Uniform in bandwidth exact rates for a class of kernel estimators

Abstract

Given an i.i.d sample (Y _i , Z _i ), taking values in ${\mathbb{R}^{d'}\times\mathbb{R}^d}$, we consider a collection Nadarya–Watson kernel estimators of the conditional expectations ${\mathbb{E}( <\,c_g(z),g(Y)>+d_g(z)\mid Z=z)}$, where z belongs to a compact set ${H\subset \mathbb{R}^d}$, g a Borel function on ${\mathbb{R}^{d'}}$ and c _g (·), d _g (·) are continuous functions on ${\mathbb{R}^d}$. Given two bandwidth sequences ${h_n<\mathfrak{h}_n}$ fulfilling mild conditions, we obtain an exact and explicit almost sure limit bounds for the deviations of these estimators around their expectations, uniformly in ${g\in\mathcal{G},\;z\in H}$ and ${h_n\le h\le \mathfrak{h}_n}$ under mild conditions on the density f _Z , the class ${\mathcal{G}}$, the kernel K and the functions c _g (·), d _g (·). We apply this result to prove that smoothed empirical likelihood can be used to build confidence intervals for conditional probabilities ${\mathbb{P}( Y\in C\mid Z=z)}$, that hold uniformly in ${z\in H,\; C\in \mathcal{C},\; h\in [h_n,\mathfrak{h}_n]}$. Here ${\mathcal{C}}$ is a Vapnik–Chervonenkis class of sets.