Weighted Local Linear Estimator Of Conditional Quantile Under Left-truncated And Dependent Data

The Quantile Regression(QR)method proposed by Koenker and Basset(1978)has been widely used in statistical research due to its own excellent properties.Compared with traditional mean regression,quantile regression has strong robustness,because it does not need to make specific assumptions about the distribution of errors,so it is often used to deal with biased data,such as data with outliers and abnormal distributions.The research of the quantile regression of complete data has a large amount of literature both in theory and application.However,incomplete data,like truncation,censorship and missing,is common in many practical applications.And QR method shows excellent properties when dealing with these incomplete data.As one of the important representatives of incomplete data,left-truncated data often appear in the fields of astronomy,epidemiology,biomedicine,and finance.This article considers the estimation of the conditional quantile function of dependent lefttruncated data.According to the existing research,the asymptotic normality of the local linear conditional quantile estimation of left-truncated data with multiple covariates has not been discussed yet.Therefore,this article will establish a non-parametric estimator of the conditional quantile function of the above data under the setting of the left-truncated data with multivariate covariates,and extend the result to the dependent situation.The estimator is a kernel-weighted local linear estimator.In order to study the convergence and asymptotic normality of the estimator under the hypothesis of ?-mixing dependence,and obtain the Bahadur representation of the estimator,Bernstein's Method,moment inequality and other methods will be used.Finally,the Mento-Carlo simulation is used to study the performance of the estimator under limited sample size with different settings,and the simulation results show that the estimator proposed in this paper has good properties.