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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