Testing multivariate normality by zeros of the harmonic oscillator in characteristic function spaces

TL;DR

This paper introduces a new class of affine invariant, consistent tests for multivariate normality based on a PDE characterization involving the harmonic oscillator, with proven asymptotic properties and strong finite-sample power.

Contribution

It proposes a novel normality test using PDE characterization linked to the harmonic oscillator, extending the toolkit for multivariate normality testing.

Findings

Tests are affine invariant and consistent.

Asymptotic distribution derived under null and alternative hypotheses.

Strong finite-sample power demonstrated on real data.

Abstract

We study a novel class of affine invariant and consistent tests for normality in any dimension. 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 Schr\"odinger 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.