On assessing multivariate normality based on Shapiro-Wilk W statistic (Q1819499)

\textit{S. S. Shapiro} and \textit{M. B. Wilk}'s W statistic [An analysis of variance test for normality (complete samples). Biometrika 52, 591-611 (1965)] has been found to be the best omnibus test for detecting departures from univariate normality. \textit{J. P. Royston} [J. R. Stat. Soc., Ser. C 32, 121-133 (1983; Zbl 0536.62043)] extends the application of W to testing multivariate normality but the procedure involves a certain approximation which needs to be justified. The procedures proposed in the present paper do not need such an approximation. The asymptotic null distributions are also given. Finally, a numerical example is used to illustrate the procedures.