Plotting squared radii

Abstract

One way of testing data for multivariate normality is to plot the ordered squared radii against their expected values. It is suggested that the approximation to the null distribution offered by the beta distribution is a useful improvement on that afforded by the chi-squared distribution. Some key words: Multivariate normality; Order statistics; Squared radius. The scaled residuals of n observations yi on a ^-dimensional random variable y may be defined as where # and S are respectively the sample mean vector and covariance matrix. Then the squared radii are r? = zjzt = (y<-One aspect of the multivariate normality of y can thus be tested (Healy, 1968) by plotting the ordered r \\ against their expected order statistics, or an approximation thereto. Under the hypothesis that y has a multivariate normal distribution the marginal distribu-tions of the r \\ are proportional to that of a beta variable of the first kind, being (Gnanadesikan & Kettenring, 1972) (n — l)in~1B{^p,^(n — p — 1)}. For sufficiently large n, the correlations