Polya tree-based nearest neighborhood regression

Parametric regression, such as linear regression, plays an important role in statistics. The use of parametric regression models typically involves the specification of a regression function of the covariates, the distribution of response and the link between the response and covariates, which are commonly at the risk of misspecification. In this paper, we introduce a fully nonparametric regression model, a Polya tree (PT)-based nearest neighborhood regression. To approximate the true conditional probability measure of the response given the covariate value, we construct a PT-distributed probability measure of the response in the nearest neighborhood of the covariate value of interest. Our proposed method gives consistent and robust estimators, and has a faster convergence rate than the kernel density estimation. We conduct extensive simulation studies and analyze a Combined Cycle Power Plant dataset to compare the performance of our method relative to kernel density estimation, PT density estimation, and linear dependent tail-free process (LDTFP). The studies suggest that the proposed method exhibits the superiority to the kernel and PT density estimation methods in terms of the estimation accuracy and convergence rate and to LDTFP in terms of robustness.