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Nonparametric estimates of regression quantiles and their local Bahadur representation. (English) Zbl 0728.62042
Summary: Let (X,Y) be a random vector such that X is d-dimensional, Y is real valued and \(Y=\theta (X)+\epsilon\), where X and \(\epsilon\) are independent and the \(\alpha\) th quantile of \(\epsilon\) is 0 (\(\alpha\) is fixed 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 a pointwise estimate \(\hat T_ n\) of T(\(\theta\)), based on a set of i.i.d. observations \((X_ 1,Y_ 1),...,(X_ n,Y_ n)\), that achieves the optimal nonparametric rate of convergence \(n^{-r}\) under appropriate regularity conditions. Further, a local Bahadur type representation is shown to hold for the estimate \(\hat T_ n\) and this is used to obtain some useful asymptotic results.

MSC:
62G07 Density estimation
62G20 Asymptotic properties of nonparametric inference
62G35 Nonparametric robustness
62E20 Asymptotic distribution theory in statistics
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