Gaussian and Robust Kronecker Product Covariance Estimation: Existence\n and Uniqueness

2015/12/01 by Ilya Soloveychik, Dmitry Trushin, Soloveychik, Ilya +1 · 1 citation
Computer Science · Mathematics · #Advanced Statistical Methods and Models #Applications (stat.AP) #Bayesian Modeling and Causal Inference #FOS: Computer and information sciences #Statistical Methods and Bayesian Inference #Statistical Methods and Inference

paper · pdf · doi:10.48550/arxiv.1512.00336

openalex publication_date 2015/12/01 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

We study the Gaussian and robust covariance estimation, assuming the true\ncovariance matrix to be a Kronecker product of two lower dimensional square\nmatrices. In both settings we define the estimators as solutions to the\nconstrained maximum likelihood programs. In the robust case, we consider\nTyler's estimator defined as the maximum likelihood estimator of a certain\ndistribution on a sphere. We develop tight sufficient conditions for the\nexistence and uniqueness of the estimates and show that in the Gaussian\nscenario with the unknown mean, p/q+q/p + 2 samples are almost surely enough\nto guarantee the existence and uniqueness, where p and q are the dimensions\nof the Kronecker product factors. In the robust case with the known mean, the\ncorresponding sufficient number of samples is \max[p/q, q/p] + 1.\n

Citations

• A Distribution-Free M-Estimator of Multivariate Scatter
• Robust Estimation of a Location Parameter

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