Kernel and nearest-neighbor estimation of a conditional quantile. (English) Zbl 0706.62040
Summary: Let $(X\sb 1,Z\sb 1),(X\sb 2,Z\sb 2),...,(X\sb n,Z\sb n)$ be iid as (X,Z), Z taking values in $R\sp 1$, and for $0<p<1$, let $\xi\sb p(x)$ denote the conditional p-quantile of Z given $X=x$, i.e., $P(Z\le \xi\sb p(x)\vert X=x)=p$. Kernel and nearest-neighbor estimators of $\xi\sb p(x)$ 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.
|62J02||General nonlinear regression|
|62G20||Nonparametric asymptotic efficiency|
|62G30||Order statistics; empirical distribution functions|
|60F17||Functional limit theorems; invariance principles|