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Testing multivariate normality by zeros of the harmonic oscillator in characteristic function spaces
Author(s) -
Dörr Philip,
Ebner Bruno,
Henze Norbert
Publication year - 2021
Publication title -
scandinavian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.359
H-Index - 65
eISSN - 1467-9469
pISSN - 0303-6898
DOI - 10.1111/sjos.12477
Subject(s) - mathematics , asymptotic distribution , normality , invariant (physics) , normality test , local asymptotic normality , affine transformation , harmonic oscillator , multivariate normal distribution , statistics , mathematical analysis , statistical hypothesis testing , multivariate statistics , pure mathematics , estimator , physics , quantum mechanics , mathematical physics
Abstract We study a novel class of affine invariant and consistent tests for normality in any dimension in an i.i.d.‐setting. The tests are based on a characterization of the standard d ‐variate normal distribution as the unique solution of an initial value problem of a partial differential equation motivated by the harmonic oscillator, which is a special case of a Schrödinger operator. We derive the asymptotic distribution of the test statistics under the hypothesis of normality as well as under fixed and contiguous alternatives. The tests are consistent against general alternatives, exhibit strong power performance for finite samples, and they are applied to a classical data set due to R.A. Fisher. The results can also be used for a neighborhood‐of‐model validation procedure.