廣義推斷在系統(tǒng)可靠性問題中的應用
發(fā)布時間：2019-04-25 20:14
【摘要】：廣義推斷是基于廣義檢驗變量或廣義樞軸量的統(tǒng)計推斷方法,由廣義置信區(qū)間估計和廣義p值檢驗組成.它的提出是為了解決小樣本中存在討厭參數(shù)的統(tǒng)計推斷問題,而這恰是經(jīng)典頻率學中的方法有時候也無法給出相應的較好解決方案的問題.系統(tǒng)可靠性的統(tǒng)計推斷是現(xiàn)代社會發(fā)展中不可忽視的重要問題,尤其是在工業(yè)農(nóng)業(yè)各個生產(chǎn)線以及高新科技產(chǎn)業(yè)等復雜、高風險的系統(tǒng)中.然而,大多數(shù)學者都只是針對個別分布做了可靠性問題的廣義推斷,且僅僅給出了數(shù)據(jù)模擬結果,并未給出相應的頻率性質(zhì)的證明.同時,用廣義推斷來研究系統(tǒng)可靠性問題的文章也很少.鑒于此,本學位論文的主要研究工作如下:1.研究了含有多個元件的串聯(lián)系統(tǒng)的可靠性問題的廣義推斷.分別在平衡與不平衡狀態(tài)下通過樞軸方程構造出了可靠性函數(shù)的廣義樞軸量,進而給出了廣義p值與廣義置信區(qū)間.并且相應的給出在對數(shù)正態(tài)分布、指數(shù)分布、Weibull分布下的結果.分別證明了小樣本與大樣本定理,說明了其犯第一類錯誤的概率接近于名義水平.最后,采取了蒙特卡羅模擬方法對我們所研究的問題做出了數(shù)據(jù)模擬,其結果顯示廣義推斷方法在小樣本且含討厭參數(shù)的串聯(lián)系統(tǒng)可靠性問題中表現(xiàn)優(yōu)秀.2.給出了含有多個元件的并聯(lián)系統(tǒng)的可靠性問題的廣義推斷.通過樞軸方程給出了可靠性函數(shù)的廣義樞軸量.并聯(lián)系統(tǒng)當中,本文直接在不平衡狀態(tài)下給出了可靠性函數(shù)的廣義p值以及廣義置信區(qū)間,并分別給出了三種不同分布下的結果,即對數(shù)正態(tài)分布、指數(shù)分布、Weibull分布.相應的證明了大樣本以及小樣本定理,說明其犯第一類錯誤的概率趨近于名義水平.最后通過蒙特卡羅模擬方法給出了犯第一類錯誤的概率、置信區(qū)間與平均長度,有力地證明了廣義推斷方法在小樣本并聯(lián)系統(tǒng)可靠性問題中有著良好的應用.3.在前兩部分的基礎之上,本文將可靠性問題的廣義推斷方法推廣到了包含m個串聯(lián)元件和m個并聯(lián)元件的混合系統(tǒng)當中.直接給出了不平衡狀態(tài)時對數(shù)正態(tài)分布、指數(shù)分布、Weibull分布下的可靠性函數(shù)的假設檢驗與置信區(qū)間.同樣,給出了小樣本與大樣本定理的證明,說明其犯第一類錯誤的概率趨近于名義水平.最后,通過蒙特卡羅模擬方法給出了犯第一類錯誤的概率、置信區(qū)間及其平均長度.從結果可以看出,廣義推斷方法在小樣本混合系統(tǒng)可靠性問題中有著優(yōu)秀表現(xiàn).
