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In this paper, we study the nonparametric maximum likelihood estimator (MLE)of a convex hazard function. We show that the MLE is consistent and convergesat a local rate of $n^{2/5}$ at points $x_0$ where the true hazard function ispositive and strictly convex. Moreover, we establish the pointwise asymptoticdistribution theory of our estimator under these same assumptions. One notablefeature of the nonparametric MLE studied here is that no arbitrary choice oftuning parameter (or complicated data-adaptive selection of the tuningparameter) is required.

Nonparametric estimation of a convex bathtub-shaped hazard function