Positive-definite thresholding estimators of covariance matrices with zeros

Abstract

A positive definite estimator of a covariance matrix with zero entries provides a valid covariance matrix that can be used an input in almost any area of multivariate statistical analysis. However, most current approaches do not yet guarantee positive definiteness or deal with the asymptotic efficiency of the covariance estimator. Focusing on the classical setting when the number of Gaussian variables is fixed and the sample size increases, we construct a positive definite and asymptotically efficient estimator by the iterative conditional fitting algorithm (Chaudhuri et al., 2007) when the location of the zero entries is known. If the location of the zero entries is unknown, we further construct a positive definite thresholding estimator by combining the iterative conditional fitting algorithm with thresholding. We prove our thresholding estimator is asymptotically efficient with probability tending to one. In simulation studies, we show our estimator more closely matches the true covariance and more correctly identifies the non-zero entries than competing estimators. We apply our estimator to a neuroimaging study of Huntington disease to detect non-zero correlations among brain regional volumes. Such correlations are timely for ongoing treatment studies to inform how different brain regions are likely to be affected by these treatments.