Hermite normality tests

We present in this paper an extended overview of the Hermite Normality test. This test makes use of the Hermite polynomials and a modified sphericity statistic to determine whether a unidimensional, standardised and white sample is normal or not. Its major advantage is to yield not a single test but a real class of test statistics which allows us to match the normality test to the data. We give the limit distribution of the Hermite tests both for the null and nonnull hypothesis and especially for those built with two polynomials. We have determined the tests asymptotically the most powerful for some fixed alternative distributions and made extensive simulations to compare the Hermite tests with three others. The results are good and encourage us to go further with the generalisation of the Hermite test to correlated and multivariate data. (C) 1998 Published by Elsevier Science B.V. All rights reserved.