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

TL;DR

This paper introduces a new statistical test for Gaussianity in Hilbert spaces, based on the empirical characteristic functional, with proven asymptotic properties and demonstrated effectiveness through simulations.

Contribution

It develops a novel test for Gaussianity in Hilbert spaces using the empirical characteristic functional, including asymptotic analysis and bootstrap approximation.

Findings

Test statistic based on deviation of empirical and Gaussian characteristic functionals.

Asymptotic distribution derived under null hypothesis.

Simulation results show competitive performance.

Abstract

Let $X_1,X_2, \ldots$ be independent and identically distributed random elements taking values in a separable Hilbert space $\mathbb{H}$. With applications for functional data in mind, $\mathbb{H}$ 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 $H_0$ that $X_1$ has some unspecified non-degenerate Gaussian distribution. The test statistic $T_n=T_n(X_1,\ldots,X_n)$ is based on a measure of deviation between the empirical characteristic functional of $X_1,\ldots,X_n$ and the characteristic functional of a suitable Gaussian random element of $\mathbb{H}$. We derive the asymptotic distribution of $T_n$ as $n \to \infty$ under $H_0$ and provide a consistent bootstrap approximation thereof. Moreover, we obtain an almost sure limit of $T_n$ as well as a normal limit distribution of $T_n$ under alternatives to Gaussianity. Simulations show that the new test is competitive with respect to the hitherto few competitors available.