Lower Bounds for Estimating a Hazard

Abstract

Publisher Summary Two very active areas of statistical research are non-parametric function estimation and the analysis of censored survival data. The techniques originally developed for density estimation, such as the kernel estimator approach introduced by Rosenblatt and Parzen, have now been applied to the estimation of densities and hazard rates for randomly right censored data. This chapter focuses on the kernel estimator and its asymptotic optimality and discusses the problem of the best obtainable asymptotic rates of convergence under a weighted integrated quadratic risk, for the global estimation of a hazard function with randomly right censored data, knowing that the hazard function belongs to a given regularity class, such as a Lipschitz or Sobolev class. The perturbation methods developed by Bretagnolle and Huber are used in the case of hazard rate estimation with censored data. Particular use is made of the hypercubes of Assouad, who gave an elegant general method for deriving lower bounds for functional estimation. This method differs from that of Donoho and Low for establishing the rates of convergence in the uncensored case. It is shown that this lower bound is achieved, in particular, by the Ramlau-Hansen kernel estimator, using results of Pons, thus establishing a minimax result.