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

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  • