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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 $$\hat T_ n$$ of $$T(\theta)$$, based on a set of i.i.d. observations $$(X_ 1,Y_ 1),\dots,(X_ n,Y_ n)$$, that achieves the optimal nonparametric rate of convergence $$n^{- r}$$ in $$L_ q$$-norms $$(1\leq q<\infty)$$ restricted to compacts under appropriate regularity conditions. Further, it has been shown that there exists an estimate $$\hat T_ n$$ of $$T(\theta)$$ that achieves the optimal rate $$(n/\log n)^{-r}$$ in $$L_ \infty$$-norm restricted to compacts.

MSC:
 62G07 Density estimation 62G20 Asymptotic properties of nonparametric inference 62G35 Nonparametric robustness
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References:
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