[Abstract]:Generalized inference is a statistical inference method based on generalized test variables or generalized pivot variables. It consists of generalized confidence interval estimates and generalized p-value tests. It is proposed in order to solve the problem of statistical inference of unwanted parameters in small samples, and this is the problem that the classical frequency method sometimes can not give the corresponding better solutions. The statistical inference of system reliability is an important problem that can not be ignored in the development of modern society, especially in the complex and high-risk systems such as industrial agricultural production lines, high-tech industries and other complex and high-tech industries. However, most scholars have only made the generalized inference for the reliability of individual distributions, and only given the results of data simulation, and did not give the corresponding proof of the frequency properties. At the same time, there are few papers using generalized inference to study the problem of system reliability. In view of this, the main research work of this dissertation is as follows: 1. In this paper, the generalized inference of reliability problem for series systems with multiple components is studied. The generalized pivot quantity of reliability function is constructed by the pivot equation in the equilibrium and unbalanced states, and then the generalized p-value and the generalized confidence interval are given. The corresponding results under logarithmic normal distribution, exponential distribution and Weibull distribution are given. We prove the theorems of small sample and large sample respectively, and show that the probability of making the first kind of error is close to the nominal level. Finally, the Monte Carlo simulation method is used to simulate the problems we have studied. The results show that the generalized inference method performs well in the reliability problems of series systems with small samples and hateful parameters. 2. In this paper, the generalized inference of reliability problem for parallel systems with multiple components is given. The generalized pivot quantity of reliability function is given by the pivot equation. In parallel systems, the generalized p-value and the generalized confidence interval of the reliability function are given directly in the unbalanced state, and the results under three different distributions are given, namely the logarithmic normal distribution, the exponential distribution and the Weibull distribution. The corresponding theorems of large samples and small samples are proved, which shows that the probability of making the first kind of errors tends to be close to the nominal level. Finally, the probability, confidence interval and average length of the first kind of errors are given by Monte Carlo simulation method. It is proved that the generalized inference method has a good application in the reliability problem of small sample parallel systems. On the basis of the first two parts, the generalized inference method of reliability problem is extended to the hybrid system with m series elements and m parallel elements. The hypothesis test and confidence interval of reliability function under the conditions of logarithmic normal distribution, exponential distribution and Weibull distribution are given directly. In the same way, the proof of small sample theorem and large sample theorem is given, which shows that the probability of making the first kind of error is close to the nominal level. Finally, the probability, confidence interval and average length of making the first kind of error are given by Monte Carlo simulation method. It can be seen from the results that the generalized inference method has excellent performance in the reliability problem of small sample hybrid systems.
【學位授予單位】：北京建筑大學
【學位級別】：碩士
【學位授予年份】：2015
【分類號】：O213.2
本文編號：2465426
[Abstract]:Generalized inference is a statistical inference method based on generalized test variables or generalized pivot variables. It consists of generalized confidence interval estimates and generalized p-value tests. It is proposed in order to solve the problem of statistical inference of unwanted parameters in small samples, and this is the problem that the classical frequency method sometimes can not give the corresponding better solutions. The statistical inference of system reliability is an important problem that can not be ignored in the development of modern society, especially in the complex and high-risk systems such as industrial agricultural production lines, high-tech industries and other complex and high-tech industries. However, most scholars have only made the generalized inference for the reliability of individual distributions, and only given the results of data simulation, and did not give the corresponding proof of the frequency properties. At the same time, there are few papers using generalized inference to study the problem of system reliability. In view of this, the main research work of this dissertation is as follows: 1. In this paper, the generalized inference of reliability problem for series systems with multiple components is studied. The generalized pivot quantity of reliability function is constructed by the pivot equation in the equilibrium and unbalanced states, and then the generalized p-value and the generalized confidence interval are given. The corresponding results under logarithmic normal distribution, exponential distribution and Weibull distribution are given. We prove the theorems of small sample and large sample respectively, and show that the probability of making the first kind of error is close to the nominal level. Finally, the Monte Carlo simulation method is used to simulate the problems we have studied. The results show that the generalized inference method performs well in the reliability problems of series systems with small samples and hateful parameters. 2. In this paper, the generalized inference of reliability problem for parallel systems with multiple components is given. The generalized pivot quantity of reliability function is given by the pivot equation. In parallel systems, the generalized p-value and the generalized confidence interval of the reliability function are given directly in the unbalanced state, and the results under three different distributions are given, namely the logarithmic normal distribution, the exponential distribution and the Weibull distribution. The corresponding theorems of large samples and small samples are proved, which shows that the probability of making the first kind of errors tends to be close to the nominal level. Finally, the probability, confidence interval and average length of the first kind of errors are given by Monte Carlo simulation method. It is proved that the generalized inference method has a good application in the reliability problem of small sample parallel systems. On the basis of the first two parts, the generalized inference method of reliability problem is extended to the hybrid system with m series elements and m parallel elements. The hypothesis test and confidence interval of reliability function under the conditions of logarithmic normal distribution, exponential distribution and Weibull distribution are given directly. In the same way, the proof of small sample theorem and large sample theorem is given, which shows that the probability of making the first kind of error is close to the nominal level. Finally, the probability, confidence interval and average length of making the first kind of error are given by Monte Carlo simulation method. It can be seen from the results that the generalized inference method has excellent performance in the reliability problem of small sample hybrid systems.
【學位授予單位】：北京建筑大學
【學位級別】：碩士
【學位授予年份】：2015
【分類號】：O213.2
【參考文獻】
相關期刊論文 前3條
1 ;A fiducial argument for generalized p-value[J];Science in China(Series A:Mathematics);2007年07期
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3 ;GENERALIZED CONFIDENCE REGIONS OF FIXED EFFECTS IN THE TWO-WAY ANOVA[J];Journal of Systems Science and Complexity;2008年02期
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