Spherical harmonics in quadratic forms for testing multivariate normality

We study two statistics for testing goodness of fit to the null hypothesis of multivariate normality, based on averages over the standardized sample of multivariate spherical harmonics, radial functions and their products. These statistics (of which one was studied in the two-dimensional case in Quiroz and Dudley, 1991) have, as limiting distributions, linear combinations of chi-squares. In arbitrary dimension, we obtain closed form expressions for the coefficients that describe the limiting distributions, which allow us to produce Monte Carlo approximate limiting quantiles. We also obtain Monte Carlo approximate finite sample size quantiles and evaluate the power of the statistics presented against several alternatives of interest. A power comparison with other relevant statistics is included. The statistics proposed are easy to compute (with Fortran code available from the authors) and their finite sample quantiles converge relatively rapidly, with increasing sample size, to their limiting values, a behaviour that could be explained by the large number of orthogonal functions used in the quadratic forms involved.