High-dimensional regression with a count response

Abstract.: We consider high-dimensional regression with a count response modeled by a Poisson or negative binomial generalized linear model (GLM). We propose a penalized maximum likelihood estimator with a properly chosen complexity penalty and establish its adaptive minimaxity across models of variing sparsity. To make the procedure computationally feasible for high-dimensional data, we investigate its LASSO and SLOPE convex surrogates and also present the FISTA algorithm for their numerical solution. Their performance is illustrated through simulated and real-data examples.