Asymptotic distribution of the Shapiro-Wilk W for testing for normality. (English) Zbl 0622.62019

Leslie, J. R.; Stephens, M. A.; Fotopoulos, S.

Ann. Stat. 14, 1497-1506 (1986).

The Shapiro-Wilk statistic W, which is popularly used for a test of normality, is given by \(W=(x'v^{-1}m)^ 2/\{m'v^{- 2}m\sum^{n}_{1}(x_ i-\bar x)^ 2\}\), where \(x=(x_ 1,...,x_ n)'\), \(x_ 1\leq x_ 2\leq...\leq x_ n\), is the vector of order statistics from the sample, \(\bar x\) is the sample mean, m is the mean vector and v is the covariance matrix of standard normal order statistics.
In this paper the asymptotic distribution of W is derived. It is shown that n(W-EW)\(\to -\xi\) in distribution, where \(\xi =\sum^{\infty}_{3}(y^ 2_ i-1)/i\) and \(\{y_ i\), \(i\geq 3\}\) is a sequence of i.i.d. N(0,1) variables. It follows that \(n(1-W^{1/2})^ 2\to 0\) in probability.

Reviewer: K.Alam

MSC:

62E20	Asymptotic distribution theory in statistics
62F05	Asymptotic properties of parametric tests
62G30	Order statistics; empirical distribution functions
62H15	Hypothesis testing in multivariate analysis

Keywords:

consistency; goodness of fit; Shapiro-Wilk statistic; test of normality; standard normal order statistics