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

Abstract

Let X 1 ,X 2 ,… be independent and identically distributed random elements taking values in a separable Hilbert space H. With applications for functional data in mind, 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 nondegenerate Gaussian distribution. The test statistic T n = T n (X 1 ,… ,X n ) is based on a measure of deviation between the empirical characteristic functional of X 1 ,… ,X n and the characteristic functional of a suitable Gaussian random element of H. We derive the asymptotic distribution of T n as n→∞ under H 0 and provide a consistent bootstrap approximation thereof. Moreover, we obtain an almost sure limit of T n and the limit distributions of T n under fixed and contiguous alternatives to Gaussianity. Simulations show that the new test is competitive with respect to the hitherto few competitors available.

Statistical Depth Meets Machine Learning: Kernel Mean Embeddings and Depth in Functional Data Analysis

“…The key link is that if k(x, y) = µ(x−y) for some µ ∈ P(X ) then the representation (14) shows that MMD, and hence the MMD-based tests, can be interpreted in terms of a distance between characteristic functions. Therefore, tests based on the empirical approximations of characteristic functions, such as those presented in [48] and [43], are in fact MMD-based tests. We will discuss two examples, one for two-sample testing for functional data in Section 6.1 and one for testing Gaussianity of random functions in Section 6.2.…”

“…A test for Gaussianity of functional data performed in [43] uses empirical characteristic functions. That may be viewed as an MMD-based test of normality studied in [51].…”

“…∼ P for some P ∈ P(X ) and we want to test whether P is Gaussian. The following test statistic was proposed in [43,Equation (4)]…”

“…k(X i , y)dN mn,Σn (y) + Using the observed link to MMD means that an empirical average to approximate the integral in (31) is not required. Such an empirical approximation was used in the numerical experiments in [43], introducing an unnecessary extra computational cost. In summary, the test for Gaussianity of functional data based on the empirical characteristic function presented in [43] coincides with the MMD-based tests for normality in [51] when using an SE-C 1/2 kernel.…”

On combining the zero bias transform and the empirical characteristic function to test normality

“…For a survey of classical methods see del Barrio et al (2000), section 3, and Henze (1994), and for comparative simulation studies, see Baringhaus et al (1989); Farrell and Rogers-Stewart (2006); Landry and Lepage (1992); Pearson et al (1977); Romão et al (2010); Shapiro et al (1968); Yap and Sim (2011). For a survey on tests of multivariate normality see Henze (2002), for recent multivariate tests see , and for new developments on normality tests for Hilbert space valued random elements, see Henze and Jiménez-Gamero (2021); Kellner and Celisse (2019).…”