Kernel and Nearest-Neighbor Estimation of a Conditional Quantile

Abstract

Let $(X_1, Z_1), (X_2, Z_2), \ldots, (X_n, Z_n)$ be iid as $(X, Z), Z$ taking values in $R^1$, and for $0 < p < 1$, let $\xi_p(x)$ denote the conditional $p$-quantile of $Z$ given $X = x,$ i.e., $P(Z \leq \xi_p(x)\mid X = x) = p$. In this paper, kernel and nearest-neighbor estimators of $\xi_p(x)$ are proposed. In order to study the asymptotics of these estimates, Bahadur-type representations of the sample conditional quantiles are obtained. These representations are used to examine the important issue of choosing the smoothing parameter by a local approach (for a fixed $x$) based on weak convergence of these estimators with varying $k$ in the $k$-nearest-neighbor method and with varying $h$ in the kernel method with bandwidth $h$. These weak convergence results lead to asymptotic linear models which motivate certain estimators.