An asymptotically optimal detector for Gaussianity testing

The aim of this paper is to provide a novel approach to the classical problem of testing the Gaussianity of a stationary process where the data can be assumed to be independent and identically distributed. Some results of Large Deviations theory are used to select a set of asymptotically optimal test statistics so that their combination results in a suitable detection procedure. The Johnson system of distributions has been used as the basis for the design of the detector. The fact that the good asymptotical properties can be extended to finite samples with small sizes provides the motivation for the rest of the work. In this respect, the performance of the proposed procedure is also compared with that of some known Gaussianity tests. The possibility to directly calculate a reliable measure of support such as the Bayesian posterior probabilities, as opposed to P-values, is an additional advantage. Furthermore, it will be shown how the proposed procedure can provide some extra information concerning the type of deviation from Gaussianity, which is present in the data.