High-Dimensional Data Analysis for Elliptically Symmetric Distributions

2026-04-15

AI Summary

High-dimensional data often exhibit heavy tails, heterogeneity, and non-Gaussian characteristics, which undermine the validity of conventional covariance-based methods. This work proposes a unified robust inference framework tailored to elliptically symmetric distributions, integrating techniques such as spatial signs, spatial ranks, multivariate Kendall’s tau matrices, shape matrix estimation, and adaptive testing. By relaxing the restrictive Gaussian assumption, the framework achieves robust and efficient performance across a range of high-dimensional tasks—including location inference, covariance and precision matrix estimation, factor model testing, discriminant analysis, and dimension reduction—thereby substantially enhancing adaptability to heavy-tailed high-dimensional data.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Intelligent Robots: State Estimation
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• 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
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

High-dimensional data arise routinely in modern statistics, econometrics, finance, genomics, and machine learning. While a large body of existing methodology is developed under Gaussian or light-tailed assumptions, many real data sets exhibit heavy tails, heterogeneity, and departures from classical covariance-based models. This book provides a systematic treatment of high-dimensional data analysis under elliptically symmetric distributions, with an emphasis on robust inference based on spatial signs, spatial ranks, multivariate Kendall's tau matrices, and related shape-based methods.The book covers the basic theory of elliptical symmetry, high-dimensional location inference, estimation and testing for covariance and precision matrices, sphericity and proportionality testing, high-dimensional alpha testing in factor pricing models, change-point analysis, white-noise and independence testing, high-dimensional discriminant analysis, and dimension reduction through principal component analysis and factor models. Throughout, we review classical low-dimensional and high-dimensional benchmark methods and then develop robust alternatives tailored to elliptical models. Particular attention is paid to the interplay between sum-type, max-type, and adaptive procedures, as well as to the role of scatter, shape, and rank-based dependence measures in heavy-tailed settings. This book is intended as a unified overview of robust high-dimensional methods under elliptical symmetry and as a synthesis of the author's recent research contributions in this area. It is written for researchers and graduate students in statistics, econometrics, and related fields who are interested in modern high-dimensional inference beyond the Gaussian paradigm.

Problem

• high-dimensional data
• elliptically symmetric distributions
• heavy tails
• robust inference
• non-Gaussian

Innovation

• elliptically symmetric distributions
• high-dimensional robust inference
• spatial signs and ranks
• multivariate Kendall's tau
• shape-based methods