A Central Limit Theorem in Non‐parametric Regression with Truncated, Censored and Dependent Data

Abstract

On the basis of the idea of the Nadaraya–Watson (NW) kernel smoother and the technique of the local linear (LL) smoother, we construct the NW and LL estimators of conditional mean functions and their derivatives for a left‐truncated and right‐censored model. The target function includes the regression function, the conditional moment and the conditional distribution function as special cases. It is assumed that the lifetime observations with covariates form a stationary α‐mixing sequence. Asymptotic normality of the estimators is established. Finite sample behaviour of the estimators is investigated via simulations. A real data illustration is included too.

Weighted estimation of conditional mean function with truncated, censored and dependent data

“…The estimators m N W (x) and m LL (x) are constructed for the first time in Liang et al (2015), who investigated the asymptotic normality of m N W (x) and m LL (x) under dependent assumptions.…”

“…In this section, we conduce a simulated study to investigate In order to obtain an α-mixing observed sequence {X i , Z i , T i , δ i }, we generate the data as in the simulated study by Liang et al (2015), which is as follows.…”

“…In this section, we apply the new NWP regression estimator to the analysis of Spanish unemployment data, and we compare its performance with the performance of NW and LL estimators which were studied in Liang et al (2015).…”

“…The random variable Y can be regarded as the lifetime of a device, the survival time of a patient or the duration time of some economic variable. El Ghouch and Van Keilegom (2008) Liang et al (2015) constructed the NW and the local linear (LL) estimators of m(x) and its derivatives for the left truncated and right censored (LTRC) model, and established asymptotic normality for the NW and LL estimators of m(x) under dependent observations. It can be seen easily that the implementation of the NW estimate of the regression function is much easier than the LL method, and the estimated values of the regression function are always within the range of the response variable.…”

“…See de Uña-Álvarez and Iglesias-Pérez (2010) andLiang et al (2015) for further description of this data set.A regression analysis on the covariate 'age (in years) when entering the unemployed stock' (X) was performed. NW and LL estimators of the conditional mean function m(x) were computed byLiang et al (2015) using the biweight kernel and their bandwidths being the minimizers of the crossvalidation (CV) criterion introduced in their Section 3 (with r = 3). Specifically, h n = 50 for both estimators, and a n took values 10.8 for the NW estimator and 10.1 for the LL estimator.…”