On Minimax Estimation of a Sparse Normal Mean Vector

Abstract

Mallows has conjectured that among distributions which are Gaussian but for occasional contamination by additive noise, the one having least Fisher information has (two-sided) geometric contamination. A very similar problem arises in estimation of a nonnegative vector parameter in Gaussian white noise when it is known also that most [i.e., $(1 - \varepsilon)$] components are zero. We provide a partial asymptotic expansion of the minimax risk as $\varepsilon \rightarrow 0$. While the conjecture seems unlikely to be exactly true for finite $\varepsilon$, we verify it asymptotically up to the accuracy of the expansion. Numerical work suggests the expansion is accurate for $\varepsilon$ as large as 0.05. The best $l_1$-estimation rule is first- but not second-order minimax. The results bear on an earlier study of maximum entropy estimation and various questions in robustness and function estimation using wavelet bases.