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 modi ed sphericity statistic and we have called it Hermite Normality Test (SH). We present brie y its asymptotical distribution both under the null and nonnull hypothesis. Large simulations have been made to compare some speci c Hermite tests to three other taken in the litterature. If our test is better for a lot nonnormal populations but works worse for some other, the main point is that we de ned in fact a wide range of tests which may match di erent nonnormal distributions.