We consider the classical problem of estimating a vector µ = (µ1,...,µn) based on independent observations Yi ∼ N(µi,1), i = 1,...,n. Suppose µi, i = 1,...,n are independent realizations from a completely unknown G. We suggest an easily computed estimator ˆµ, such that the ratio of its risk E(ˆµ − µ) 2 with that of the Bayes procedure approaches 1. A related compound decision result is also obtained. Our asymptotics is of a triangular array; that is, we allow the distribution G to depend on n. Thus, our theoretical asymptotic results are also meaningful in situations where the vector µ is sparse and the proportion of zero coordinates approaches 1. We demonstrate the performance of our estimator in simulations, emphasizing sparse setups. In “moderately-sparse ” situations, our procedure performs very well compared to known procedures tailored for sparse setups. It also adapts well to nonsparse situations.

We consider a joint processing of n independent similar sparse regression problems. Each is based on a sample (yi1, xi1)..., (yim, xim) of m i.i.d. observations from yi1 = x T i1 βi + εi1, yi1 ∈ R, xi1 ∈ R p, and εi1 ∼ N(0, σ 2), say. The dimension p is large enough so that the empirical risk minimizer is not feasible. We consider, from a Bayesian point of view, three possible extensions of the lasso. Each of the three estimators, the lassoes, the group lasso, and the RING lasso, utilizes different assumptions on the relation between the n vectors β1,..., βn. “... and only a star or two set sparsedly in the vault of heaven; and you will find a sight as stimulating as the hoariest summit of the Alps. ” R. L. Stevenson 1