[R] Consistency of m-shapiro.test

From: Manuel Rivas <Mrivas03_at_MIT.EDU>
Date: Sat, 26 Apr 2008 11:42:53 -0400

Hello all,  

I tried several experiments with the mshapiro.test package in R and compared it with the energy package to test for multivariate normality and find that the mshapiro.test is not consistent which is a bit concerning and has suspicious behavior. On the other hand the energy test seems to be a more appropriate test for testing multivariate normality in any dimension. I looked for the reference literature and either seem to reference the original literature on the univariate Shapiro Test or implementation algorithm of the Shapiro test. I was wondering if the procedure used in the mshapiro.test is published elsewhere.  

For direct comparison we selected three alternative distributions used in Quiroz and Manzotti (2001):

  1. Gaussian mixture-bimodal (GM) - (1=2)N (0; Iq) + (1=2)N (3; Iq),
  2. Uniformdistribution on the unit cube [0; 1]q, and
  3. distribution having i.i.d. coordinates with the

Logistic distribution.

We evaluate the power of each statistic(1) energy statistic of Szekely and Rizzo, and 2) Multivariate Shapiro Wilk, on the set of alternatives considered, from sets of 1000 samples with the alternative distribution at each dimension q = 2; 3; 4; 5 and each sample size n = 20; 50; 100. We use the R implementation of the

energy test and multivariate Shapiro-Wilk test.  

It seems as if the Multivariate Shapiro-Wilk test exhibits good power against alternatives that are Gaussian Mixture. My numbers do not agree with the power reportedby Szekely and Rizzo in their paper for the Gaussian Mixture distribution (but this is of less importance). I find behavior that are kind of suspect with the Shapiro-Wilk method, such as increase power with smaller sample size for U[0; 1]q and increase power across all alternatives for increasing q. On the other hand, the multivariate energy test exhibits consistent power against all alternatives, i.e. power increases with sample size which is what is desired from any test.  

Best Regards,

Manuel Rivas

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