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

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 standardd-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 Schrodinger 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.

Automated summary

A novel class of affine invariant and consistent tests for normality has been studied, showing consistency against general alternatives, strong power performance for finite samples, and successful application to classical datasets. The results are also useful for model validation procedures.