Nonparametric Estimation of a Convex Bathtub-Shaped Hazard Function

In this paper we study the nonparametric MLE and LSE of a convex hazard function. Our estimators are shown to be consistent and to converge at rate $n^{2/5}$. Moreover we establish the pointwise asymptotic distribution theory of both estimators under the assumption that the true hazard function is positive with positive second derivative at the fixed point. The same problems for a convex hazard function under right censoring and for the Poisson process with a convex rate are also discussed.