Spectral Analysis of Gram Matrices with Missing at Random Observations: Convergence, Central Limit Theorems, and Applications in Statistical Inference

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 $\text{mb}{S}_n=\frac{1}{n}\text{mb}Z\text{mb}{Z}^{\ast }$, where $\text{mb}Z=\text{mb}D\circ (\text{mb}X)$ represents a Hadamard random matrix with entries determined by independent Bernoulli variables $\text{mb}D$. Operating within the high-dimensional framework, we establish the convergence of the empirical spectral distribution of $\text{mb}{S}_n$ 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 ${\mathbf{S}}_n$. We establish the central limit theorem for the linear spectral statistics of ${\mathbf{S}}_n$, 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.