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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.

62G05Nonparametric estimation
62G07Density estimation
62J02General nonlinear regression
62G20Nonparametric asymptotic efficiency
62G30Order statistics; empirical distribution functions
60F17Functional limit theorems; invariance principles
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