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
Externally publishedYes

Fingerprint

Kernel Density Estimator
Weak Convergence
Kernel Estimator
Probability Density
Asymptotic Normality
Function Space
Topology
Converge
Estimator
Kernel density
Weak convergence
Impossibility
Kernel estimator
Asymptotic normality

Keywords

  • Kernel estimator
  • Weak convergence in L Space

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

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