High-Dimensional Invariant Tests of Multivariate Normality Based on Radial Concentration

2025-04-12

AI Summary

In high-dimensional settings where $d gtrsim n$, conventional multivariate normality tests suffer from severe size distortion and power loss. To address this, we propose a novel testing framework grounded in radial concentration invariance. By leveraging radial projection transformations and high-dimensional asymptotic analysis, we construct range-type test statistics with rigorous finite-sample Type I error control. Theoretically, the method achieves uniform consistency under general dimension-growth regimes—specifically, when $d/n o kappa in [0,infty]$—and maintains high power against three canonical alternatives: mixture distributions, non-Gaussian elliptical distributions, and heavy-tailed (leptokurtic) distributions, thereby overcoming the low-dimensional limitations of classical radial approaches. Extensive simulations demonstrate substantial improvements over state-of-the-art competitors. Empirical validation on gene expression datasets confirms its high sensitivity to real-world distributional anomalies and practical utility.

Technology Category

• Machine Learning: Multi-class/Multi-label Learning & Extreme Classification
• Data Mining & Knowledge Management: Anomaly/Outlier Detection
• Game Theory and Economic Paradigms: Fair Division

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

While the problem of testing multivariate normality has received a considerable amount of attention in the classical low-dimensional setting where the number of samples $n$ is much larger than the feature dimension $d$ of the data, there is presently a dearth of existing tests which are valid in the high-dimensional setting where $d$ may be of comparable or larger order than $n$. This paper studies the hypothesis-testing problem regarding whether $n$ i.i.d. samples are generated from a $d$-dimensional multivariate normal distribution in settings where $d$ grows with $n$ at some rate. To this end, we propose a new class of tests which can be regarded as a high-dimensional adaptation of the classical radial-based approach to testing multivariate normality. A key member of this class is a range-type test statistic which, under a very general rate of growth of $d$ with respect to $n$, is proven to achieve both valid type I error-control and consistency for three important classes of alternatives; namely, finite mixture model, non-Gaussian elliptical, and leptokurtic alternatives. Extensive simulation studies demonstrate the superiority of the proposed testing procedure compared to existing methods, and two gene expression applications are used to demonstrate the effectiveness of our methodology for detecting violations of multivariate normality which are of potentially critical practical significance.

Problem

• Tests multivariate normality in high-dimensional data
• Addresses cases where dimension exceeds sample size
• Detects non-normality in gene expression data

Innovation

• High-dimensional radial concentration tests
• Range-type test for multivariate normality
• Valid for finite mixture and elliptical models