Uniform-in-bandwidth kernel estimation for censored data

Abstract

We present a sharp uniform-in-bandwidth functional limit law for the increments of the Kaplan-Meier empirical process based upon right-censored random data. We apply this result to obtain limit laws for nonparametric kernel estimators of local functionals of lifetime densities, which are uniform with respect to the choices of bandwidth and kernel. These are established in the framework of convergence in probability, and we allow the bandwidth to vary within the complete range for which the estimators are consistent. We provide explicit values for the asymptotic limiting constant for the sup-norm of the estimation random error.AMS 2010 Classification: 62G15, 62G20, 62G30, 60F15, 60F17, 60N01, 60N02. Keywords and phrases: Functional limit laws, nonparametric kernel density estimation, right censorship model, kernel lifetime density estimators, kernel failure rate estimators, Kaplan-Meier empirical process, uniform empirical process, convergence in probability.

Functional limit laws for local empirical processes in a spatial setting

“…We will now prove (25). By another almost sure monotonicity argument, we only need to prove the convergence to zero, as k → ∞, of…”

“…II Some other works are directly intended to obtain strong consistency and to derive explicit constants in the rates of convergences. The first works in that direction are (to the best of our knowledge) due to Deheuvels [5], and was then followed by a series of related works by several researchers [4,12,25,28,29,33]. [29]).…”

On the study of the Beran estimator for generalized censoring indicators

“…The most popular and well known non-parametric approach leads to the Kaplan and Meier (1958) product-limit estimator. This latter has been extensively studied during the past decades and shows appealing properties; its asymptotic efficiency was proved in Földes et al (1981) and Wellner (1982) and extented with the study of the limit law process, as in Deheuvels and Einmahl (2000) or Ouadah (2013). It is also worth mentioning that its convergences rates have been widely studied, among others in Horváth (1983), Chen and Lo (1997) or Wellner (2008) with exponential bounds for the empirical process.…”