Asymptotic distribution of the Hermite normality test

This paper presents some asymptotical results of the Hermite normality test previously introduced. We show that the Hermite statistic S_H is distributed under the null hypothesis as a quadratic form of normal variates and under the nonnull hypothesis as normal. The special case of tests with two polynomials is studied in detail. Finally, we give some considerations for the choice of the best Hermite test when prior knowledge is available and especially we determine the test asymptotically the most powerful for a fixed alternative distribution (the uniform distribution). Those results are supported by simulations