Graphical Estimators From Probability Plots With Right-Censored Data

Abstract

Probability plots are popular graphical tools used by reliability engineers and other practitioners for assessing parametric distributional assumptions. They are particularly well suited for location-scale families or those that can be transformed to such families. When the plot indicates a reasonable conformity to the assumed family, it is common to estimate the underlying location and scale parameters by fitting a line through the plot. This quick-and-easy method is especially useful with censored data. Indeed, the current version of a popular statistical software package uses this as the default estimation method. In this article we investigate the properties of graphical estimators with multiply right-censored data and compare their performance with that of maximum likelihood estimators. Large-sample results on consistency, asymptotic normality, and asymptotic variance expressions are obtained. Small-sample properties are studied through simulation for selected distributions and censoring patterns. The results presented in this article extend the work of Nair (1984) to right-censored data.

A Comparison of Maximum Likelihood and Median-Rank Regression for Weibull Estimation

“…It would be of interest to conduct similar studies for the other widely used distributions, especially the lognormal distribution. The results in Somboonsavatdee et al (2007) suggest that there could be some interesting differences. .…”

“…The points are plotted using the approximate median-rank positions (i À 0.3)=(n þ 0.4). Note that the ML line would have agreed better with the points had the plotting positions (i À 0.5)=n been displayed instead (as in Lawless 2003;Meeker and Escobar 1998;and Somboonsavatdee et al 2007). Figure 3 compares ML and MRR Weibull distribution estimates for a simulated type 1 censored sample that resulted in seven failures before the censoring time of 100 time hours.…”

“…Nair (1984) also reviewed earlier theoretical work that studies properties of linear estimators based on a subset of the order statistics (which arise in certain kinds of failure censoring). This work was extended in Somboonsavatdee et al (2007), who considered random right censoring with specified censoring distributions. Such censoring schemes would mimic censoring arising from random phenomena like competing failure modes, random entry into a study, and variation in use rates.…”

Comments

“…Because of the prevalence of staggered entry and random censored reliability data the results with m > 2 would complement the random censoring study in Somboonsavatdee et al (2007).…”

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

“…The remaining six engines are right censored. Both Hossain and Zimmer (2002) and Somboonsavatdee et al (2007) considered right-censored data in their studies.…”