含桥联复杂系统的Bayes可靠性分析及Bayes统计决策问题中的几个结论
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摘要
本文由独立的两部分组成。
第一部分研究了含桥联复杂系统的Bayes可靠性问题。
本文在分析现有研究的基础上，借鉴了已有的复杂系统可靠性的Bayes分析方法，研究了串联系统的可靠性，并从统计学的角度研究了新的模型——桥联系统、串—桥联系统以及桥—串联系统的可靠性，在这三种新模型下得出了便于操作的Bayes可靠性分析方法和步骤，以及准确的结果。尤其是在这三种模型中还独创性地研究了由相依(即不独立)单元组成的系统的可靠性，并首次将最大熵方法引入含桥联系统模型中，从而将单元可靠性信息综合为系统验前分布，最后就有无系统级信息这两种情况，分别进行讨论，得到了相应的统计分析方法。
第二部分本文首先在样本总体服从均匀分布，参数θ的先验分布是pareto分布的条件下，求得参数的Bayes估计，并在取一种新的损失函数的条件下，得出了平方损失函数的Bayes估计。通过分析，本文给出了平方损失函数的Bayes估计是保守估计的判断标准，以及风险函数的Bayes估计是保守估计的判断标准，并分别给予了证明。接着，作者给出了参数在加权平方损失函数下的Bayes估计和风险函数的Bayes估计。最后本文从矩估计、极大似然估计等角度研究了共轭先验分布Gamma的超参数，给出了几种确定超参数的方法。
This paper contains two parts.
In the first part, the Bayesian reliablity of complex systems is discussed. On the basis of our analysis to the existing research on the complex system, the related analysis methods are optimized in this paper, resulting one effective new method that the maximum entropy principle is used to analysing the systems with components are dependent each other. In this paper, the following systems are studied: the series system, the bridge system, the series-bridge system and the bridge-series system.Moreover, the above systems are discussed from the two sides: one is that the components are dependent each other; especially the other is that the components are independent each other.
In the second part, the super-parametre of conjugate prior distribution Gamma is discussed from the view of maximum likelihood estimation and torque estimatation.This paper shows the method of getting its Bayesian estimation when the samples submit to homogeneous distribution and the prior distribution of parameter 6 is Pareto distribution. This paper gives the yardstick of verdicting risk function that is conservative estimation or not. At the same time, this paper gives the yardstick of verdicting the Bayesian estimation of quadratic loss function that is conservative estimation or not. In the end, The Bayesian estimation of the parameter when it is in the complexion of weighted quadratic loss function is given in this paper.
第一部分研究了含桥联复杂系统的Bayes可靠性问题。
本文在分析现有研究的基础上，借鉴了已有的复杂系统可靠性的Bayes分析方法，研究了串联系统的可靠性，并从统计学的角度研究了新的模型——桥联系统、串—桥联系统以及桥—串联系统的可靠性，在这三种新模型下得出了便于操作的Bayes可靠性分析方法和步骤，以及准确的结果。尤其是在这三种模型中还独创性地研究了由相依(即不独立)单元组成的系统的可靠性，并首次将最大熵方法引入含桥联系统模型中，从而将单元可靠性信息综合为系统验前分布，最后就有无系统级信息这两种情况，分别进行讨论，得到了相应的统计分析方法。
第二部分本文首先在样本总体服从均匀分布，参数θ的先验分布是pareto分布的条件下，求得参数的Bayes估计，并在取一种新的损失函数的条件下，得出了平方损失函数的Bayes估计。通过分析，本文给出了平方损失函数的Bayes估计是保守估计的判断标准，以及风险函数的Bayes估计是保守估计的判断标准，并分别给予了证明。接着，作者给出了参数在加权平方损失函数下的Bayes估计和风险函数的Bayes估计。最后本文从矩估计、极大似然估计等角度研究了共轭先验分布Gamma的超参数，给出了几种确定超参数的方法。
This paper contains two parts.
In the first part, the Bayesian reliablity of complex systems is discussed. On the basis of our analysis to the existing research on the complex system, the related analysis methods are optimized in this paper, resulting one effective new method that the maximum entropy principle is used to analysing the systems with components are dependent each other. In this paper, the following systems are studied: the series system, the bridge system, the series-bridge system and the bridge-series system.Moreover, the above systems are discussed from the two sides: one is that the components are dependent each other; especially the other is that the components are independent each other.
