Affiliation:
1. Department of Statistics and Data Science, Cornell University , Ithaca, NY 14853 , USA
2. Department of Statistics, University of California , Los Angeles, Los Angeles, CA 90095 , USA
Abstract
Abstract
In this paper, we propose a new framework to construct confidence sets for a $d$-dimensional unknown sparse parameter ${\boldsymbol \theta }$ under the normal mean model ${\boldsymbol X}\sim N({\boldsymbol \theta },\sigma ^{2}\bf{I})$. A key feature of the proposed confidence set is its capability to account for the sparsity of ${\boldsymbol \theta }$, 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 ${\boldsymbol \theta }$ 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 ${\boldsymbol \theta }$ 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 for detecting non-zero $\theta _{j}$’s. To exploit the sparsity of ${\boldsymbol \theta }$, we allow the confidence interval for $\theta _{j}$ to degenerate 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. Numerical studies are conducted to verify the theoretical results.
Funder
National Science Foundation
Office of Naval Research
Adobe Data Science Award
Publisher
Oxford University Press (OUP)
Subject
Applied Mathematics,Computational Theory and Mathematics,Numerical Analysis,Statistics and Probability,Analysis
Reference34 articles.
1. Adapting to unknown sparsity by controlling the false discovery rate;Abramovich;Ann Stat,2006
2. Global testing under sparse alternatives: Anova, multiple comparisons and the higher criticism;Arias-Castro;Ann Stat,2011
3. Some nonasymptotic results on resampling in high dimension, i: confidence regions;Arlot;Ann Stat,2010
4. Non-asymptotic minimax rates of testing in signal detection;Baraud;Bernoulli,2002
5. Uniform post-selection inference for least absolute deviation regression and other z-estimation problems;Belloni;Biometrika,2014