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On moment inequalities of the supremum of empirical processes with applications to kernel estimation

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  • Ahmad, Ibrahim A.

Abstract

Let X1,...,Xn be a random sample from a distribution function F. Let Fn(x)=(1/n)[summation operator]i=1nI(Xi[less-than-or-equals, slant]x) denote the corresponding empirical distribution function. The empirical process is defined by In this note, upper bounds are found for E(Dn) and for E(etDn), where Dn=supx Dn(x). An extension to the two sample case is indicated. As one application, upper bounds are obtained for E(Wn), where, with is the celebrated "kernel" density estimate of f(x), the density corresponding to F(x) and an optimal bandwidth is selected based on Wn. Analogous results for the kernel estimate of F are also mentioned.

Suggested Citation

  • Ahmad, Ibrahim A., 2002. "On moment inequalities of the supremum of empirical processes with applications to kernel estimation," Statistics & Probability Letters, Elsevier, vol. 57(3), pages 215-220, April.
  • Handle: RePEc:eee:stapro:v:57:y:2002:i:3:p:215-220
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    Cited by:

    1. Elisa Molanes-López & Ricardo Cao, 2008. "Plug-in bandwidth selector for the kernel relative density estimator," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 60(2), pages 273-300, June.

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