Exponential dimensional dependence in high-dimensional Hermite method of moments

TL;DR

This paper rigorously proves that Hermite-based moment tests for Gaussianity become exponentially unreliable as the highest moment order increases, requiring exponentially larger sample sizes for accuracy.

Contribution

It provides the first rigorous exponential bounds for Hermite-based moment tests, clarifying their limitations at high moment orders.

Findings

Sample size must grow exponentially with moment order d

Bounds apply to classical tests like Shenton-Bowman

Results hold under both convex and Kolmogorov-Smirnov distances

Abstract

It is numerically well known that moment-based tests for Gaussianity and estimators become increasingly unreliable at higher moment orders; however, this phenomenon has lacked rigorous mathematical justification. In this work, we establish quantitative bounds for Hermite-based moment tests, with matching exponential upper and lower bounds. Our results show that, even under ideal conditions with i.i.d. standard normal data, the sample size must grow exponentially with the highest moment order $d$ used in the test. These bounds, derived under both the convex distance and the Kolmogorov-Smirnov distance, are applied to classical procedures, such as the Shenton-Bowman test.