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Convergence rates for pointwise curve estimation with a degenerate design

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).
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Contributor : Stéphane Gaïffas Connect in order to contact the contributor
Submitted on : Friday, January 13, 2006 - 9:43:13 PM
Last modification on : Sunday, June 26, 2022 - 5:14:03 AM
Long-term archiving on: : Thursday, September 23, 2010 - 3:59:57 PM



Stéphane Gaiffas. Convergence rates for pointwise curve estimation with a degenerate design. 2006. ⟨hal-00003086v3⟩



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