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Computable error bounds for high-dimensional Edgeworth expansions in sphericity testing under two-step monotone incomplete data

Tetsuya Sato, Tomoyuki Nakagawa

Abstract

In this paper, we consider the sphericity test for a one-sample problem under high-dimensional two-step monotone incomplete data. Existing asymptotic expansions for the null distributions of the likelihood ratio test (LRT) statistic and modified LRT statistic are inaccurate in high-dimensional settings. Therefore, we derive Edgeworth expansions for the null distribution of the LRT statistic in such settings and obtain computable error bounds. Furthermore, we demonstrate that our proposed Edgeworth expansions provide better approximation accuracy than the existing asymptotic expansions. We also conduct numerical experiments using Monte Carlo simulations to evaluate the maximum absolute error (MAE) between the distribution function of the standardized test statistic and Edgeworth expansions for the null distribution of the LRT statistic, as well as to assess the performance of the computable error bounds.

Computable error bounds for high-dimensional Edgeworth expansions in sphericity testing under two-step monotone incomplete data

Abstract

In this paper, we consider the sphericity test for a one-sample problem under high-dimensional two-step monotone incomplete data. Existing asymptotic expansions for the null distributions of the likelihood ratio test (LRT) statistic and modified LRT statistic are inaccurate in high-dimensional settings. Therefore, we derive Edgeworth expansions for the null distribution of the LRT statistic in such settings and obtain computable error bounds. Furthermore, we demonstrate that our proposed Edgeworth expansions provide better approximation accuracy than the existing asymptotic expansions. We also conduct numerical experiments using Monte Carlo simulations to evaluate the maximum absolute error (MAE) between the distribution function of the standardized test statistic and Edgeworth expansions for the null distribution of the LRT statistic, as well as to assess the performance of the computable error bounds.

Paper Structure

This paper contains 12 sections, 3 theorems, 99 equations, 2 figures, 9 tables.

Table of Contents

  1. Introduction
  2. Likelihood ratio test for sphericity
  3. Edgeworth expansions
  4. Error bounds
  5. Numerical experiments
  6. Comparison of approximation accuracy
  7. Evaluation of error bounds and maximum absolute error
  8. Conclusions
  9. Proof of Lemma \ref{['lem:1']}
  10. Proof of gamma kj = O(m)
  11. Proof of (A.13)
  12. Tables

Key Result

Theorem 1

When $\tau_1=N_1/N\to\delta_1\in(0,1]~(N_1\to\infty)$, asymptotic expansions for the null distributions of the LRT statistic and modified LRT statistic are given for large $N_1$ as where $G_k(x)$ is the distribution function of the $\chi^{2}$ distribution with $k$ degrees of freedom, $f=(p+2)(p-1)/2$, $M=\rho N$ and $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Box plot of the biases $B_{prop}$ and $B_{SYS}$. The red wavy line indicates the case where the bias is zero.
  • Figure 2: Line graph of the biases $B_{prop}$ and $B_{SYS}$ for $N_1=N_2=200$ and $p_1=p_2=2,20,40,80$. The red lines represent $B_{prop}$ and blue lines represent $B_{SYS}$. The solid, wavy and dotted lines indicate $\alpha=0.10,0.05$ and $0.01$, respectively.

Theorems & Definitions (3)

  • Theorem 1: Sato2025sphericity
  • Lemma 1
  • Theorem 2