In the second part, the super-parametre of conjugate prior distribution Gamma is discussed from the view of maximum likelihood estimation and torque estimatation.This paper shows the method of getting its Bayesian estimation when the samples submit to homogeneous distribution and the prior distribution of parameter 6 is Pareto distribution. This paper gives the yardstick of verdicting risk function that is conservative estimation or not. At the same time, this paper gives the yardstick of verdicting the Bayesian estimation of quadratic loss function that is conservative estimation or not. In the end, The Bayesian estimation of the parameter when it is in the complexion of weighted quadratic loss function is given in this paper.
引文
1 B．L．阿姆斯塔特．可靠性数学．北京：科学出版社，1998，p．192-193
2 Berger J O.Statistical decision theory and Bayesian analysis.2nded.New York:Springer-Verlag,1985．中译本：贾乃光译．统计决策理论和贝叶斯分析．北京：中国统计出版社，1998
3 曹晋华，程侃．可靠性数学引论．北京：科学出版社，1986
4 程侃．两相依部件的系统可靠性分析．数学进展，11(1982)，P．206-207
5 陈希儒．数理统计引论．北京：科学出版社，1982
6 丁元耀．指数分布寿命试验的Bayes可靠性分析．宁波大学学报(理工版)，Vol．11 No．2 June 1998
7 范大茵．基于无失效数据指数寿命型单元并联系统可靠性置信限[J]．数理统计与应用概率，Vol．8 No．4 1993．p．93-100
8 高社生，张玲霞．可靠性理论与工程应用．北京：国防工业出版社，2002，p．47-63
9 李杨．正态参数估计的损失函数和风险函数的Bayes推断．沈阳化工学院学报，2000，14(2)p．127-129．
10 刘丹，尹智刚，董晓刚．POISSON分布无失效的BAYES可靠性验证试验．长春工业大学学报，Vol．24 No．2 June 2003，p．44-47
11 茆诗松，王静龙，淮晓龙．高等数理统计．上海：华东师范大学出版社，1998，p．303-389
12 茆诗松，王玲玲．可靠性统计．上海：华东师范大学出版社，1984，p．400-420
13 谭文芬．复杂系统可靠性的Bayes分析．西安：西安交通大学学报，2002，p．26-30
14 唐雪梅，张金槐，邵凤昌，李荣．武器装备小子样试验分析与评估．国防工业出版社．2001．p．202-205
15 屠庆慈，陆廷孝．系统可靠性分析与设计．北京：中国航空学会科普与教育工作委员会，1984，p．25-36
16 吴喜之．现代贝叶斯统计学[M]．北京：中国统计出版社，2000 p．58-66
17 项志华．参数估计的损失函数和风险函数的Bayes推断[J]．数理统计与应用概率1993，8(3)p．25-30．
18 姚俊．指数分布寿命试验Bayes可靠性评估．沈阳工业学院学报．Vol．13 No．4 Dec．1994，p．13—17
19 张尧庭，陈汉峰．贝叶斯统计推断．北京：科学出版社，1991
20 张士峰，樊树江，王慧频．复杂系统的Bayes可靠性评估．航天控制，2(2000)，p．72-79
21 张之华，指数分布场合下步进应力加速寿命试验的Bayes分析．高校应用数学学报，1997，12(2)，p．175-181
22 张之华．无失效数据的统计分析．数理统计与应用概率，Vol．10 No．1 1995，p．94-101
23 张金槐．指数寿命型可靠性增长试验的Bayes分析．飞行器测控学报，Vol．22，No．2，June 2003
24 张士峰，李荣，樊树江．Bayes可靠性评估方法述评．飞行器测控学报．Vol．19，No．2 Jun 2000，p．28-34
25 张金槐．分布参数可变时的Bayes估计．飞行器测控学报．Vol．18，No．1 Jun 2001，p．28-34
26 张金槐，唐雪梅．Bayes方法．长沙：国防科技大学出版社，1993
27 张金槐．Bayes可靠性鉴定方案的设计与分析．质量与可靠性．Vol．96．No．6 Jun 2001 p．22-25
28 周广涛．计算机辅助可靠性工程．北京：宇航出版社，1990，p．443-458
29 周源泉，翁朝曦．可靠性评定．北京：科学出版社，1990
30 周源泉，翁朝曦．可靠性基础入门．北京：中国统计出版社，1990
31 Ammar M. Sarhan. Empirical Bayes estimates in exponential reliability model.Applied Mathematics and Computation 135(2003)p.319-332
32 Ammar M. Sarhan, Ahmed H. El-Bassiouny. Estimation of components reliability in a parallel system using masked system life data. Applied Mathematics and computation, 138(2003),p.61-75
33 A.M.Sarhan. The Bayes procedure in exponential reliability family models using conjugate convex tent prior family. Reliability Engineering System Safety 71 (2001) p.97-102
34 A. Madans 峙.Approximate confidence limits for the reliability of series and parallel systems. TeTechnometrics,Vol 7,1965 Nov, p.495-503
35 A.Winterbottom.The interval estimation of system reliability from component test data.Operations Research, Vo 132, 1984 May/Jun, p.628-640
36 Barlow, R. E., and E. Proschan. Mathematical Theory of Reliability. New York, Wiely, 1965
37 Barlow, R. E., and Proschan, F., Statistical Theory of Reliability and Life Testing. New York, Holt Rinehart and Winston, 1975
38 Berger J. The frequentist view point and conditioning, in Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kidfer, Ed. L.Lecanm and R. olshen, 1985:p 15-44. Belmont, Calif. Wadsworth.
