Rates of convergence for pointwise curve estimation with a degenerate design
Résumé
The model of nonparametric regression with a random design is considered. We want to estimate the regression function at a point $x_0$ where the density of the design is vanishing or exploding. Depending on assumptions on the local regularity of the regression function and on the local behaviour of the design, we find several minimax rates. These rates lie in a wide range, from slow rates of order $\ell(n)$ where $\ell(n)$ is slowly varying (for instance $(\log n)^{-1}$ to fast rates of order $n^{-1/2} \ell(n)$. In particular, if the modulus of continuity at $x_0$ of the regression function can be bounded from above by a regularly varying function of index $s$, and if the density of the design is regularly varying of index $\beta$, we prove that the minimax rate of convergence at $x_0$ is of order $n^{-s/(1+2s+\beta)} \ell(n)$.