Smoothing Methods

This chapter describes how kernel estimators for the pdf and the hazard function can be constructed in the doubly truncated setting. It analyzes their asymptotic properties and illustrates the performance of the estimators with some examples. The chapter presents some selection bandwidth procedures for kernel density estimation when X is doubly truncated, and discusses bandwidth selection for the hazard rate. It aims to adapt the kernel estimator for a doubly truncated variable, and introduces some smoothing selectors such as normal reference, plug-in, cross-validation and bootstrap bandwidths. The chapter illustrates the practical performance of the several selectors for the kernel density bandwidth through the analysis of simulated and real data. Kernel density estimation has been often used for data exploration. The hazard function, or failure rate function, is often used in Survival Analysis and Reliability to represent the prognosis of a patient or the performance of a device along time.