SIGLE: a valid procedure for Selective Inference with the Generalized Linear Lasso

This articles investigates the distribution of the solutions of the generalized linear lasso (GLL) sharing the same support. In this framework of post-selection inference (PSI), we investigate the selected and the saturated models: two different paradigms that determine the hypothesis being tested. Based on a new conditional Maximum Likelihood Estimator (MLE) approach, we introduce a procedure (referred to as SIGLE) to obtain asymptotically valid PSI confidence regions and simple hypothesis testing procedures for Generalized Linear Models (GLMs).In a second stage, we focus on the sparse logistic regression and we exhibit conditions ensuring that our conditional MLE method is valid. We present extensive numerical simulations supporting our theoretical results.