Convergence rates for pointwise curve estimation with a degenerate design

Published in Mathematical Methods of Statistics, 2005

S. Gaïffas

The nonparametric regression with a random design model is considered. We want to recover the regression function at a point $x_0$ where the design density is vanishing or exploding. Depending on assumptions on 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 $\ell(n)$ rates, where $\ell$ is slowly varying (for instance $(\log n)^{−1}$), to fast $n^{−1/2} \ell(n)$ rates. If the continuity modulus of the regression function at $x_0$ can be bounded from above by an $s$-regularly varying function, and if the design density is $\beta$-regularly varying, we prove that the minimax convergence rate at $x_0$ is $n^{−s / (1 + 2 s + \beta)} \ell(n)$.

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