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.

• The paper's main contribution is the derivation of computable Edgeworth expansions with explicit error bounds for high-dimensional sphericity testing under incomplete data.
• It employs cumulant calculations and Monte Carlo simulations to demonstrate improved finite-sample accuracy over classical asymptotic methods.
• The methodology provides a practical framework for rigorously controlling test size in multivariate analyses with missing data.

Computable Error Bounds for High-Dimensional Edgeworth Expansions in Sphericity Testing with Two-Step Monotone Incomplete Data

Introduction

This work undertakes a rigorous analysis of the sphericity test for a single multivariate normal sample under high-dimensional, two-step monotone incomplete data. The null hypothesis, $H_0: \Sigma = \sigma^2 I_p$, with alternative $H_1: \Sigma \neq \sigma^2 I_p$, is foundational in multivariate analysis, with implications for methods such as PCA and tests of independence. In high-dimensional settings and presence of incomplete data, traditional large-sample asymptotic results for the likelihood ratio test (LRT) under $H_0$ are inadequate, exhibiting significant inaccuracies in both test level and power.

The principal contribution here is the derivation of Edgeworth expansions for the null distribution of the LRT statistic under two-step monotone incomplete data, with computable explicit error bounds. Edgeworth expansions provide higher-order correction terms to asymptotic approximations, which are crucial for finite-sample accuracy in high dimensions. Theoretical findings are supported by Monte Carlo simulations that systematically compare Edgeworth expansions against conventional asymptotic results, demonstrating significantly improved accuracy, especially with increasing $p$.

High-Dimensional LRT under Two-Step Monotone Incomplete Data

The two-step monotone missing framework considered splits the data into $N_1$ samples observed on all $p$ variables and $N_2$ additional samples observed only on a subset $p_1 < p$, with $N = N_1 + N_2$, $p = p_1 + p_2$. This is a highly relevant scenario for real-world high-dimensional datasets.

The authors generalize the LRT for sphericity to this setting, carefully deriving MLE of the covariance blocks and reduced forms for the test statistic. Previous expansions, e.g., John’s test and its corrections, or first-order chi-square approximations as in Mauchly and Gleser, prove empirically deficient as $H_1: \Sigma \neq \sigma^2 I_p$0 increases. The deficiencies are quantitatively confirmed via simulations, particularly in maintaining test size.

Edgeworth Expansion Derivation and Properties

Edgeworth expansion for the null distribution of $H_1: \Sigma \neq \sigma^2 I_p$1 is developed via precise calculation of cumulants to arbitrarily high order, leveraging properties and expansions of the multivariate gamma function and polygamma functions associated with the LRT's distribution. For the standardized statistic $H_1: \Sigma \neq \sigma^2 I_p$2, cumulants $H_1: \Sigma \neq \sigma^2 I_p$3 are explicitly bounded in closed form, enabling rigorous control of the remainder in the Edgeworth series.

The expansion is

$H_1: \Sigma \neq \sigma^2 I_p$4

where $H_1: \Sigma \neq \sigma^2 I_p$5 are functions of standardized cumulants and $H_1: \Sigma \neq \sigma^2 I_p$6 is the Hermite polynomial of order $H_1: \Sigma \neq \sigma^2 I_p$7.

Bounding arguments show, under regularity ($H_1: \Sigma \neq \sigma^2 I_p$8), all standardized cumulants decay with rate $H_1: \Sigma \neq \sigma^2 I_p$9, where $H_0$0 grows with $H_0$1 (sample size for the complete part) and $H_0$2. This guarantees the asymptotic validity of the Edgeworth expansion, with remaining error shrinking polynomially.

Explicit Computable Error Bounds

An outstanding feature of this work is the explicit computable upper bounds for the error of the Edgeworth expansion, derived by analytic estimation of the Fourier inversion integral separating the characteristic functions of the exact and expanded statistics.

The bounds are constructed as

$H_0$3

where $H_0$4, $H_0$5, $H_0$6 are prescribed in terms of the problem parameters and numerical quadratures, and the minimization is numerically tractable over a grid of $H_0$7.

Multiple upper bounds (BOUND1–BOUND4) are presented, differing in the degree of numerical and analytic tightness or simplifications adopted, all convergent as $H_0$8 when regularity conditions are satisfied.

Numerical Experiments

Comprehensive Monte Carlo analysis is performed with $H_0$9 replicates per configuration, systematically varying $p$0. For test significance levels $p$1, and over a wide range of high-dimensional and high-missing setups, the Edgeworth approximations yield Type I error biases ($p$2) systematically near zero, while classical (systematic) expansions ($p$3) show marked deviations that grow with $p$4.

Figure 1: Box plot of the biases $p$5 and $p$6, demonstrating the uniform superiority of the Edgeworth expansion in controlling the error across simulation scenarios.

This improved accuracy is maintained as either the proportion of missingness or dimensionality increases. For visual clarity, line graphs of $p$7 and $p$8 traced over varying $p$9 are provided.

Figure 2: Line graph of $N_1$0 and $N_1$1 for fixed $N_1$2 and varying $N_1$3, emphasizing growing bias in classical approximations and persistent accuracy of the Edgeworth method.

Extensive tabulated results confirm that the maximum absolute error (MAE) between the empirical and Edgeworth approximation is always smaller than the provided error bounds ($N_1$4–$N_1$5), validating the non-asymptotic sharpness of the theoretical guarantees.

Theoretical and Practical Implications

This work establishes that Edgeworth expansions with rigorously computable bounds are uniquely suited to provide usable distributional approximations in the finite-sample, high-dimensional, incomplete-data sphericity testing context—where classical first- or second-order expansions systematically fail. The methodology makes it possible to rigorously control nominal test levels, enhancing reliability for high-throughput data or experimental designs with partially observed factors.

Error bounds provided are not only theoretically convergent but practical in actual analysis, computable with standard quadrature and optimization libraries. This enables their immediate application in clinical, genetic, or high-dimensional scientific studies where missing data is the norm.

The formal methods developed can be extended to several other high-dimensional hypothesis testing scenarios with complex data missingness, e.g., MANOVA, covariance structure testing, or multi-sample generalizations. The explicitness of the error bounds makes these results suitable as a template for deriving Cornish-Fisher expansions for critical values, offering another avenue for improvement in test-type control.

Conclusion

An exact Edgeworth expansion for the null distribution of the sphericity LRT under two-step monotone incomplete data has been derived, with non-asymptotic computable error bounds that are both theoretical and practical. Simulations evidence the superiority of these expansions over asymptotic alternatives across a full range of dimensional and missing data regimes. The ability to control and numerically evaluate approximation errors positions these results as highly relevant for modern high-dimensional multivariate analysis, providing a pathway for principled inference even when classical theory fails.