Consider the following nonparametric regression model with random censored data:
where 
is an unknown smoothing function, 
is a positive function representing the standard deviation and 
is the random error with mean 0 and variance 1. Let C denote the censoring variable, whose distribution may depend on U, where U is vector of observed covariates. In this paper, we focus on random right censoring, we only observe the triples
, where 
and 
are the observed response variable and the censoring indicator respectively, where 
is the survival time.
Censored quantile regression was first studied by [
Intuitively, the composite quantile regression (CQR) should provide estimation efficiency gain over a single quantile regression; see [
The paper is organized as follows. In Section 2, local composite quantile regression for nonparametric model with censored data is introduced, and the main theoretical results are also given in this section. Both simulation examples and a real data application are given in Section 3 to illustrate the proposed procedures. Final remarks are given in Section 4. The technical proofs are deferred to the Appendix.
, the conditional cumulative distribution function of the survival time 
given
, is assumed to be known. In this case, we define the following weight function:
. In reality, 
is unknown and has to be estimated. We propose to estimate 
nonparametrically using the local Kaplan-Meier estimator
and
, where
is a smooth kernel function, 
is the bandwidth converging to zero as
. By plugging 
into (2), we obtain the estimated local weights
. Consider estimating the value of 
at
. The LCQRC procedure estimates
, defined by
, via minimizing the locally weighted objective function
, be q check loss functions at q quantile positions: 
and 
is any value sufficiently large to exceed all
.
can see Remark 1 of [
the marginal density function of the covariate
, 
and
. To prove main results in this paperthe following technical conditions are imposed.
and 
have first derivatives with respect to
, denoted as 
and
, which are uniformly bounded away from infinity. In addition, 
and 
have bounded second order partial derivatives with respect to U.
is positive definite matrix.
has a continuous second derivative in the neighborhood of
.
is differentiable and positive in the neighborhood of
.
is continuous in the neighborhood of
.
.
and 
in the neighborhood of
, and to obtain the uniform consistency and the linear representation of
, which are needed for deriving the asymptotic normality result. Assumption A2 ensures that the expectation of the estimating function has a unique zero, and it is needed to establish the asymptotic distribution. Assumptions A3- A6 are the same conditions for establishing the asymptotic normality of local composite quantile regression ([
in the following theorem.
constitute and i.i.d. multivariate random sample, and that the censoring variable 
is independent of 
conditional on the covariate
. Suppose that 
is an interior of the support of
. Under the regularity conditions A1-A6, if 
and
, then
stands for convergence in distribution and
, where
.
and
. The bandwidth h* can be obtained by 10-fold cross-validation method (see [
(see [
is uniformly distributed on 
and 
is i.i.d. standard normal random variables. The censoring variable 
and
. The value of the constant c in the model determines the censoring proportion. In our simulations, we consider three censoring rates (CR): 20%, 30% and 40%. For each censoring rate, the sample sizes are taken to be 100 and 200. To evaluate the finite sample performance of our estimator. Two distance measures are approximated, the first one the mean absolute deviation error (MADE) is given by
, and the second one the mean squared error (MSE) defines as
. Furthermore, we define the rate of MADE and MSE which are
and
.
close to 1 may not be identifiable due to censorship. In our similations, we consider 
for LCQR and LCQRC estimators. The means and standard deviations of MADE, MSE, RMADE and RMSE are respectively reported in 
, are generated from following model
and 
are generated following the same way as in Example 1. The means and standard deviations of MADE, MSE, RMADE and RMSE are respectively reported in 
with n = 100 for Example 1.
with n = 200 for Example 1.
with n = 100 for Example 2.
with n = 200 for Example 2.
. We are interested in estimating the conditional of survival time (in days) given age (in years). Here, we use model (1) to fit the lung cancer data, where 
is the 
(survival time) and U is the age/100. To evaluate the performance of our estimator. Two distance measures were approximatedthe first one the mean absolute deviation error
given by
, and the second one the mean squared error 
defined as
, where
. Furthermore, we define the rate of 
and 
which are
and
. Next, we report and compare results with LCQR and NQRC for estimating the survival time. The simulation results for the LCQR, LCQRC and NQRC are given in 
for lung cancer data.
.
,
, 
,
,
is the minimizer of the following criterion:
and
. To apply the identity ([
is any value sufficiently large to exceed all
, 
and
, then
.
, where
.
and 
given
, we have
, then
.
, then it follows that
, then
