A test for Gaussianity in Hilbert spaces via the empirical characteristic functional

Abstract

AbstractLet X1,X2,… be independent and identically distributed random elements taking values in a separable Hilbert space . With applications for functional data in mind, may be regarded as a space of square‐integrable functions, defined on a compact interval. We propose and study a novel test of the hypothesis H0 that X1 has some unspecified nondegenerate Gaussian distribution. The test statistic Tn = Tn(X1,…,Xn) is based on a measure of deviation between the empirical characteristic functional of X1,…,Xn and the characteristic functional of a suitable Gaussian random element of . We derive the asymptotic distribution of Tn as n→∞ under H0 and provide a consistent bootstrap approximation thereof. Moreover, we obtain an almost sure limit of Tn and the limit distributions of Tn under fixed and contiguous alternatives to Gaussianity. Simulations show that the new test is competitive with respect to the hitherto few competitors available.