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 $\theta$ under the normal mean model $X\sim N(\theta,\sigma^2I)$. A key feature of the proposed confidence set is its capability to account for the sparsity of $\theta$, thus named as {\em sparse} confidence set. This is in sharp contrast with the classical methods, such as 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 $\theta$ 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 (TNR) for detecting nonzero $\theta_j$'s. To exploit the sparsity of $\theta$, we define that the confidence interval for $\theta_j$ degenerates to a single point 0 for any $j\notin S$. 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 and SNR. Numerical studies are conducted to verify the theoretical results.