Tables and Large-Sample Distribution Theory for Censored-Data Correlation Statistics for Testing Normality

Abstract

Plotting order statistics versus some variant of the normal scores is a standard graphical technique for assessing the assumption of normality. To obtain an objective evaluation of the normal assumption, it is customary to calculate the correlation coefficient associated with this plot. The Shapiro—Francia statistic is the square of the correlation between the observed order statistics and the expected values of standard normal order statistics, whereas the Shapiro—Wilk statistic also involves the covariances of the standard normal order statistics. In a wide variety of applications, an investigation of the plausibility of the normal (or lognormal) model is needed when the observations on strength or life length are right-censored. The plotting procedure still applies if the observations are censored at a fixed order statistic or a fixed time. Here, the corresponding distribution theory for some modified versions of the Shapiro—Wilk correlation statistic is investigated. Because the asymptotic theory used in this article shows a surprisingly slow rate of convergence even for complete samples, a table of critical values based on a Monte Carlo study is provided. Results from an empirical power study are also presented. Finally, large-sample critical values are obtained and compared with the Monte Carlo values.

Key Words:

• Asymptotic distributions
• Modified Shapiro—Wilk statistics
• Normal probability plot
• Small-sample power