Risk bounds for sparse sequences

This version includes some results on Besov spaces at the end! 1 The single sequence problem Consider the estimation of a sequence of means μ = (μ1, μ2, . . . , μn) given independent observations Xi ∼ N(μi, 1). Define ŵ to be the marginal maximum likelihood estimator of the parameter w based on X1, X2, . . . , Xn, defined subject to the constraint that the corresponding threshold t̂ of the posterior median thresholding function satisfies t̂ ≤ √ 2 log n. We now estimate the μi by using a prior distribution (1− ŵ)δ(0) + ŵγa, where γa is the density of the Laplace distribution with parameter a, so that γa(u) = 1 2 a exp−a|u|. Let μ̂i be the posterior median of the distribution given the observation Xi. We define the mean square error loss R(μ̂, μ) = n−1 n ∑ i=1 E(μ̂i − μi). (1) The following theorem then gives bounds on the loss function. Theorem 1 There is a constant C0 such that, for all n and for all n-vectors μ, R(μ̂, μ) ≤ C0.