Kernel and Nearest-Neighbor Estimation of a Conditional Quantile

Let (X₁, Z₁), (X₂, Z₂), …, (Xₙ, Zₙ) be iid as $(X, Z), Z$ taking values in R¹, and for $0 < p < 1$, let ξₚ(x) denote the conditional p-quantile of Z given $X = x,$ i.e., P(Z ≤ ξₚ(x) X = x) = p. In this paper, kernel and nearest-neighbor estimators of ξₚ(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.