Hermite normality tests

This paper introduces a new test statistic of normality which evaluates the cross covariances between choosen Hermite polynomials which are zero under the null hypothesis. The special form of the test leads to a modified sphericity statistic and we have called it the Hermite normality test (S_H). We present its asymptotical distribution both under the null and nonnull hypothesis. A large number of simulations have been made to compare some specific Hermite tests to three others taken form the literature. If our test is better for a lot of nonnormal populations but works worse for some other, the main point is that we defined a wide range of tests which may match different nonnormal distributions