利用複式方法估計Burr分配百分位數之信賴區間

在現今競爭日益激烈的市場中，由於產品的壽命週期逐漸縮短，顧客的期望與故障的成本隨之提高，產品可靠度就成為一個非常重要的品質指標。可靠度是指一個產品或服務之功能在要求之壽命或週期中可以正常運作的機率，故可靠度的高低會影響客戶對商品或服務的品質滿意度，而可靠度可視為產品壽命週期的保證。而在可靠度分析領域當中，可以使用Burr分配來建構可靠度模型，為了研究產品可靠度，工業界通常會注重產品壽命的極小值或其低百分位數。因此本論文之主要目的是應用複式方法模擬可靠度資料，再建構Burr分配百分位數的複式信賴區間，故利用複式信賴區間以增加估計Burr分配低百分位數之準確性。本研究利用MLE方法及三種複式信賴區間(PB, BCPB , BCa)來估計Burr分配之低百分位數。並利用蒙地卡羅模擬法模擬Burr分配在不同之參數值組合及不同樣本大小，並使用了三個衡量指標:Coverage Performance, Interval Mean及Interval Standard Deviation來比較各種信賴區間的優劣，本研究之模擬結果顯示，不論Burr分配之母體參數設定值為何，若希望覆蓋率達到90%以上，本研究建議使用PB法來估計Burr分配百分位數信賴區間較佳，但當樣本數≥15時，則三種信賴區間之估計準確度均很相似且相當不錯。若希望信賴區間之長度和變異減少，增加樣本數即可。本研究方法撰寫成程式供業界應用，對可靠度管理應有相當幫助。

關鍵字

Burr分配；百分位數；複式模擬法；複式信賴區間；可靠度

並列摘要

Under the global competition in industry, as the life cycle of products gradually shortens, customer expectations and the cost of failures increase. Product reliability has become a very important quality indicator. Reliability refers to whether a product or service can function normally during the required life time, so the reliability can be regarded as a guarantee of the product life cycle. Burr distribution can be used to construct a reliability model. The value of product life can be utilized as an index for product reliability. Therefore, the main objective of this study is to use the Bootstrap methods to simulate the reliability data, and then construct the Bootstrap confidence interval for the percentile of the Burr distribution. The traditional method of estimating the low percentile of a Burr can only provide point estimation. To increase the accuracy of estimating the low percentile of the Burr distribution, this study uses the MLE method and three Bootstrap confidence intervals (PB, BCPB, BCa) to estimate the percentile of the Burr distribution. In the sensitivity analysis, this study used three measures: Coverage Performance, Interval Mean, and Interval Standard Deviation as the standards for comparing various confidence intervals. The Monte Carlo simulation method to simulate Burr distribution under different value combinations parameter and different sample sizes. The results of the sensitivity indicated that regardless of parameter values of the Burr distribution, if the coverage rate is more than 90%, PB method is used. When sample sizes is greater than 15, all three methods are equally good. This result of this study can be written into a program for practice use in industry.