Local asymptotics of selection models

Selection models are ubiquitous in statistics. In recent years, they have regained considerable popularity as the working inferential models in many selective inference problems. In this paper, we derive an asymptotic expansion of the local likelihood ratios of selection models. We show that under mild regularity conditions, they are asymptotically equivalent to a sequence of Gaussian selection models. This generalizes the Local Asymptotic Normality framework of Le Cam (1960). Furthermore, we derive the asymptotic shape of Bayesian posterior distributions constructed from selection models, and show that they can be significantly miscalibrated in a frequentist sense.