Rates of convergence for nearest neighbor estimators with the smoother regression function

TL;DR

This paper establishes the optimal convergence rates for nearest neighbor estimators in regression when the regression function is smooth, resolving an open problem for certain smoothness levels.

Contribution

It proves that the optimal rate can be achieved by nearest neighbor estimators for functions with smoothness parameter p in (1,1.5], an open problem in the field.

Findings

Optimal convergence rates are achieved for p in (1,1.5].

The open problem regarding the rate for certain smoothness levels is solved.

Expected L2 error is used as the error criterion.

Abstract

In regression analysis one wants to estimate the regression function from a data. In this paper we consider the rate of convergence for the nearest neighbor estimator in case that the regression function is $(p,C)$-smooth. It is an open problem whether the optimal rate can be achieved by some nearest neighbor estimator in case that $p$ is on (1,1.5]. We solve the problem affirmatively. This is the main result of this paper. Throughout this paper, we assume that the data is independent and identically distributed and as an error criterion we use the expected $L_2$ error.