Nonparametric Estimation of Hazard Rate under the Constraint of Monotonicity

We show how to smoothly`monotonise' standard kernel estimators of hazard rate, using bootstrap weights. Our method takes a variety of forms, depending on choice of kernel estimator and on the distance function used to deene a certain constrained optimisation problem. We connne attention to a particularly simple kernel approach, and explore a range of distance functions. It is straightforward to reducèquadratic' inequality constraints tòlinear' equality constraints, and so our method may be implemented using little more than conventional Newton-Raphson iteration. Thus, the necessary computational techniques are very familiar to statisticians. We show both numerically and theoretically that monotonicity, in either direction, can generally be imposed on a kernel hazard rate estimator regardless of the monotonicity or otherwise of the true hazard rate. Our methods have straightforward extension to the problem of testing for monotonicity of hazard rate, where the distance function plays the role of a test statistic.