Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Testing Elliptical Models in High Dimensions

Siyao Wang, Miles E. Lopes

TL;DR

This work develops a high-dimensional goodness-of-fit test for elliptical distributions by leveraging a coordinate-wise kurtosis identity to construct two estimators, \tilde{\kappa} and \check{\kappa}, whose discrepancy signals departures from ellipticity. A central limit theorem is established for the test statistic T_n, with a fully data-driven, ratio-consistent variance estimator \widehat{\sigma}_n^2 that accommodates an unrestricted covariance matrix. The theory hinges on a novel adaptation of Isserlis' theorem to elliptical vectors and a careful analysis of high-order moments, including a U-statistics treatment for one estimator. Numerical experiments show accurate level control and superior power relative to a state-of-the-art high-dimensional normality test, complemented by real-data demonstrations in finance and genomics. The approach provides a theoretically grounded, practical tool for validating elliptical models in settings where p grows with n.

Abstract

Due to the broad applications of elliptical models, there is a long line of research on goodness-of-fit tests for empirically validating them. However, the existing literature on this topic is generally confined to low-dimensional settings, and to the best of our knowledge, there are no established goodness-of-fit tests for elliptical models that are supported by theoretical guarantees in high dimensions. In this paper, we propose a new goodness-of-fit test for this problem, and our main result shows that the test is asymptotically valid when the dimension and sample size diverge proportionally. Remarkably, it also turns out that the asymptotic validity of the test requires no assumptions on the population covariance matrix. With regard to numerical performance, we confirm that the empirical level of the test is close to the nominal level across a range of conditions, and that the test is able to reliably detect non-elliptical distributions. Moreover, when the proposed test is specialized to the problem of testing normality in high dimensions, we show that it compares favorably with a state-of-the-art method, and hence, this way of using the proposed test is of independent interest.

Testing Elliptical Models in High Dimensions

TL;DR

This work develops a high-dimensional goodness-of-fit test for elliptical distributions by leveraging a coordinate-wise kurtosis identity to construct two estimators, \tilde{\kappa} and \check{\kappa}, whose discrepancy signals departures from ellipticity. A central limit theorem is established for the test statistic T_n, with a fully data-driven, ratio-consistent variance estimator \widehat{\sigma}_n^2 that accommodates an unrestricted covariance matrix. The theory hinges on a novel adaptation of Isserlis' theorem to elliptical vectors and a careful analysis of high-order moments, including a U-statistics treatment for one estimator. Numerical experiments show accurate level control and superior power relative to a state-of-the-art high-dimensional normality test, complemented by real-data demonstrations in finance and genomics. The approach provides a theoretically grounded, practical tool for validating elliptical models in settings where p grows with n.

Abstract

Due to the broad applications of elliptical models, there is a long line of research on goodness-of-fit tests for empirically validating them. However, the existing literature on this topic is generally confined to low-dimensional settings, and to the best of our knowledge, there are no established goodness-of-fit tests for elliptical models that are supported by theoretical guarantees in high dimensions. In this paper, we propose a new goodness-of-fit test for this problem, and our main result shows that the test is asymptotically valid when the dimension and sample size diverge proportionally. Remarkably, it also turns out that the asymptotic validity of the test requires no assumptions on the population covariance matrix. With regard to numerical performance, we confirm that the empirical level of the test is close to the nominal level across a range of conditions, and that the test is able to reliably detect non-elliptical distributions. Moreover, when the proposed test is specialized to the problem of testing normality in high dimensions, we show that it compares favorably with a state-of-the-art method, and hence, this way of using the proposed test is of independent interest.
Paper Structure (21 sections, 31 theorems, 235 equations, 7 figures, 4 tables)

This paper contains 21 sections, 31 theorems, 235 equations, 7 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Method
  3. Estimating the variance of $T_n$
  4. Theory
  5. Numerical experiments
  6. Assessment of level
  7. Assessment of power
  8. Stock market data
  9. Breast cancer data
  10. Preliminaries
  11. Asymptotic normality of $T_n$
  12. Asymptotic normality of $T_{n,1}$
  13. Negligible terms associated with $T_{n,1}$
  14. Main terms of $T_{n,1}$
  15. Asymptotic normality of $T_{n,2}$
  16. ...and 6 more sections

Key Result

Theorem 1

If Assumption Data generating model holds, then as $n\to\infty$ we have and hence, for any fixed $\alpha\in (0,1)$, the level of the test satisfies

Figures (7)

  • Figure 1: Results for $y_{11}\sim \frac{\textup{Laplace}(0,1)}{\sqrt{2}}$ when $p/n = 0.5$ and $p=200$.
  • Figure 2: Results for $y_{11}\sim \frac{\textup{Beta}(2,1.5) - 4/7}{\sqrt{8/147}}$ when $p/n = 0.5$ and $p=200$.
  • Figure 3: Results for $y_{11}\sim \frac{\textup{Laplace}(0,1)}{\sqrt{2}}$ when $p/n = 1$ and $p = 400$.
  • Figure 4: Results for $y_{11}\sim \frac{\textup{Beta}(2,1.5) - 4/7}{\sqrt{8/147}}$ when $p/n = 1$ and $p=400$.
  • Figure 5: Results for $y_{11}\sim \frac{\textup{Laplace}(0,1)}{\sqrt{2}}$ when $p/n = 1.5$ and $p=600$.
  • ...and 2 more figures

Theorems & Definitions (31)

  • Theorem 1
  • Proposition 1
  • Proposition 2
  • Lemma B.1
  • Lemma B.2
  • Lemma B.3
  • Lemma B.4
  • Lemma B.5
  • Proposition 3
  • Lemma B.6
  • ...and 21 more