Convergence rates for pointwise curve estimation with a degenerate design

TL;DR

This paper investigates the rates at which the regression function can be estimated at a point in nonparametric regression models with irregular design densities, deriving minimax convergence rates under various local regularity conditions.

Contribution

It establishes new minimax convergence rates for pointwise estimation in nonparametric regression with degenerate design densities, extending understanding of estimation difficulty in such settings.

Findings

Derived minimax rates ranging from slow to fast depending on local regularity.

Proved that the convergence rate depends on the regularity of the regression function and the design density.

Provided explicit formulas for the minimax rates under specific regularity assumptions.

Abstract

The nonparametric regression with a random design model is considered. We want to recover the regression function at a point x where the design density is vanishing or exploding. Depending on assumptions on the regression function local regularity and on the design local behaviour, we find several minimax rates. These rates lie in a wide range, from slow l(n) rates where l(.) is slowly varying (for instance (log n)^(-1)) to fast n^(-1/2) * l(n) rates. If the continuity modulus of the regression function at x can be bounded from above by a s-regularly varying function, and if the design density is b-regularly varying, we prove that the minimax convergence rate at x is n^(-s/(1+2s+b)) * l(n).