## Adaptive function estimation in nonparametric regression with one-sided errors.(English)Zbl 1305.62172

Summary: We consider the model of nonregular nonparametric regression where smoothness constraints are imposed on the regression function $$f$$ and the regression errors are assumed to decay with some sharpness level at their endpoints. The aim of this paper is to construct an adaptive estimator for the regression function $$f$$. In contrast to the standard model where local averaging is fruitful, the nonregular conditions require a substantial different treatment based on local extreme values. We study this model under the realistic setting in which both the smoothness degree $$\beta>0$$ and the sharpness degree $$\mathfrak{a}\in(0,\infty)$$ are unknown in advance. We construct adaptation procedures applying a nested version of Lepski’s method and the negative Hill estimator which show no loss in the convergence rates with respect to the general $$L_{q}$$-risk and a logarithmic loss with respect to the pointwise risk. Optimality of these rates is proved for $$\mathfrak{a}\in(0,\infty)$$. Some numerical simulations and an application to real data are provided.

### MSC:

 62G08 Nonparametric regression and quantile regression 62G32 Statistics of extreme values; tail inference
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