ON TESTS OF MULTIVARIATE NORMALITY

In multivariate analysis, it is quite often assumed that the observations are normally distributed. However, the departures from normality may affect the use and application of the methodology. Thus it would be desirable to check this assumption so that some remedial steps can be taken such as transforming the data. It has also been pointed out by Gnanadesikan (1977, Wiley Series) that there is no optimal test and so a number of test procedures will be suggested in this thesis.;Six more tests which make use of the Johnson's (1949, Biometrika) transformation are proposed with approximate distributions given. Three of them are based on the sum of the individual transformed skewness and kurtosis and the rest are based on the maximum of the individual transformed skewness and kurtosis. Some numerical examples are given and it is found that they have the same conclusion as some other statistics such as those suggested by Small (1980, Applied Statistics).;The univariate Shapiro-Wilk (1965, Biometrika) and Vasicek (1976, JRSS (B)) statistics are also generalized to the multivariate case. These statistics are then compared with the generalized Shapiro-Wilk statistic suggested by Malkovich and Afifi (1973, JASA) by Monte Carlo methods. It has been found that the new statistics perform very well in most of the cases. Two further test procedures which are based on the transformation of the new generalized Shapiro-Wilk statistic are suggested with approximate distributions given. All the procedures are illustrated with example.;The univariate skewness and kurtosis statistics provide a direct measure of departure from normality and they are generalized to test the multivariate normality by making use of the Principal Component techniques suggested by Srivastava (1982, U.T. Technical Report). The comparison of these statistics with those similar statistics suggested by Mardia (1970, Biometrika) and, Malkovich and Afifi (1973, JASA) are carried out by Monte Carlo methods. It has been found that the new statistics perform pretty good especially for large (rho) and N. The asymptotic distributions of these statistics are also derived. Some numerical examples are used to illustrate all the procedures proposed.