型 I 設限樣本下不同韋伯低百分位數估計方法之比較

Comparison on Various Estimation Methods of Weibull Lower Percentiles for Type I Censored Data

黃芸琪(Yun-Chi Huang)

指導教授 ： 唐麗英 ； 李榮貴

國立交通大學/管理學院/工業工程與管理系所/碩士(2020年)

摘要

在可靠度與故障分析領域中，韋伯機率分佈常用來適配產品的壽命資料。由於產品或材料失效通常是發生在其最弱處，因此，工業界常利用產品壽命或失效 資料的低百分位數來代表產品的品質，現已有文獻提出估計韋伯分佈低百分位數 的方法，但這些方法均屬點估計方法，無法提供估計誤差。此外，對有高可靠度 要求的產品，在做可靠度測試時，可能會發生測試時間不足的情況，而導致無法 蒐集到產品所有的觀察值，此種樣本稱為型 I 設限樣本(Type-I censored sample)。 因此，本論文之主要目的是應用複式模擬法模擬產品失效設限樣本，分別利用最 大概似法(Maximum Likelihood Estimation, MLE)、中間點近似法(Mid-Point Approximation method, MPA)、動差法(Method of Moments, MOM)來估計韋伯百分位數，再建構其複式信賴區間以找出各種估計式之估計誤差，並決定何種估計 方法較佳。本研究使用三個衡量複式信賴區間的指標:覆蓋率(Coverage Performance)、區間平均(Interval Mean)及區間標準差(Interval Standard Deviation) 來比較各信賴區間的準確性，本研究最後以第十百分位數為例，利用蒙地卡羅模 擬法比較不同估計微薄百分位數方法，發現在不同之參數值組合、樣本大小、失效率下，MOM 法之表現最優。本研究之成果可供業界快速準確選擇適當之估計公式來估計韋伯分佈之低百分位數。

關鍵字

韋伯分佈 ； 百分位數 ； 設限樣本 ； 複式信賴區間 ； 可靠度分析

並列摘要

In the field of reliability and failure analysis, the Weibull probability distribution is commonly used to construct a prediction model of product life. Since the failure of a product or material usually occurs at its weakest point, estimation of the low percentile of products becomes a very important index for assessing the true quality of a product. In addition, for products with high reliability requirements, it may be impossible to collect all the reliability related measures due to time. In a reliability test, such data are called Type-I censored sample. Therefore, this study uses bootstrap method to simulate failure-time censored data of product. The Maximum Likelihood Estimation method (MLE), the Mid-Point Approximation method (MPA), and the Method of Moments (MOM) were then utilized to estimate the Weibull lower percentile, respectively. Because these estimators are point estimators and therefore cannot provide the error of estimation, the objective of this study is to construct confidence intervals for Weibull lower percentile estimators using bootstrap methods. This study uses three estimation methods (MLE, MPA, MOM) and PTBT bootstrap confidence interval to compare the estimation error of the Weibull lower percentile. Three measures: Coverage Performance, Interval Mean, and Interval Standard Deviation are utilized to compare the performance of the bootstrap confidence intervals in the sensitivity analysis. The result indicates that MOM is an appropriate estimator for Weibull lower percentile under various values of Weibull parameters, sample sizes, and failure rates.

並列關鍵字

Weibull distribution ； Percentiles ； Censored samples ； Bootstrap confidence intervals ； Reliability analysis

參考文獻

[1] 游雅婷(2018)。「應用最大概似法估計韋伯低百分位數及建構韋伯新製程低 百分位數複式管制圖」。交通大學工業工程與管理學系碩士論文。

[2] 林慶泓(2019)。「應用最大概似估計法及複式模擬法建構韋伯分佈新製程百 分位數之 Tukey 管制圖」。交通大學工業工程與管理學系碩士論文。

[3] Aggarwala, R. (2001). Progressive interval censoring: some mathematical results with applications to inference. Communications in Statistics-Theory and Methods, 30(8-9), 1921-1935.

[4] Balakrishnan, N., & Aggarwala, R. (2000). Progressive censoring: theory, methods, and applications. Springer Science & Business Media.

[5] Barndorff-Nielsen, O. E. (1986). Infereni on full or partial parameters based on