Spectral Analysis of Gram Matrices with Missing at Random Observations: Convergence, Central Limit Theorems, and Applications in Statistical Inference
Abstract
Motivated by the statistical inference using the Gram matrix in the context of missing at random observations, this paper investigates the spectral properties of the random matrices mb Sn=1/nmb Zmb Z*, where mb Z=mb D∘(Σ1/2mb X) represents a Hadamard random matrix with entries determined by independent Bernoulli variables mb D. Operating within the high-dimensional framework, we establish the convergence of the empirical spectral distribution of mb Sn to a well-defined limiting distribution. In addition, we explore the impact of the missing mechanism on the second-order properties of the spectral distribution of the Gram matrix Sn. We establish the central limit theorem for the linear spectral statistics of Sn, shedding light on their fluctuations. Surprisingly, our analysis reveals that even in the ideal Gaussian distribution scenario, the fluctuations of statistics generated by eigenvalues are influenced by the eigenvectors of the population covariance matrix in the missing-at-random case. This discovery uncovers a remarkable phenomenon that starkly contrasts with the classical case. Subsequently, we demonstrate the practical application of our central limit theorem in hypothesis testing for the population covariance matrix.
Versions
Related
- Statistical inference and applications of Kumaraswamy Odd Perks power Lomax distribution
- Statistical inference for low-rank tensors: Heteroskedasticity, subgaussianity, and applications
- Statistical Testing for Multiple Instance Learning via Selective Inference with Applications to Computational Pathology
- The Wrapped Q-Lindley Distribution: A New Circular Model with Statistical Properties, Classical and Bayesian Inference, and Applications to Directional Data
- Semiparametrically efficient semi-supervised statistical inference
BibTeX
@misc{li2026spectral,
title = {{Spectral Analysis of Gram Matrices with Missing at Random Observations: Convergence, Central Limit Theorems, and Applications in Statistical Inference}},
author = {Huiqin Li and Guangming Pan and Yanqing Yin and Wang Zhou},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02024},
eprint = {2609.02024},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02024},
}
RIS
TY - UNPB
TI - Spectral Analysis of Gram Matrices with Missing at Random Observations: Convergence, Central Limit Theorems, and Applications in Statistical Inference
AU - Huiqin Li
AU - Guangming Pan
AU - Yanqing Yin
AU - Wang Zhou
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02024
UR - https://arxiv.org/abs/2609.02024
AB - Motivated by the statistical inference using the Gram matrix in the context of missing at random observations, this paper investigates the spectral properties of the random matrices mb S_n=1/nmb Zmb Z^*, where mb Z=mb D∘(Σ^1/2mb X) represents a Hadamard random matrix with entries determined by independent Bernoulli variables mb D. Operating within the high-dimensional framework, we establish the convergence of the empirical spectral distribution of mb S_n to a well-defined limiting distribution. In addition, we explore the impact of the missing mechanism on the second-order properties of the spectral distribution of the Gram matrix S_n. We establish the central limit theorem for the linear spectral statistics of S_n, shedding light on their fluctuations. Surprisingly, our analysis reveals that even in the ideal Gaussian distribution scenario, the fluctuations of statistics generated by eigenvalues are influenced by the eigenvectors of the population covariance matrix in the missing-at-random case. This discovery uncovers a remarkable phenomenon that starkly contrasts with the classical case. Subsequently, we demonstrate the practical application of our central limit theorem in hypothesis testing for the population covariance matrix.
ER -