Non parametric estimation of the conditional density function with right-censored and dependent data

Abstract

In this paper, we study the local constant and the local linear estimators of the conditional density function with right-censored data which exhibit some type of dependence. It is assumed that the observations form a stationary α−mixing sequence. The asymptotic normality of the two estimators is established, which combined with the condition that lim n→∞ nh n b n = ∞ implies the consistency of the two estimators and can be employed to construct confidence intervals for the conditional density function. The result on the local linear estimator of the conditional density function in Kim et al. (2010) is relaxed from the i.i.d. assumption to the α−mixing setting, and the result on the local linear estimator of the conditional density function in Spierdijk (2008) is relaxed from the ρ-mixing assumption to the α−mixing setting. Finite sample behavior of the estimators is investigated by simulations.In Survival Analysis or Reliability, right-censored data are often encountered. There is substantial literature on nonparametric modelling of right-censored data. For example, Guessoum and Ould Saïd (2008, 2012 and Liang and Iglesias-Pérez (2018) considered the estimation of the conditional mean function. This paper focuses on the estimation of the conditional density function, and there are some papers on this topic such as Spierdijk (2008), Kim et al. (2010), Liang and Peng (2010) and Khardani and Semmar (2014). Spierdijk (2008) studied one version of local linear estimators of the conditional density function with stationary ρ−mixing observations, while Kim et al. (2010) proposed another version of local linear estimators with different weights in the independent and identical distributed (i.i.d.) case. The weights are random quantities determined by the Kaplan-Meier estimation of the survival function of the censoring times, and they indeed include the equal weights used by Spierdijk (2008). Those weights have been also adopted by many papers including Guessoum and Ould Saïd (2008, 2012, Liang and Peng (2010) andLiang and de Uña-Álvarez (2011). Especially, Liang and Peng (2010) derived the asymptotic normality and the Berry-Esseen type bound for the kernel estimator of the conditional density function with stationary α-mixing observations. It should be noted that the kernel estimator proposed by Liang and Peng (2010) is a single-smoothing estimator (smoothing the covariates only) of the conditional density function, while the local linear estimator proposed by Kim et al. (2010) is a double-smoothing estimator (smoothing both the response variable and the covariates). In fact, compared with the single-smoothing estimator, the double-smoothing estimator not only appears closer to a conditional density function, but also has more flexibility to reduce the meansquared error when the optimal bandwidths are selected. However, the asymptotic distribution of the local linear estimator of the conditional density function in Kim et al. (2010) was investigated in the i.i.d. setting (although Kim et al. (2010) established the asym...