A test for Gaussianity in Hilbert spaces via theempirical characteristic functional

Abstract

LetX1,X2,...be independent and identically distributedrandom elements taking values in a separable HilbertspaceH. With applications for functional data in mind,Hmay be regarded as a space of square-integrable func-tions, defined on a compact interval. We propose andstudy a novel test of the hypothesisH0thatX1has someunspecified nondegenerate Gaussian distribution. Thetest statisticTn=Tn(X1,...,Xn) is based on a measureof deviation between the empirical characteristic func-tional ofX1,...,Xnand the characteristic functional ofa suitable Gaussian random element ofH.Wederivethe asymptotic distribution ofTnasn→∞underH0andprovide a consistent bootstrap approximation thereof.Moreover, we obtain an almost sure limit ofTnandthe limit distributions ofTnunder fixed and contiguousalternatives to Gaussianity. Simulations show that thenew test is competitive with respect to the hitherto fewcompetitors available.