We tackle the problem of the estimation of a vector of means from a single
vector-valued observation y. Whereas previous work reduces the size of the
estimates for the largest (absolute) sample elements via shrinkage (like
James-Stein) or biases estimated via empirical Bayes methodology, we take a
novel approach. We adapt recent developments by Lee et al (2013) in post
selection inference for the Lasso to the orthogonal setting, where sample
elements have different underlying signal sizes. This is exactly the setup
encountered when estimating many means. It is shown that other selection
procedures, like selecting the K largest (absolute) sample elements and the
Benjamini-Hochberg procedure, can be cast into their framework, allowing us to
leverage their results. Point and interval estimates for signal sizes are
proposed. These seem to perform quite well against competitors, both recent and
more tenured.
Furthermore, we prove an upper bound to the worst case risk of our estimator,
when combined with the Benjamini-Hochberg 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.Comment: 27 pages, 13 figure