Sparse confidence sets for normal mean models

In this paper, we propose a new framework to construct confidence sets for a d-dimensional unknown sparse parameter ? under the normal mean model X similar to N(theta, sigma I-2). A key feature of the proposed confidence set is its capability to account for the sparsity of theta, thus named as sparse confidence set. This is in sharp contrast with the classical methods, such as the Bonferroni confidence intervals and other resampling-based procedures, where the sparsity of theta is often ignored. Specifically, we require the desired sparse confidence set to satisfy the following two conditions: (i) uniformly over the parameter space, the coverage probability for 0 is above a pre-specified level; (ii) there exists a random subset S of {1, ..., d} such that S guarantees the pre-specified true negative rate for detecting non-zero theta(j)'s. To exploit the sparsity of theta, we allow the confidence interval for theta(j) to degenerate to a single point theta for any j theta/ S. Under this new framework, we first consider whether there exist sparse confidence sets that satisfy the above two conditions. To address this question, we establish a non-asymptotic minimax lower bound for the non-coverage probability over a suitable class of sparse confidence sets. The lower bound deciphers the role of sparsity and minimum signal-to-noise ratio (SNR) in the construction of sparse confidence sets. Furthermore, under suitable conditions on the SNR, a two-stage procedure is proposed to construct a sparse confidence set. To evaluate the optimality, the proposed sparse confidence set is shown to attain a minimax lower bound of some properly defined risk function up to a constant factor. Finally, we develop an adaptive procedure to the unknown sparsity. Numerical studies are conducted to verify the theoretical results.