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

2019/10/24 by Norbert Henze, Henze, Norbert, M.D. Jiménez–Gamero +1 · 1 citation
Computer Science · Decision Sciences · Mathematics · #62G20 #62H15 #Bayesian Methods and Mixture Models #FOS: Mathematics #Probability and Risk Models #Statistical Methods and Inference #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.1910.10924

openalex publication_date 2019/10/24 · openalex created_date 2022/07/28 · openalex updated_date 2026/07/28

Abstract

Let X₁,X₂, \… be independent and identically distributed random\nelements taking values in a separable Hilbert space \ℍ. With\napplications for functional data in mind, \ℍ may be regarded as a\nspace of square-integrable functions, defined on a compact interval. We propose\nand study a novel test of the hypothesis H₀ that X₁ has some unspecified\nnon-degenerate Gaussian distribution. The test statistic\nT_n=T_n(X₁,\…,X_n) is based on a measure of deviation between the\nempirical characteristic functional of X₁,\…,X_n and the characteristic\nfunctional of a suitable Gaussian random element of \ℍ. We derive the\nasymptotic distribution of T_n as n \→ \∞ under H₀ and provide a\nconsistent bootstrap approximation thereof. Moreover, we obtain an almost sure\nlimit of T_n as well as a normal limit distribution of T_n under\nalternatives to Gaussianity. Simulations show that the new test is competitive\nwith respect to the hitherto few competitors available.\n

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