【摘要】：Bebbington等(2007)提出一個(gè)兩參數(shù)修正的Weibull(簡(jiǎn)記為FW)分布,該分布具有一個(gè)簡(jiǎn)單的失效率函數(shù).當(dāng)給定不同參數(shù)值時(shí),其失效率可以是單調(diào)增加的,亦可以是修正浴盆狀的.記錄值作為特殊的次序統(tǒng)計(jì)量,由于其具有重要的理論意義和應(yīng)用價(jià)值,自提出以來(lái)被廣泛的應(yīng)用到可靠性分析等諸多領(lǐng)域.本文主要運(yùn)用經(jīng)典方法和Bayes方法研究該分布的可靠性統(tǒng)計(jì)分析.首先給出記錄值的概念、性質(zhì)以及似然函數(shù).其次介紹了FW分布失效率的性質(zhì)并給出WPP圖法理論,然后討論了FW分布參數(shù)、可靠度和失效率的回歸估計(jì)、逆矩估計(jì)、MLE以及基于觀測(cè)信息矩陣下的置信區(qū)間,并運(yùn)用Bootstrap法構(gòu)造了相應(yīng)的置信區(qū)間.在假定兩參數(shù)先驗(yàn)為Gamma先驗(yàn)和無(wú)信息先驗(yàn)的條件下證明了FW分布參數(shù)的滿條件后驗(yàn)分布均為對(duì)數(shù)凹函數(shù),這樣就可以運(yùn)用Gibbs抽樣方法來(lái)獲得Markov Chain Monte Carlo(MCMC)樣本,從而做出參數(shù)、可靠度和失效率的Bayes估計(jì)和后驗(yàn)可信區(qū)間.最后通過(guò)實(shí)例模擬獲得各種估計(jì)的結(jié)果,并基于Bootstrap法和Gibbs抽樣法構(gòu)造的均方誤差和置信區(qū)間來(lái)比較參數(shù)、可靠度和失效率的回歸估計(jì)、逆矩估計(jì)、MLE和Bayes估計(jì)的優(yōu)良性.結(jié)果表明當(dāng)選取合適的先驗(yàn)時(shí),Bayes估計(jì)優(yōu)于其他估計(jì),MLE優(yōu)于逆矩估計(jì)和回歸估計(jì),而回歸估計(jì)的精度最差,而且比較基于觀測(cè)信息矩陣和Bootstrap法得到的置信區(qū)間發(fā)現(xiàn)基于觀測(cè)信息矩陣得到的結(jié)果較好.第四章主要基于記錄值樣本研究FW分布的可靠性統(tǒng)計(jì).首先給出參數(shù)、可靠度和失效率的MLE,并基于觀測(cè)信息矩陣構(gòu)造了置信區(qū)間.其次給出不同先驗(yàn)下FW分布參數(shù)的滿條件分布并證明其為對(duì)數(shù)凹函數(shù),于是運(yùn)用Gibbs抽樣方法得出參數(shù)、可靠度和失效率的Bayes估計(jì)和后驗(yàn)可信區(qū)間.最后通過(guò)數(shù)值模擬得出一致的結(jié)論,在選取合適的先驗(yàn)時(shí),Bayes結(jié)果優(yōu)于MLE。
[Abstract]:Bebbington et al. (2007) proposed a two-parameter modified Weibull (FW) distribution, which has a simple failure rate function. When given different parameter values, the failure rate can be monotonously increased, or it can be modified bathtub. As a special order statistic, recording value has been widely used in many fields such as reliability analysis because of its important theoretical significance and application value. In this paper, the classical method and Bayes method are used to study the reliability statistical analysis of the distribution. Firstly, the concept, properties and likelihood function of record value are given. Secondly, the properties of FW distribution failure rate are introduced and the theory of WPP graph method is given. Then the FW distribution parameters, regression estimation of reliability and failure rate, inverse moment estimation, MLE and confidence interval based on observation information matrix are discussed, and the corresponding confidence interval is constructed by Bootstrap method. Under the assumption that the two parameter priori is Gamma prior and no information prior, it is proved that the full conditional posterior distribution of FW distribution parameters is logarithmic concave function, so that the Gibbs sampling method can be used to obtain Markov Chain Monte Carlo (MCMC) samples, and the Bayes estimation and posterior confidence interval of parameters, reliability and failure rate can be obtained. Finally, various estimation results are obtained by example simulation, and the mean square error and confidence interval constructed by bootstrapper method and Gibbs sampling method are compared to compare the parameters, regression estimation of reliability and failure rate, inverse moment estimation, MLE and Bayes estimation. The results show that when a suitable priori is selected, Bayes estimation is superior to other estimates, MLE is superior to inverse moment estimation and regression estimation, and the accuracy of regression estimation is the worst, and the confidence interval based on observation information matrix and bootstrapper method is better than that based on observation information matrix. The fourth chapter mainly studies the reliability statistics of FW distribution based on record value samples. Firstly, the MLE, of parameters, reliability and failure rate are given, and the confidence interval is constructed based on the observation information matrix. Secondly, the full conditional distribution of FW distribution parameters under different priori is given and proved to be logarithmic concave function, so the Bayes estimation and posterior confidence interval of parameters, reliability and failure rate are obtained by using Gibbs sampling method. Finally, the consistent conclusion is obtained by numerical simulation. When selecting the appropriate prior, the Bayes result is better than that of MLE..
【學(xué)位授予單位】：內(nèi)蒙古工業(yè)大學(xué)
【學(xué)位級(jí)別】：碩士
【學(xué)位授予年份】：2017
【分類號(hào)】：O21

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