Finite Dimensional Statistical Inference

TL;DR

This paper derives explicit eigenvalue distribution series for finite-dimensional Wishart models, extending free probability tools to finite cases and providing algorithms for moment computation, especially where asymptotic methods fail.

Contribution

It introduces finite-dimensional eigenvalue distribution expansions for Wishart models using extended free probability techniques and develops algorithms for moment calculation in finite settings.

Findings

Explicit series expansions for eigenvalue distributions of Wishart models.

Algorithms for computing moments in finite-dimensional cases.

Discussion of scenarios where asymptotic results are not applicable.

Abstract

In this paper, we derive the explicit series expansion of the eigenvalue distribution of various models, namely the case of non-central Wishart distributions, as well as correlated zero mean Wishart distributions. The tools used extend those of the free probability framework, which have been quite successful for high dimensional statistical inference (when the size of the matrices tends to infinity), also known as free deconvolution. This contribution focuses on the finite Gaussian case and proposes algorithmic methods to compute the moments. Cases where asymptotic results fail to apply are also discussed.