Weak Convergence of $k$-NN Density and Regression Estimators with Varying $k$ and Applications

Abstract

In both density and regression estimation problems, the $k$-nearest neighbor estimators with $k$ varying in an appropriate range, when transformed to continuous time stochastic processes, are shown to have a common limiting structure under the usual second-order smoothness conditions as the sample size tends to $\infty$. These results lead to asymptotic linear models in which BLUE's and suitably biased linear combinations are considered.