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Asymptotic distributions of four linear hypotheses test statistics under generalized spiked model

Zhijun Liu, Jiang Hu, Zhidong Bai, Zhihui Lv

TL;DR

The paper advances high-dimensional hypothesis testing by establishing a central limit theorem for linear spectral statistics under a generalized spiked covariance model, allowing spiked eigenvalues to be either bounded or diverging. It then derives asymptotic distributions for four classical linear-test statistics—Wilks' $U$, Lawley-Hotelling $W$, Bartlett-Nanda-Pillai $V$, and Roy's largest root $R$ (the latter used for comparison)—using Taylor expansions to handle complex test functions. The results are complemented by explicit asymptotic means and variances that incorporate spike structure and bulk behavior, as well as asymptotic power functions illustrating when each test is most powerful relative to Roy's root. Numerical simulations validate the theoretical approximations under several covariance models and distributions, highlighting the practical relevance for high-dimensional inference in applications with possibly unbounded population spectra.

Abstract

In this paper, we establish the Central Limit Theorem (CLT) for linear spectral statistics (LSSs) of large-dimensional generalized spiked sample covariance matrices, where the spiked eigenvalues may be either bounded or diverge to infinity. Building upon this theorem, we derive the asymptotic distributions of linear hypothesis test statistics under the generalized spiked model, including Wilks' likelihood ratio test statistic U, the Lawley-Hotelling trace test statistic W, and the Bartlett-Nanda-Pillai trace test statistic V. Due to the complexity of the test functions, explicit solutions for the contour integrals in our calculations are generally intractable. To address this, we employ Taylor series expansions to approximate the theoretical results in the asymptotic regime. We also derive asymptotic power functions for three test criteria above, and make comparisons with Roy's largest root test under specific scenarios. Finally, numerical simulations are conducted to validate the accuracy of our asymptotic approximations.

Asymptotic distributions of four linear hypotheses test statistics under generalized spiked model

TL;DR

The paper advances high-dimensional hypothesis testing by establishing a central limit theorem for linear spectral statistics under a generalized spiked covariance model, allowing spiked eigenvalues to be either bounded or diverging. It then derives asymptotic distributions for four classical linear-test statistics—Wilks' , Lawley-Hotelling , Bartlett-Nanda-Pillai , and Roy's largest root (the latter used for comparison)—using Taylor expansions to handle complex test functions. The results are complemented by explicit asymptotic means and variances that incorporate spike structure and bulk behavior, as well as asymptotic power functions illustrating when each test is most powerful relative to Roy's root. Numerical simulations validate the theoretical approximations under several covariance models and distributions, highlighting the practical relevance for high-dimensional inference in applications with possibly unbounded population spectra.

Abstract

In this paper, we establish the Central Limit Theorem (CLT) for linear spectral statistics (LSSs) of large-dimensional generalized spiked sample covariance matrices, where the spiked eigenvalues may be either bounded or diverge to infinity. Building upon this theorem, we derive the asymptotic distributions of linear hypothesis test statistics under the generalized spiked model, including Wilks' likelihood ratio test statistic U, the Lawley-Hotelling trace test statistic W, and the Bartlett-Nanda-Pillai trace test statistic V. Due to the complexity of the test functions, explicit solutions for the contour integrals in our calculations are generally intractable. To address this, we employ Taylor series expansions to approximate the theoretical results in the asymptotic regime. We also derive asymptotic power functions for three test criteria above, and make comparisons with Roy's largest root test under specific scenarios. Finally, numerical simulations are conducted to validate the accuracy of our asymptotic approximations.

Paper Structure

This paper contains 17 sections, 11 theorems, 104 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model
  3. Main results
  4. Asymptotic results for test statistics $U,W,V$
  5. Power analysis
  6. Numerical studies
  7. Technical proofs
  8. Preliminary results
  9. Proof of Theorem \ref{['U']}
  10. Proof of Corollary \ref{['Ucase1cor']}
  11. Proof of Theorem \ref{['W']}
  12. Proof of Corollary \ref{['Wcase1cor']}
  13. Proof of Theorem \ref{['V']}
  14. Proof of Corollary \ref{['Vcase1cor']}
  15. Proof of Theorem \ref{['thm1']}
  16. ...and 2 more sections

Key Result

Theorem 3.1

Under Assumptions ass1 and ass2 with $c_{n}=p/n\rightarrow c\in\left(0,1 \right)$, we have under $H_{1}$ in (hyapp), where

Figures (6)

  • Figure 1: Comparisons between empirical distributions of statistics $U$ and standard normal curves under Models 1--4, respectively, when samples are from $Dt_1$
  • Figure 2: Comparisons between empirical distributions of statistics $W$ and standard normal curves under Models 1--4, respectively, when samples are from $Dt_1$
  • Figure 3: Comparisons between empirical distributions of statistics $V$ and standard normal curves under Models 1--4, respectively, when samples are from $Dt_1$
  • Figure 4: Comparisons between empirical distributions of statistics $U$ and standard normal curves under Models 1--4, respectively, when samples are from $Dt_2$
  • Figure 5: Comparisons between empirical distributions of statistics $W$ and standard normal curves under Models 1--4, respectively, when samples are from $Dt_2$
  • ...and 1 more figures

Theorems & Definitions (28)

  • Theorem 3.1: U statistics
  • Theorem 3.2: W statistics
  • Theorem 3.3: V statistic
  • Remark 3.1
  • Corollary 3.1: Power function of CUT
  • Corollary 3.2: Power function of CWT
  • Corollary 3.3: Power function of CVT
  • Remark 3.2
  • Lemma 3.1: Theorem 2.7 in DingY18N
  • Lemma 3.2: Theorem 4.5 in Liu23
  • ...and 18 more