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We tackle the problem of the estimation of a vector of means from a singlevector-valued observation $y$. Whereas previous work reduces the size of theestimates for the largest (absolute) sample elements via shrinkage (likeJames-Stein) or biases estimated via empirical Bayes methodology, we take anovel approach. We adapt recent developments by Lee et al (2013) in postselection inference for the Lasso to the orthogonal setting, where sampleelements have different underlying signal sizes. This is exactly the setupencountered when estimating many means. It is shown that other selectionprocedures, like selecting the $K$ largest (absolute) sample elements and theBenjamini-Hochberg procedure, can be cast into their framework, allowing us toleverage their results. Point and interval estimates for signal sizes areproposed. These seem to perform quite well against competitors, both recent andmore 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 withina constant multiple of the minimax risk over a rich set of parameter spacesmeant to evoke sparsity.

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