Renewable Composite Quantile Method and Algorithm for Nonparametric Models with Streaming Data

We are interested in renewable estimation and algorithms for nonparametric models with streaming data. We express the parameter of interest through a functional depending on a weight function and a conditional distribution function (CDF). By renewable kernel estimations combined with function interpolations, we obtain renewable estimator for the CDF and propose the method of renewable weighted composite quantile regression (WCQR). By fully using the model structure, we propose new weight selectors, by which the WCQR can achieve asymptotic unbiasness when estimating specific functions in the model. We also propose practical bandwidth selectors for streaming data and find the optimal weight function minimizing the asymptotic variance. Our asymptotical results show that our estimator is almost equivalent to the oracle estimator obtained from the entire data together. And our method also enjoys adaptiveness to error distributions, robustness to outliers, and efficiency in both estimation and computation. Simulation studies and real data analyses further comfirm our theoretical findings.