Some asymptotic properties for a smooth kernel estimator of the conditional mode under random censorship

Abstract

a b s t r a c tLet (T i ) 1≤i≤n be a sample of independent and identically distributed (iid) random variables (rv) of interest and (X i ) 1≤i≤n be a corresponding sample of covariates. In censorship models the rv T is subject to random censoring by another rv C . Let θ (x) be the conditional mode function of the density of T given X = x. In this work we define a new smooth kernel estimatorθ n (x) of θ (x) and establish its almost sure convergence and asymptotic normality.An application to prediction and confidence bands is also given. Simulations are drawn to lend further support to our theoretical results for finite sample sizes.

Modal regression using kernel density estimation: A review

“…Equation has been generalized to the case of censored response variables (Khardani, Lemdani, & Saïd, , ; Ould‐Saïd & Cai, ). Suppose that instead of observing the response variables Y 1 , ⋯, Y n , we observe T i = min { Y i , C i } and an indicator δ i = I ( T i = Y i ) that informs whether Y i is observed or not and C i is a random variable that is independent of X i and Y i .…”

“…Under the assumptions on conditional density, the covariate is more responsible for the smoothing effect than the response. Various studies have reported the convergence of unimodal regression with dependent covariates (Attaoui, 2014;Collomb et al, 1986;Dabo-Niang & Laksaci, 2010;Khardani et al, 2010Khardani et al, , 2011Ould-Saïd, 1993, 1997Ould-Saïd & Cai, 2005). Strong consistency was investigated in Collomb et al (1986)) and Ould-Saïd (1993, 1997).…”

“…Note that the joint PDF can be estimated by other approaches such as local polynomial estimation as well (Einbeck & Tutz, 2006;Fan, Yao, & Tong, 1996;Fan & Yim, 2004). Equation (4) has been generalized to the case of censored response variables (Khardani, Lemdani, & Saïd, 2010, 2011Ould-Saïd & Cai, 2005). Suppose that instead of observing the response variables Y 1 , Á Á Á, Y n , we observe T i = min {Y i , C i } and an indicator δ i = I(T i = Y i ) that informs whether Y i is observed or not and C i is a random variable that is independent of X i and Y i .…”

Randomly censored quantile regression estimation using functional stationary ergodic data

“…An appropriate estimator of the conditional df F(t|x) for censored data is, then, obtained by adapting Equation (5) in order to put more emphasis on large values of the interest random variable T which are more censored than a small one. Based on the same idea as in Carbonez, Downloaded by [Selcuk Universitesi] at 11:29 03 January 2015 Györfi, and van der Meulen (1995) and Khardani, Lemdani, and Ould-Said (2010), we consider the following weights:…”

A strong linear representation for the maximum conditional hazard rate estimator in survival analysis

“…Much research has been devoted to the estimation of the unconditional and conditional mode. We focus on recent papers based on conditional mode estimators to mention [25,29,22,26,4,9,15,16].…”