CAJ | 학술논문

针对贝叶斯分析中平方误差损失存在的“高估和低估同等重要”问题，提出了一种基于熵损失函数的贝叶斯可靠性分析方法。利用该方法，分别在无信息先验和共轭先验分布下，推导出逆威布尔部件参数、可靠度函数及失效率的 Bayes 估计，并证明了形如[cT（x）＋d]－1的一类估计具有容许性。为了比较不同估计结果的忧劣，文中还给出了逆威布尔部件参数的一致最小方差无偏估计（UMVUE）。最后运用 Monte Carlo 方法对各种估计的均方误差进行了模拟比较。结果表明，当样本量比较小时，Bayes 估计的均方误差小于 UMVUE 的均方误差。随着样本量的增加，各个估计的均方误差都减小，但在共轭先验下 Bayes 估计的均方误差最小。
침대패협사분석중평방오차손실존재적“고고화저고동등중요”문제，제출료일충기우적손실함수적패협사가고성분석방법。이용해방법，분별재무신식선험화공액선험분포하，추도출역위포이부건삼수、가고도함수급실효솔적 Bayes 고계，병증명료형여[cT（x）＋d]－1적일류고계구유용허성。위료비교불동고계결과적우렬，문중환급출료역위포이부건삼수적일치최소방차무편고계（UMVUE）。최후운용 Monte Carlo 방법대각충고계적균방오차진행료모의비교。결과표명，당양본량비교소시，Bayes 고계적균방오차소우 UMVUE 적균방오차。수착양본량적증가，각개고계적균방오차도감소，단재공액선험하 Bayes 고계적균방오차최소。
The mean square error loss in the Bayes estimation has the problem of " equal importance of overestima?tion and underestimation" . Hence we propose the Bayes reliability analysis method based on the entropy loss func?tion. With this method, we derive respectively the parameters, reliability function and failure rate function of the in?verse Weibull component under non?informative priori distribution and conjugate priori distribution. We also prove that the estimation of the class [cT(x)+ d] -1 has admissibility. In order to compare the advantages and disadvanta?ges of different estimation results, we derive the uniform minimum variance unbiased estimate (UMVUE) of the pa?rameters of the inverse Weibull component. Finally, we use the Monte Carlo method to carry out the calculation of the mean square errors of various estimations to analyze the influence of different sample sizes on the accuracy of different estimation results and to compare the effects of the Bayes estimation under non?informative priori distribu?tion and conjunctional prior distribution respectively. The calculation results, given in Table 1, and their analysis show preliminarily that: (1) when the sample size is relatively small, the mean square error of the Bayes estimation is smaller than that of UMVUE; (2) the mean square error of each estimation decreases with increasing sample size; (3) under conjugate priori distribution, the Bayes estimation has minimum mean square error.