Sparse covariance matrix estimation in high-dimensional deconvolution

2017/10/30 by Belomestny, Denis, Trabs, Mathias, Tsybakov, Alexandre B.
#62G05 #FOS: Computer and information sciences #FOS: Mathematics #Methodology (stat.ME) #Primary 62H12 #Statistics Theory (math.ST) #secondary 62F12

paper · doi:10.48550/arxiv.1710.10870

Abstract

We study the estimation of the covariance matrix Σ of a p-dimensional normal random vector based on n independent observations corrupted by additive noise. Only a general nonparametric assumption is imposed on the distribution of the noise without any sparsity constraint on its covariance matrix. In this high-dimensional semiparametric deconvolution problem, we propose spectral thresholding estimators that are adaptive to the sparsity of Σ. We establish an oracle inequality for these estimators under model miss-specification and derive non-asymptotic minimax convergence rates that are shown to be logarithmic in n/log p. We also discuss the estimation of low-rank matrices based on indirect observations as well as the generalization to elliptical distributions. The finite sample performance of the threshold estimators is illustrated in a numerical example.

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