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Uniform central limit theorems for kernel density estimators

Abstract

Let \({\mathbb{P}}_{n} \ast K_{h_{n}}(x) = n^{-1}h_{n}^{-d}\sum_{i=1}^{n}K\left((x-X_{i})/h_{n}\right)\) be the classical kernel density estimator based on a kernel K and n independent random vectors X i each distributed according to an absolutely continuous law \({\mathbb{P}}\) on \({\mathbb{R}}^{d}\) . It is shown that the processes \(f \longmapsto \sqrt{n}\int fd({\mathbb{P}}_{n} \ast K_{h_{n}}-{\mathbb{P}})\) , \(f \in {\mathcal{F}}\) , converge in law in the Banach space \(\ell ^{\infty }({\mathcal{F}})\) , for many interesting classes \({\mathcal{F}}\) of functions or sets, some \({\mathbb{P}}\) -Donsker, some just \({\mathbb{P}}\) -pregaussian. The conditions allow for the classical bandwidths h n that simultaneously ensure optimal rates of convergence of the kernel density estimator in mean integrated squared error, thus showing that, subject to some natural conditions, kernel density estimators are ‘plug-in’ estimators in the sense of Bickel and Ritov (Ann Statist 31:1033–1053, 2003). Some new results on the uniform central limit theorem for smoothed empirical processes, needed in the proofs, are also included.

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Correspondence to Evarist Giné.

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Giné, E., Nickl, R. Uniform central limit theorems for kernel density estimators. Probab. Theory Relat. Fields 141, 333–387 (2008). https://doi.org/10.1007/s00440-007-0087-9

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Mathematics subject classification (2000)

  • Primary: 62G07
  • Secondary: 60F05

Keywords

  • Kernel density estimation
  • Uniform central limit theorem
  • Plug-in property
  • Smoothed empirical processes