A New Approach to the BHEP Tests for Multivariate Normality

LetX1, …, Xnbe i.i.d. randomd-vectors,d⩾1, with sample meanXand sample covariance matrixS. For testing the hypothesisHdthat the law ofX1is some nondegenerate normal distribution, there is a whole class of practicable affine invariant and universally consistent tests. These procedures are based on weighted integrals of the squared modulus of the difference between the empirical characteristic function of the scaled residualsYj=S−1/2(Xj−X) and its almost sure pointwise limit exp(−‖t‖2/2) underHd. The test statistics have an alternative interpretation in terms ofL2-distances between a nonparametric kernel density estimator and the parametric density estimator underHd, applied toY1, …, Yn. By working in the Fréchet space of continuous functions on Rd, we obtain a new representation of the limiting null distributions of the test statistics and show that the tests have asymptotic power against sequences of contiguous alternatives converging toHdat the raten−1/2, independent ofd.