Extreme smoothing and testing for multivariate normality

Recently, Bowman and Foster (1993) proposed to base a test for multivariate normality on a L2 distance between a nonparametric kernel density estimator and the parametric density estimator under normality, applied to the empirically standardized data. We show that, for a fixed bandwidth (not depending on the sample size), the test of Bowman and Foster is a member of the class of invariant and universally consistent procedures suggested by Henze and Zirkler (1990). Moreover, we identify and study the tests for multivariate normality obtained by letting the bandwidth tend to zero and to infinity. While the former test statistic is based solely on the Euclidean norm of the standardized data, letting the bandwidth tend to infinity yields a weighted sum of Mardia's time-honoured measure of multivariate skewness and a sample version of a recently introduced skewness measure of Mori, Rohatgi and Szekely (1993).

[1]  N. Henze,et al.  A consistent test for multivariate normality based on the empirical characteristic function , 1988 .

[2]  Ludwig Baringhaus,et al.  Limit Distributions for Mardia's Measure of Multivariate Skewness , 1992 .

[3]  K. Mardia.  Measures of multivariate skewness and kurtosis with applications , 1970 .

[4]  T. W. Epps,et al.  A test for normality based on the empirical characteristic function , 1983 .

[5]  Locally best invariant tests for multivariate normality in curved families with $\mu$ known , 1994 .

[6]  Stephen Warwick Looney,et al.  A comparison of tests for multivariate normality that are based on measures of multivariate skewness and kurtosis , 1992 .

[7]  A. Viollaz.  Convergence in weighted supremum norms of the skorokhod representation of the estimated empirical process , 1995 .

[8]  A. Paulson,et al.  Some empirical distribution function tests for multivariate normality , 1987 .

[9]  Sung K. Ahn.  F-probability plot and its application to multivariate normality , 1992 .

[10]  A. Bowman,et al.  Adaptive Smoothing and Density-Based Tests of Multivariate Normality , 1993 .

[11]  Kai-Tai Fang,et al.  A test for multivariate normality based on sample entropy and projection pursuit , 1995 .

[12]  Jorge Luis Romeu,et al.  A comparative study of goodness-of-fit tests for multivariate normality , 1993 .

[13]  Calyampudi R. Rao,et al.  Multivariate environmental statistics , 1995 .

[14]  S. Kotz,et al.  Symmetric Multivariate and Related Distributions , 1989 .

[15]  K. Naito.  On weighting the studentized empirical characteristic function for testing normality , 1996 .

[16]  Sándor Csörgő,et al.  Consistency of some tests for multivariate normality , 1989 .

[17]  Norbert Henze,et al.  Limit laws for multivariate skewness in the sense of Móri, Rohatgi and Székely , 1997 .

[18]  Deo Kumar Srivastava,et al.  Some p-variate adaptations of the shapiro-wilk test of normality , 1995 .

[19]  Norbert Henze,et al.  On Mardia’s kurtosis test for multivariate normality , 1994 .

[20]  Tamás F. Móri,et al.  On Multivariate Skewness and Kurtosis , 1994 .

[21]  James A. Koziol,et al.  A class of invariant procedures for assessing multivariate normality , 1982 .

[22]  Deo Kumar Srivastava,et al.  A test of p-variate normality , 1992 .

[23]  C. Versluis.  Comparison of tests for bivariate normality with unknown parameters by transformation to an univariate statistic , 1996 .

[24]  Norbert Henze,et al.  A class of invariant consistent tests for multivariate normality , 1990 .

[25]  John C. W. Rayner,et al.  Interpreting the skewness coefficient , 1995 .

[26]  A. Öztürk,et al.  A new method for assessing multivariate normality with graphical applications , 1992 .

[27]  James A. Koziol,et al.  Probability plots for assessing multivariate normality , 1993 .