39 Calabria R. and Pulcini G. An engineering approach to Bayes estimation for Weibull distribution. Microelection. Reliab. 1994, 5, p.789-802
40 Craig J. Willits Dennis C.Dietz Albert H.Moore. Series-system Reliability-Estimation Using Very Small Binomial Samples. IEEE TRANSACTIONS ON RELIABILIT, VOL. 45, NO. 2, JUNE 1997
41 Chang E Y and Thompson W E. Bayes analysis of reliability for complex system. Operation Research, 1976, 24(1), p.156-168
42 David Robinson, Duane Dietrich. A nonparameric-Bayes Reliability-Growth Model. IEEE TRANSACTIONS ON RELIABILILY. VOL.38.NO.5,DECEMBER
43 Franciszek Grabski and Ammar Sarhan. Empirical Bayes estimation in the case of exponential reliability Reliability Engineering and System Safety, 53(1996), p. 105-113
44 Gelman A, Carlin J B, Stem H S, Bubin D B. Bayesian data analysis. New York: Chapman-Hall, 1995
45 Jun-ichi, Takeuchi. Characterization of the Bayes Estimator and the MDL Estimator for Exponential Families IEEE TRANSACTIONS ON INFORMATION THEORE,Vol.43 No.4 JULY 1997,p. 1165-1174
46 Kleyner A.et al.Bayesian techniques to reduce the sample size in automotive electronics attribute testing. Mmicroelectron. Reliab. 1997,37(6), p.879-883
47 Kiefei J.Conditional confidence statements and confidence estimators[J].J.Am.Statist.Assoc,1977,(72) p.789-827
48 Lin Mei,Min Xie.Bayes Reliability Demonstration Test Plan for Series-Systemswith Binomial Subsystem Data. 1998 PROCEEDINGS Annuaireliability AND maintainability YMPOSIUM p.241-246
49 Louis Hart. A Bayes Approach to Simultaneous Evaluation of Similar Assem Blies[J].IEEE Transactions on Reliability, 1989,38(4)p.483-484
50 Martz H F. and Waller R.A. The basic of Bayesian reliability estimation from attribute test data. Los Alamos Scientific Laboratory, Report UC-79p, February, 1976
51 Martz H F, Waller R.A and Ficks E T. Bayesian reliability analysis of series systems or bionmial subsystems and components. Technometrics, 1988, 30(5), p. 143-154
52 Olcay Akman, Longcheen Huwang.Bayes Computation for Reliability Estimation. IEEE TRANSACTIONS ON RELIABILITY.VOL. 46. NO. 1. 1997 MARCH
53 Press S J.Bayesian Statiatics. New York:John Wiley-Sons,1989.中译本：廖文，陈安贵译．贝叶斯统计学．北京：中国统计出版社，1992
54 Rukhin A L. Estimated loss and admissible loss estimators[J].In Proceedings of Forth Purdue Symposium on Decision Theory, Ed. J.O.Berger and S. S. Gupta, Berlin: Springer—Verlag, 1987,(1)p.365-375.
