The Two-interval Line-segment Problem

In this paper we define and study the non-parametric maximum likelihood estimator (NPMLE) in the one-dimensional line-segment problem, where we observe line-segments on the real line through an interval with a gap which is smaller than the two remaining intervals. We define the self-consistency equations for the NPMLE and provide a quick algorithm for solving them. We prove supremum norm weak convergence to a Gaussian process and efficiency of the NPMLE. The problem has a geological application in the study of the lifespan of species