Model selection and sharp asymptotic minimaxity

Abstract

We obtain sharp minimax results for estimation of an n-dimensional normal mean under quadratic loss. The estimators are chosen by penalized least squares with a penalty that grows like ck log(n/k), for k equal to the number of nonzero elements in the estimating vector. For a wide range of sparse parameter spaces, we show that the penalized estimator achieves the exact minimax rate with the correct multiplication constant if and only if c equals 2. Our results unify the theory obtained by many other authors for penalized estimation of normal means. In particular we establish that a conjecture by Abramovich et al. (Ann Stat 34:584–653, 2006) is true.