On pointwise adaptive curve estimation with a degenerate random design

Abstract

We consider the nonparametric regression with a random design model, and we are interested in the adaptive estimation of the regression at a point $x\_0$ where the design is degenerate. When the design density is $\beta$-regularly varying at $x\_0$ and $f$ has a smoothness $s$ in the H\"{o}lder sense, we know from Ga\"{i}ffas (2004)\nocite{gaiffas04a} that the minimax rate is equal to $n^{-s/(1+2s+\beta)} \ell(1/n)$ where $\ell$ is slowly varying. In this paper we provide an estimator which is adaptive both on the design and the regression function smoothness and we show that it converges with the rate $(\log n/n)^{s/(1+2s+\beta)} \ell(\log n/n)$. The procedure consists of a local polynomial estimator with a Lepski type data-driven bandwidth selector similar to the one in Goldenshluger and Nemirovski (1997)\nocite{goldenshluger\_nemirovski97} or Spokoiny (1998)\nocite{spok98}. Moreover, we prove that the payment of a $\log$ in this adaptive rate compared to the minimax rate is unavoidable.