. Soliman, Comparison of Linex and quadratic Bayes estimators for the Rayleigh distribution, Communication in Statistics Theory and Methods, 2000,29(1), 95-107
[22] Chaudhuri G., A new approach to system reliability, IEEE Transactions on Reliability, 2001, 50(1), 75-84
[23] Levitin G., Reliability evaluation for linear consecutively-connected systems with multistate elements and retransmission delays, Quality and Reliability Engineering International 2001, 17: 373-378
[24] Yimin Shi, Yong Xu, Huiguang Kang, A strong limit theorem on random selection, Southeast Asian Bulletin of Mathematics, 2001, VOl.25, NO.3
[25] Malay Ghosh, Dal Ho Kim. Robust Bayes analysis with partially exchangeable priors[J].. Journal of Statistical Planning and Inference, 2002(102), 99-107
[26] Huiguang Kang, Yimin Shi, Xiaoshan Zhao, Empirical Bayes estimation for the parameter of two dimension one-side truncated distribution families with Linex loss, Chinese Quarterly Journal of Mathematics, 2001, VOl.16, NO.3.
[27] Yong Xu, The Applications of Statistical Principles to Information Fusion, The Fifth International Conference on Information Fusion, 7-11 July 2002.
[28] Zehua Chen, Component reliability analysis of k-out-of-n system with censored data [J]. Journal of Statistical Planning and Inference. 2002,12,1-11.
[29] Ahmed A. Soliman, Reliability estimation in a generalized life model with application to the Burr-ⅩⅡ, IEEE Transactions on Reliability, 2002, 51 (3) 337-343
[30] Zehua chen, Component reliability analysis of k-out of n system with censored data. J. of statistical planning and inference. 2003, (1), 1-11
[31] Yimin Shi, Limei Ren, Xuyan Wang, The extension of fixed point theorems for set valued mapping, Journal of Applied Mathematics and Computing, 2003, 14(1-2).
[32] Yimin Shi, Yong Xu, Xiaolin Shi, Convergence rate of empirical Bayes test with truncation parameters under the Linex loss, Southeast Asian Bulletin of Mathematics, 2003,27: 705-717
[33] Yimin Shi, Xiaolin Shi, Shesheng Gao, Empirical Bayes estimation of the truncation parameter with asymmetric loss function using NA samples, Journal of Applied Mathematics and Computing, 2004, 14(1-2),
[34] 赵林城.一类离散分布参数的经验Baye估计的收敛速度[J].数学研究与评论,1981,1:59～69
[35] 成平.经验贝叶斯估计[J].数理统计与应用概率,1985,3(2),156-163.
[36] 唐雪梅.复杂系统可靠性鉴定方法[J].航天控制,1994,4,61-66.
[37] 师义民,黄希利,指数分布下恒定应力加速寿命试验的贝叶斯统计分析,工程数学学报,1995,vol.12,NO.2.
[38] 师义民,宋乾坤,一类单边截断型分布族参数的EB检验,纯粹数学与应用数,1996,vol..12,NO.1.
[39] 施军,邓克绪,段文颖.K/n(G)系统可靠性评定的熵近似限[J].航空学报,1998,19(1),30-34.
[40] 赵勇辉,李国英,基于Bayes估计的系统可靠性综合方法,科学通报,1999,44(10),1038-1042
[41] 师义民,加速寿命试验中产品可靠度的贝叶斯估计,西北大学学报,1999,vol.26
[42] 师义民,刘小冬,截断型分布族参数的EB检验,西北大学学报,1999,vol.25
[43] 韩明.无失效数据失效率的综合多层Bayes估计[J].运筹与管理,,1999,8(1),1-5.
[44] 张金槐.Bayes试验分析中验前分布的表示[J].国防科技大学学报.1999,21(6):109-113.
[45] 宋笔锋,李为吉,吉国明,大型结构可靠性优化设计的大系统方法,力学进展,30(1),2000
[46] 韦来生,刻度指数族参数的经验Bayes检验问题:NA样本情形,应用数学学报,2000,23(3),403-412
[47] 师义民,双边截断型分布族参数的经验Bayes估计,高校应用数学学报2000,VOl.15.NO.4
[48] 张士峰,蔡洪.Bayes分析中的多源信息融合问题[J].系统仿真学报,2000,1(12):54-57.
[49] 师义民,魏玲,肖华勇.冷储备系统可靠性指标的估计,西北工业大学学报,2001,NO.1,EIP0139661390
[50] 张士峰等,复杂系统的Bayes可靠性评估,航天控制,2000(2),72-79
[51] 张士峰等,多源验前信息的融合方法,飞行器控制学报,2000,19(1),26-30
[52] 张金槐.验前分布的稳健性[J].国防科技大学学报,2000,22(6),17-22.
[53] 张士峰,樊树江,王慧频.复杂系统的Bayes可靠性评估[J].航天控制,2000,2:72-78.
[54] 许勇,师义民,单边截断型分布族参数的EB检验:NA样本情形,应用数学,2001,NO.4
[55] 康会光,师义民,Linex损失下单边截断型分布族参数的EB估计,应用数学,2001,NO.3
[56] 孙有朝,基于信息理论的复杂产品的可靠性综合评估,应用科学学报,2001,19(2),113-116
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