## 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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