An L_1 analysis of a kernel-based hazard rate estimator

摘要

This article provides an L1 analysis of the standard nonparametric kernel-based hazard rate estimator under the random right censorship model. The analysis starts with the asymptotic formula for the integrated mean absolute error (IMAE) and then addresses the issue of bandwidth selection. In particular, we show that as a function of bandwidth, the asymptotic minimum of IMAE occurs at the minimising argument of the dominant term of the IMAE's asymptotic expression. Further, it is noted that finding the minimising argument of the dominant term of the asymptotic IMAE is very close to the similar problem in L1 density estimation except that, as one would expect, the minimising argument now depends on the functionals of the unknown hazard rate. We then use estimates of these unknown functionals in an algorithm to calculate an adaptive version of the optimal bandwidth. We also show that, asymptotically, both theoretical and adaptive forms of the bandwidths do minimise the L1 distance between the true hazard rate function and its kernel estimator. We also provide a simulation study to illustrate the methodology and compare L1 errors of hazard rate estimates which use L1 and L2 optimal bandwidths.