Performance bounds for sparse parametric covariance estimation in Gaussian models

We consider estimation of a sparse parameter vector that determines the covariance matrix of a Gaussian random vector via a sparse expansion into known “basis matrices.” Using the theory of reproducing kernel Hilbert spaces, we derive lower bounds on the variance of estimators with a given mean function. This includes unbiased estimation as a special case. We also present a numerical comparison of our lower bounds with the variance of two standard estimators (hard-thresholding estimator and maximum likelihood estimator).