55 Rukhin A L. Estimating the loss of estimators of a binomial parameters[J].Biometrika, 1988, 75(1) p.153-155
56 T. Elperin, LGertsbakh. Bayes Cridibility Estimation of an Exponential Parameter for Random Censoring&Icomplete Information. IEEE TRANSACTIONS ON RELIABILITY VOL. 39, NO. 2, JUNE 1990
57 Tang J.fang K and Thompson W E. Exact Bayesian estimation of system reliability from component test dat. 1991, working paper
58 Winterbottom A. The interval estimation of system reliability from component test data. Operations Research, 1984, 32(3), p.624-640
2 Berger J O.Statistical decision theory and Bayesian analysis.2nded.New York:Springer-Verlag,1985．中译本：贾乃光译．统计决策理论和贝叶斯分析．北京：中国统计出版社，1998
3 曹晋华，程侃．可靠性数学引论．北京：科学出版社，1986
4 程侃．两相依部件的系统可靠性分析．数学进展，11(1982)，P．206-207
5 陈希儒．数理统计引论．北京：科学出版社，1982
6 丁元耀．指数分布寿命试验的Bayes可靠性分析．宁波大学学报(理工版)，Vol．11 No．2 June 1998
7 范大茵．基于无失效数据指数寿命型单元并联系统可靠性置信限[J]．数理统计与应用概率，Vol．8 No．4 1993．p．93-100
8 高社生，张玲霞．可靠性理论与工程应用．北京：国防工业出版社，2002，p．47-63
9 李杨．正态参数估计的损失函数和风险函数的Bayes推断．沈阳化工学院学报，2000，14(2)p．127-129．
10 刘丹，尹智刚，董晓刚．POISSON分布无失效的BAYES可靠性验证试验．长春工业大学学报，Vol．24 No．2 June 2003，p．44-47
11 茆诗松，王静龙，淮晓龙．高等数理统计．上海：华东师范大学出版社，1998，p．303-389
12 茆诗松，王玲玲．可靠性统计．上海：华东师范大学出版社，1984，p．400-420
13 谭文芬．复杂系统可靠性的Bayes分析．西安：西安交通大学学报，2002，p．26-30
14 唐雪梅，张金槐，邵凤昌，李荣．武器装备小子样试验分析与评估．国防工业出版社．2001．p．202-205
15 屠庆慈，陆廷孝．系统可靠性分析与设计．北京：中国航空学会科普与教育工作委员会，1984，p．25-36
16 吴喜之．现代贝叶斯统计学[M]．北京：中国统计出版社，2000 p．58-66
17 项志华．参数估计的损失函数和风险函数的Bayes推断[J]．数理统计与应用概率1993，8(3)p．25-30．
18 姚俊．指数分布寿命试验Bayes可靠性评估．沈阳工业学院学报．Vol．13 No．4 Dec．1994，p．13—17
19 张尧庭，陈汉峰．贝叶斯统计推断．北京：科学出版社，1991
20 张士峰，樊树江，王慧频．复杂系统的Bayes可靠性评估．航天控制，2(2000)，p．72-79
21 张之华，指数分布场合下步进应力加速寿命试验的Bayes分析．高校应用数学学报，1997，12(2)，p．175-181
22 张之华．无失效数据的统计分析．数理统计与应用概率，Vol．10 No．1 1995，p．94-101
23 张金槐．指数寿命型可靠性增长试验的Bayes分析．飞行器测控学报，Vol．22，No．2，June 2003
24 张士峰，李荣，樊树江．Bayes可靠性评估方法述评．飞行器测控学报．Vol．19，No．2 Jun 2000，p．28-34
25 张金槐．分布参数可变时的Bayes估计．飞行器测控学报．Vol．18，No．1 Jun 2001，p．28-34
26 张金槐，唐雪梅．Bayes方法．长沙：国防科技大学出版社，1993
27 张金槐．Bayes可靠性鉴定方案的设计与分析．质量与可靠性．Vol．96．No．6 Jun 2001 p．22-25
28 周广涛．计算机辅助可靠性工程．北京：宇航出版社，1990，p．443-458
29 周源泉，翁朝曦．可靠性评定．北京：科学出版社，1990
30 周源泉，翁朝曦．可靠性基础入门．北京：中国统计出版社，1990
31 Ammar M. Sarhan. Empirical Bayes estimates in exponential reliability model.Applied Mathematics and Computation 135(2003)p.319-332
32 Ammar M. Sarhan, Ahmed H. El-Bassiouny. Estimation of components reliability in a parallel system using masked system life data. Applied Mathematics and computation, 138(2003),p.61-75
33 A.M.Sarhan. The Bayes procedure in exponential reliability family models using conjugate convex tent prior family. Reliability Engineering System Safety 71 (2001) p.97-102
34 A. Madans 峙.Approximate confidence limits for the reliability of series and parallel systems. TeTechnometrics,Vol 7,1965 Nov, p.495-503
35 A.Winterbottom.The interval estimation of system reliability from component test data.Operations Research, Vo 132, 1984 May/Jun, p.628-640
36 Barlow, R. E., and E. Proschan. Mathematical Theory of Reliability. New York, Wiely, 1965
37 Barlow, R. E., and Proschan, F., Statistical Theory of Reliability and Life Testing. New York, Holt Rinehart and Winston, 1975
38 Berger J. The frequentist view point and conditioning, in Proceedings of the Berkeley Conference in Honor of Jerzy Neyman and Jack Kidfer, Ed. L.Lecanm and R. olshen, 1985:p 15-44. Belmont, Calif. Wadsworth.
