Twenty years have elapsed since the Shapiro-Wilk statistic W for testing the normality of a sample first appeared. In that time a number of statistics that are close relatives of W have been found to have a common (known) asymptotic distribution. It was assumed, therefore, that W must have that asymptotic distribution. We show this to be the case and examine the norming constants that are used with all the statistics. In addition the consistency of the W test is established.
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Leslie et al. (1986) studied this question.