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