Post‐selection point and interval estimation of signal sizes in Gaussian samples

We tackle the problem of the estimation of a vector of underlying means (signal sizes) from a single vector‐valued observation y . Often one is interested in estimating only a subvector of signals corresponding to a set of selected, “interesting” sample elements. These “interesting” sample elements tend to have the largest absolute size, gleaned by applying some selection procedure like that of Benjamini & Hochberg (2015). Previous work on this estimation task proposes the reduction in size of the largest (absolute) sample elements either via shrinkage (like James–Stein) or by subtracting biases estimated using empirical Bayes methodology. We take a novel approach and adapt recent developments by Lee et al. (2016) in post‐selection inference. Adapting and applying their distributional results to our problem post‐selection point and interval estimators for underlying signal sizes are proposed. Simulations suggest that our estimator seems to perform quite well against competitors. Furthermore we prove an upper bound to the so‐called “worst case risk” of our estimator—when combined with the Benjamini–Hochberg selection procedure—and show that it is within a constant multiple of the minimax risk over a rich set of parameter spaces meant to evoke sparsity. The Canadian Journal of Statistics 45: 128–148; 2017 © 2017 Statistical Society of Canada