Display options
Cite
Mukherjee R, Pillai NS, Lin X. HYPOTHESIS TESTING FOR HIGH-DIMENSIONAL SPARSE BINARY REGRESSION. Ann Stat. 2015;43(1):352-381doi: 10.1214/14-AOS1279.
Mukherjee, R., Pillai, N. S., & Lin, X. (2015). HYPOTHESIS TESTING FOR HIGH-DIMENSIONAL SPARSE BINARY REGRESSION. Annals of statistics, 43(1), 352-381. https://doi.org/10.1214/14-AOS1279
Mukherjee, Rajarshi, et al. "HYPOTHESIS TESTING FOR HIGH-DIMENSIONAL SPARSE BINARY REGRESSION." Annals of statistics vol. 43,1 (2015): 352-381. doi: https://doi.org/10.1214/14-AOS1279
Mukherjee R, Pillai NS, Lin X. HYPOTHESIS TESTING FOR HIGH-DIMENSIONAL SPARSE BINARY REGRESSION. Ann Stat. 2015 Feb;43(1):352-381. doi: 10.1214/14-AOS1279. PMID: 26246645; PMCID: PMC4522432.
Copy Download .nbib Format: NLM
Share it on

Ann Stat. 2015 Feb;43(1):352-381. doi: 10.1214/14-AOS1279.

HYPOTHESIS TESTING FOR HIGH-DIMENSIONAL SPARSE BINARY REGRESSION.

Annals of statistics

Rajarshi Mukherjee, Natesh S Pillai, Xihong Lin

Affiliations

  1. Department of Statistics, Stanford University, Sequoia Hall, 390 Serra Mall, Stanford, California 94305-4065, USA.
  2. Department of Statistics, Harvard University, 1 Oxford Street, Cambridge, Massachusetts 01880, USA.
  3. Department of Biostatistics, Harvard University, 655 Huntington Avenue, SPH2, 4th Floor, Boston, Massachusetts 02115, USA.

PMID: 26246645 PMCID: PMC4522432 DOI: 10.1214/14-AOS1279

Abstract

In this paper, we study the detection boundary for minimax hypothesis testing in the context of high-dimensional, sparse binary regression models. Motivated by genetic sequencing association studies for rare variant effects, we investigate the complexity of the hypothesis testing problem when the design matrix is sparse. We observe a new phenomenon in the behavior of detection boundary which does not occur in the case of Gaussian linear regression. We derive the detection boundary as a function of two components: a design matrix sparsity index and signal strength, each of which is a function of the sparsity of the alternative. For any alternative, if the design matrix sparsity index is too high, any test is asymptotically powerless irrespective of the magnitude of signal strength. For binary design matrices with the sparsity index that is not too high, our results are parallel to those in the Gaussian case. In this context, we derive detection boundaries for both dense and sparse regimes. For the dense regime, we show that the generalized likelihood ratio is rate optimal; for the sparse regime, we propose an extended Higher Criticism Test and show it is rate optimal and sharp. We illustrate the finite sample properties of the theoretical results using simulation studies.

Keywords: Higher Criticism; Minimax hypothesis testing; binary regression; detection boundary; sparsity

References

  1. Nat Genet. 2014 Jan;46(1):45-50 - PubMed
  2. Ann Stat. 2015 Feb;43(1):352-381 - PubMed
  3. Am J Hum Genet. 2014 Jul 3;95(1):5-23 - PubMed
  4. Am J Cardiol. 2004 Jun 15;93(12):1473-80 - PubMed
  5. Nature. 2013 Jan 10;493(7431):216-20 - PubMed
  6. Science. 2012 Jul 6;337(6090):100-4 - PubMed
  7. Nature. 2012 Nov 1;491(7422):56-65 - PubMed

Publication Types

Grant support