Strong consistency properties of nonparametric estimators for randomly censored data, II: Estimation of density and failure rate

Abstract

This article is Part I I of a two-part study. Properties of the product-limit estimator established in the previous part [2] are now used to prove the strong consistency of some nonparametric density and failure rate estimators which ca~ be used with randomly censored data.

Probability density estimation for survival data with censoring indicators missing at random

“…. , n. This estimator was investigated extensively by Foldes et al [12], Mielniczuk [7], McNichols and Padgett [13], Marron and Padgett [8], Lo et al [9], Yang [14] and Diehl and Stute [15] among others. Other density estimators including the kth nearest uncensored neighbor estimator, histogram estimator and presmoothed kernel density estimator were also considered by Mielniczuk [7], Foldes et al [12] and Cao and Jacome [10].…”

C.440. On data- based bandwidth selection for density estimation from censored samples: a simulation

“…Kernel-type density estimators based on right-censored data have been studied by several authors, including Blum and Susarla (1980), Diehl and Stute (1988), Foldes, Rejto and Winter (1981), Marron and Padgett (1987), McNichols and Padgett (1986), Mielniczuk (1986), Padgett (1988), Padgett and McNichols (1984), and Stute (1985). In particular, Stute (1985) gave an expression for the asymptotically optimal local (or pointwise) bandwidth value for the kernel density estimator with right-censored data in the sense of mean squared error (which is also obtainable from the results of Diehl and Stute, 1988), while Marron and Padgett (1987) presented the asymptotically optimal global bandwidth formula with respect to an integrated squared error criterion.…”

A strong consistency of a nonparametric estimate of entropy under random censorship

“…We refer to [8] for a discussion on the choice of T F T G . In view of (10), (11), for each ε > 0, there exist constants a and b, such that 0…”