Uniform in bandwidth exact rates for a class of kernel estimators |
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Authors: | Davit Varron Ingrid Van Keilegom |
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Institution: | 1.Laboratoire de Mathématiques Pures et Appliquées,Université de Franche-Comté,Besan?on,France;2.Institute of Statistics,Université catholique de Louvain,Louvain-la-Neuve,Belgium |
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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. |
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