利用Bootstrap方法估計韋伯百分位數

Abstract

許多可靠度數據(如:材料能承受的最低應力)的分佈通常呈現韋伯分佈，由於產品或材料失效通常是其最弱處發生，對於製造業者來說，如何有效偵測出材料失效時間或所能承受之應力的極小值或低百分位數對於維持產品的品質非常重要。因此可靠性數據常來估計產品或材料的最弱點(weakest spot)。因此本論文之主要目的是應用bootstrap方法模擬可靠度資料，再建立韋伯分配百分位數的複式信賴區間，一些文獻針對可靠性數據，利用最大概似估計法(Maximum Likelihood Estimation, MLE)、最佳線性不偏估計(Best Linear Unbiased Estimators, BLUEs)與最佳線線性不變估計(Best Linear Invariant Estimators, BLIEs)等點估計方法，由於這些估計方法只能提供點估計，無法提供估計誤差，故利用複式信賴區間以增加估計韋伯及小值或低百分位數之準確性。本研究利用BLIEs方法及四種 bootstrap 信賴區間(SB, PB, BCPB ,BCa)來估計韋伯百分位數之不確定性，並與傳統信賴區間進行比較。本研究使用了三個衡量指標:Coverage Performance, Interval Mean及Interval Standard Deviation做為比較各種信賴區間的標準，並利用蒙地卡羅模擬法模擬韋伯分配在不同之參數值組合及不同樣本大小下各複式信賴區間之優劣。本研究之模擬結果顯示，不論韋伯分配之參數為何種設定值，只要樣本數n大於10，利用BCPB信賴區間來估計韋伯分配之低百分位數效果較佳。本研究方法可撰寫成軟體程式方便業界應用，對可靠度管理應有相當幫助。

The distribution of many reliability data such as the lowest stress that the material can withstand will exhibit Weibull distributions, The Weibull distribution can also be used to estimate the weakest point of the weakest spot of a product or material. Since the failure of the product or material is usually occurred at it's weakest spot, it is important for the manufacturer to effectively estimate the minimum or the low percentile of the failure time of the materials to maintain the quality of the product. Previous studies developed the Maximum Likelihood Estimation (MLE), the Best Linear Unbiased Estimators (BLUEs) and Best Linear Invariant Estimators (BLIEs) to estimate the Weibull percentile. Therefore, the main objective of this study is to use the bootstrap method to simulate the reliability data, and then establish the confidence interval for the Weibull low percentile. In this study, four bootstrap confidence intervals (i.e. SB, PB, BCPB, BCa) are utilized to construct the confidence simulation interval for the Weibull low-percentile uncertainty, and they are compared with the traditional confidence interval, using three criteria: coverage performance, interval mean and interval standard. A sensitivity analysis conducted under different combinations of parameter values of Weibull distributions with different sample sizes. The results indicate that as long as the sample size is greater than 10, the BCPB interval is recommended to estimate the low percentile of a Weibull distribution, regardless of the values of the parameters of a Weibull distribution. The results of this study can be written into software for industry use.