 
Summary: Two tests for multivariate normality based on
the characteristic function
Miguel A. Arcones
Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902.
Email:arcones@math.binghamton.edu
April 10, 2007
Abstract
We present two tests for multivariate normality. The presented tests are based on
the L´evy characterization of the normal distribution and on the BHEP tests. The tests
are affine invariant and consistent. We obtain the asymptotic limit null distribution of
the test statistics using some results about generalized onesample Ustatistics which
are of independent interest.
1. Introduction. A common assumption in many statistical procedures is normality of
the observations. Since departure from the model can affect statistical procedures, testing
for normality should be done before using several statistical methods. Many authors have
presented different normality tests. Reviews of normality tests are Henze (2002), Thode (2002)
and Mecklin and Mundfrom (2004). A classical normality test is the one in (SW) Shapiro
and Wilk (1965, 1968). Epps and Pulley (1983) introduced a test of normality based on the
weighted integral of the squared modulus of the difference the empirical (ch.f.) characteristic
function and the normal ch.f. which is competitive with the ShapiroWilk test. Baringhaus
