Global Bias-Corrected Divide-and-Conquer by Quantile-Matched Composite for General Nonparametric Regressions

The issues of bias-correction and robustness are crucial in the strategy of divide-and-conquer (DC), especially for asymmetric nonparametric models with massive data. It is known that quantile-based methods can achieve the robustness, but the quantile estimation for nonparametric regression has non-ignorable bias when the error distribution is asymmetric. This paper explores a global bias-corrected DC by quantile-matched composite for nonparametric regressions with general error distributions. The proposed strategies can achieve the bias-correction and robustness, simultaneously. Unlike common DC quantile estimations that use an identical quantile level to construct a local estimator by each local machine, in the new methodologies, the local estimators are obtained at various quantile levels for different data batches, and then the global estimator is elaborately constructed as a weighted sum of the local estimators. In the weighted sum, the weights and quantile levels are well-matched such that the bias of the global estimator is corrected significantly, especially for the case where the error distribution is asymmetric. Based on the asymptotic properties of the global estimator, the optimal weights are attained, and the corresponding algorithms are then suggested. The behaviors of the new methods are further illustrated by various numerical examples from simulation experiments and real data analyses. Compared with the competitors, the new methods have the favorable features of estimation accuracy, robustness, applicability and computational efficiency.