Hypothesis Testing for Sparse Binary Regression

• Activity Number: 463
• Type: Contributed
• Date/Time: Wednesday, August 6, 2014 : 8:30 AM to 10:20 AM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Rajarshi Mukherjee*+ and Xihong Lin and Natesh S. Pillai
• Companies: Harvard and Harvard School of Public Health and Harvard
• Keywords: Minimax Hypothesis Testing ; Sparse Alternatives ; Detection Boundary ; Binary Outcomes ; Higher Criticism

Abstract

In this paper, we study the detection boundary for minimax hypothesis testing in the context of high-dimensional, sparse binary regression models. We observe a new phenomenon in the behavior of detection boundary which does not occur in the case of Gaussian linear models. We derive the detection boundary as a function of two components: the minimal signal strength required for successful detection and the sparsity of the design matrix. If the design matrix with binary entries is too sparse, any test is asymptotically powerless irrespective of the magnitude of signal strength. For binary design matrices which are not too sparse, our results are parallel to the Gaussian case. In this context we derive detection boundaries for both dense and sparse regimes. For the dense regime, our results are rate optimal; for the sparse regime, we provide sharp constants. In the dense regime the generalized likelihood ratio test continues to be asymptotically powerful above the detection boundary. In the sparse regime, however, we need to design a new test which is a version of the popular Higher Criticism test. We show that this new test attains the detection boundary as a sharp upper bound.