Nonparametric Estimation and Inference on Conditional Quantile Processes

We consider the estimation and inference about a nonparametrically specified con- ditional quantile process. For estimation, a two-step procedure is proposed. The first step utilizes local linear regressions and maintains quantile monotonicity through sim- ple inequality constraints. The second step involves linear interpolation between adja- cent quantiles. When computing the estimator, the bandwidth parameter is allowed to vary across quantiles to adapt to the data sparsity. The procedure is computationally simple to implement and is feasible even for relatively large data sets. For inference, we first obtain a uniform Bahadur-type representation for the conditional quantile process. Then, we show that the estimator converges weakly to a continuous Gaussian process, whose critical values can be estimated via simulations by exploiting the fact that it is conditionally pivotal. Next, we demonstrate how to compute the optimal bandwidth, construct uniform confidence bands and test hypotheses about the quantile process. Finally, we examine the performance of the bandwidth selection rule and the uniform confidence bands through simulations.