A new goodness of fit test for multivariate normality

Abstract

This paper presents a multivariate Kolmogorov-Smirnov (MVKS) goodness of fit test for multivariate normality. The proposed test is based on the difference between the empirical distribution function and the theoretical distribution function. While calculating them in multivariate case, the problem is that the variables cannot be distribution-free as in the univariate case. Firstly, the variables are made independent to solve this problem and the Rosenblatt transform is applied for independence of variates. Then the theoretical and empirical distribution values are calculated and the MVKS test statistic is computed. It provides an easy calculation for d-dimensional data by using the same algorithm and critical table values. This paper demonstrates the effectiveness of the MVKS for different dimensions with a simulation study which also includes the comparison of the MVKS critical tables with univariate Kolmogorov-Smirnov (KS) critical table and the power comparisons of the MVKS (bivariate case) against with the existing bivariate normality tests. Lastly, the MVKS is applied to two different multivariate data sets to confirm that it achieves consistent, accurate and correct results.

Keywords

• multivariate empirical distribution function
• goodness of fit test
• elliptical distributions

References

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