  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.
