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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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