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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 $Y=\theta \left(X\right)+ϵ$, where X and $ϵ$ are independent and the $\alpha$ th quantile of $ϵ$ is 0 ($\alpha$ is fixed such that $0<\alpha <1\right)$. Assume that $\theta$ is a smooth function with order of smoothness $p>0$, and set

$r=\left(p-m\right)/\left(2p+d\right),$

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 ${\stackrel{^}{T}}_{n}$ of T($\theta$), based on a set of i.i.d. observations $\left({X}_{1},{Y}_{1}\right),···,\left({X}_{n},{Y}_{n}\right)$, 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 ${\stackrel{^}{T}}_{n}$ and this is used to obtain some useful asymptotic results.

##### MSC:
 62G07 Density estimation 62G20 Nonparametric asymptotic efficiency 62G35 Nonparametric robustness 62E20 Asymptotic distribution theory in statistics