Estimation of a star-shaped distribution function

Abstract

A life distribution function (DF) F is said to be star-shaped if F(x)/x is nondecreasing on its support. This generalises the model of a convex DF, even allowing for jumps. The nonparametric maximum likelihood estimation is known to be inconsistent. We provide a uniformly strongly consistent least-squares estimator. We also derive the convergence in distribution of the estimator at a point where F(x)/x is increasing using the arg max theorem.