Impossibility of weak convergence of kernel density estimators to a non-degenerate law in L2(ℝd)

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3 Citations (Scopus)

Abstract

It is well known that the kernel estimator, for the probability density f on ℝd has pointwise asymptotic normality and that its weak convergence in a function space, especially with the uniform topology, is a difficult problem. One may conjecture that the weak convergence in L2(ℝd) could be possible. In this paper, we deny this conjecture. That is, letting, we prove that for any sequence {rn} of positive constants such that rn = o(√n), if the rescaled residual, converges weakly to a Borel limit in L2(ℝd), then the limit is necessarily degenerate.

Original languageEnglish
Pages (from-to)129-135
Number of pages7
JournalJournal of Nonparametric Statistics
Volume23
Issue number1
DOIs
Publication statusPublished - 2011 Mar 1
Externally publishedYes

Keywords

  • Kernel estimator
  • Weak convergence in L Space

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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