Testing for large-dimensional covariance matrix under differential privacy

Abstract: The increasing prevalence of high-dimensional data across various applications has raised significant privacy concerns in statistical inference. In this paper, we propose a differentially private integrated statistic for testing large-dimensional covariance structures, enabling accurate statistical insights while safeguarding privacy. First, we analyze the global sensitivity of sample eigenvalues for sub-Gaussian populations, where our method bypasses the commonly assumed boundedness of data covariates. For sufficiently large sample size, the privatized statistic guarantees privacy with high probability. Furthermore, when the ratio of dimension to sample size, $d/n \to y \in (0, \infty)$, the privatized test is asymptotically distribution-free with well-known critical values, and detects the local alternative hypotheses distinct from the null at the fastest rate of $1/\sqrt{n}$. Extensive numerical studies on synthetic and real data showcase the validity and powerfulness of our proposed method.