Thinning a Wishart Random Matrix

• URL: http://arxiv.org/abs/2502.09957v1
• Date: Fri, 14 Feb 2025 07:34:38 GMT
• Title: Thinning a Wishart Random Matrix
• Authors: Ameer Dharamshi, Anna Neufeld, Lucy L. Gao, Daniela Witten, Jacob Bien,
• Abstract summary: We show that it is possible to generate two independent data matrices with independent $N_p(mu, Sigma)$ rows. These independent data matrices can either be used directly within a train-test paradigm, or can be used to derive independent summary statistics.
• Score: 3.734088413551237
• License:
• Abstract: Recent work has explored data thinning, a generalization of sample splitting that involves decomposing a (possibly matrix-valued) random variable into independent components. In the special case of a $n \times p$ random matrix with independent and identically distributed $N_p(\mu, \Sigma)$ rows, Dharamshi et al. (2024a) provides a comprehensive analysis of the settings in which thinning is or is not possible: briefly, if $\Sigma$ is unknown, then one can thin provided that $n>1$. However, in some situations a data analyst may not have direct access to the data itself. For example, to preserve individuals' privacy, a data bank may provide only summary statistics such as the sample mean and sample covariance matrix. While the sample mean follows a Gaussian distribution, the sample covariance follows (up to scaling) a Wishart distribution, for which no thinning strategies have yet been proposed. In this note, we fill this gap: we show that it is possible to generate two independent data matrices with independent $N_p(\mu, \Sigma)$ rows, based only on the sample mean and sample covariance matrix. These independent data matrices can either be used directly within a train-test paradigm, or can be used to derive independent summary statistics. Furthermore, they can be recombined to yield the original sample mean and sample covariance.

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