The Evaluation of Median-Rank Regression and Maximum Likelihood Estimation Techniques for a Two-Parameter Weibull Distribution

ABSTRACT

Practitioners frequently model failure times in reliability analysis via the Weibull distribution. Often risk managers must make decisions after only a few failures. Thus, an important question is how to estimate the parameters of this distribution for extremely small sample sizes and/or highly censored data. This study evaluates two methods: maximum likelihood estimation (MLE) and median-rank regression (MRR). Asymptotically, we know that MLE has superior properties; however, this study seeks to evaluate these two methods for small numbers of failures and high degrees of censoring, where one cannot depend on the asymptotic properties of maximum likelihood estimation. This research is the direct result of a practitioners question at Pratt & Whitney, where they use both methods of estimation. We evaluate the two estimation methods for extremely high censoring cases, focusing on censoring greater than 99%. Such extreme levels of censoring present certain difficulties. For example, the estimated parameters based on MLE follow extremely skewed distributions, even under a log transformation. This article compares the two estimation methods based on parameter estimation and ability to predict future failures. We provide recommendations on which method to use based on sample size, the parameter values, and the degree of censoring present in the data.

KEYWORDS:

• censoring
• maximum likelihood
• median-rank regression
• reliability

ACKNOWLEDGMENTS

A special thanks to Steve Luko, Grant Reinman, and Jim Breneman from Pratt & Whitney for their comments and suggestions. Additionally, we thank the reviewers and discussants, especially Dr. Luis Escobar and Dr. Robert Abernethy, for their detailed comments and suggestions that greatly improved upon earlier versions of this article.