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Global nonparametric estimation of conditional quantile functions and their derivatives. (English) Zbl 0739.62028
Summary: Let $(X,Y)$ be a random vector such that $X$ is $d$-dimensional, $Y$ is real valued, and $\theta(X)$ is the conditional $\alpha$th quantile of $Y$ given $X$, where $\alpha$ is a fixed number such that $0<\alpha<1$. Assume that $\theta$ is a smooth function with order of smoothness $p>0$, and set $r=(p-m)/(2p+d)$, where $m$ is a nonnegative integer smaller than $p$. Let $T(\theta)$ denote a derivative of $\theta$ of order $m$. It is proved that there exists an estimate $\hat T\sb n$ of $T(\theta)$, based on a set of i.i.d. observations $(X\sb 1,Y\sb 1),\dots,(X\sb n,Y\sb n)$, that achieves the optimal nonparametric rate of convergence $n\sp{- r}$ in $L\sb q$-norms $(1\leq q<\infty)$ restricted to compacts under appropriate regularity conditions. Further, it has been shown that there exists an estimate $\hat T\sb n$ of $T(\theta)$ that achieves the optimal rate $(n/\log n)\sp{-r}$ in $L\sb \infty$-norm restricted to compacts.

MSC:
62G07Density estimation
62G20Nonparametric asymptotic efficiency
62G35Nonparametric robustness
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References:
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