39 Calabria R. and Pulcini G. An engineering approach to Bayes estimation for Weibull distribution. Microelection. Reliab. 1994, 5, p.789-802
40 Craig J. Willits Dennis C.Dietz Albert H.Moore. Series-system Reliability-Estimation Using Very Small Binomial Samples. IEEE TRANSACTIONS ON RELIABILIT, VOL. 45, NO. 2, JUNE 1997
41 Chang E Y and Thompson W E. Bayes analysis of reliability for complex system. Operation Research, 1976, 24(1), p.156-168
42 David Robinson, Duane Dietrich. A nonparameric-Bayes Reliability-Growth Model. IEEE TRANSACTIONS ON RELIABILILY. VOL.38.NO.5,DECEMBER
43 Franciszek Grabski and Ammar Sarhan. Empirical Bayes estimation in the case of exponential reliability Reliability Engineering and System Safety, 53(1996), p. 105-113
44 Gelman A, Carlin J B, Stem H S, Bubin D B. Bayesian data analysis. New York: Chapman-Hall, 1995
45 Jun-ichi, Takeuchi. Characterization of the Bayes Estimator and the MDL Estimator for Exponential Families IEEE TRANSACTIONS ON INFORMATION THEORE,Vol.43 No.4 JULY 1997,p. 1165-1174
46 Kleyner A.et al.Bayesian techniques to reduce the sample size in automotive electronics attribute testing. Mmicroelectron. Reliab. 1997,37(6), p.879-883
47 Kiefei J.Conditional confidence statements and confidence estimators[J].J.Am.Statist.Assoc,1977,(72) p.789-827
48 Lin Mei,Min Xie.Bayes Reliability Demonstration Test Plan for Series-Systemswith Binomial Subsystem Data. 1998 PROCEEDINGS Annuaireliability AND maintainability YMPOSIUM p.241-246
49 Louis Hart. A Bayes Approach to Simultaneous Evaluation of Similar Assem Blies[J].IEEE Transactions on Reliability, 1989,38(4)p.483-484
50 Martz H F. and Waller R.A. The basic of Bayesian reliability estimation from attribute test data. Los Alamos Scientific Laboratory, Report UC-79p, February, 1976
51 Martz H F, Waller R.A and Ficks E T. Bayesian reliability analysis of series systems or bionmial subsystems and components. Technometrics, 1988, 30(5), p. 143-154
52 Olcay Akman, Longcheen Huwang.Bayes Computation for Reliability Estimation. IEEE TRANSACTIONS ON RELIABILITY.VOL. 46. NO. 1. 1997 MARCH
53 Press S J.Bayesian Statiatics. New York:John Wiley-Sons,1989.中译本：廖文，陈安贵译．贝叶斯统计学．北京：中国统计出版社，1992
54 Rukhin A L. Estimated loss and admissible loss estimators[J].In Proceedings of Forth Purdue Symposium on Decision Theory, Ed. J.O.Berger and S. S. Gupta, Berlin: Springer—Verlag, 1987,(1)p.365-375.
55 Rukhin A L. Estimating the loss of estimators of a binomial parameters[J].Biometrika, 1988, 75(1) p.153-155
56 T. Elperin, LGertsbakh. Bayes Cridibility Estimation of an Exponential Parameter for Random Censoring&Icomplete Information. IEEE TRANSACTIONS ON RELIABILITY VOL. 39, NO. 2, JUNE 1990
57 Tang J.fang K and Thompson W E. Exact Bayesian estimation of system reliability from component test dat. 1991, working paper
58 Winterbottom A. The interval estimation of system reliability from component test data. Operations Research, 1984, 32(3), p.624-640