# Testing for normality in any dimension based on a partial differential equation involving the moment generating function

## TL;DR

This paper introduces a new class of affine invariant tests for multivariate normality based on a PDE involving the moment generating function, demonstrating strong power and consistency.

## Contribution

It develops a novel testing method using PDEs for the moment generating function, applicable in any dimension, with proven null distribution and power against alternatives.

## Key findings

- Tests are affine invariant and consistent.
- Strong power against heavy-tailed distributions.
- Connection with multivariate skewness measures.

## Abstract

We use a system of first-order partial differential equations that characterize the moment generating function of the $d$-variate standard normal distribution to construct a class of affine invariant tests for normality in any dimension. We derive the limit null distribution of the resulting test statistics, and we prove consistency of the tests against general alternatives. In the case $d > 1$, a certain limit of these tests is connected with two measures of multivariate skewness. The new tests show strong power performance when compared to well-known competitors, especially against heavy-tailed distributions, and they are illustrated by means of a real data set.