Uniform in bandwidth exact rates for a class of kernel estimators
Résumé
Given an i.i.d sample (Yi, Zi), taking values in Rd′ × Rd, we consider a collection Nadarya-Watson kernel estimators of the conditional expectations E(< cg(z),g(Y) > +dg(z) | Z = z), where z belongs to a compact set H ⊂ Rd, g a Borel function on Rd′ and cg(·),dg(·) are continuous functions on Rd. Given two bandwidth sequences hn < hn fulfilling mild conditions, we obtain an exact and explicit almost sure limit bounds for the deviations of these estimators around their expectations, uniformly in g ∈ G, z ∈ H and hn ≤ h ≤ hn under mild conditions on the density fZ, the class G, the kernel K and the functions cg(·),dg(·). We apply this result to prove that smoothed empirical likelihood can be used to build confidence intervals for conditional probabilities P(Y ∈ C | Z = z), that hold uniform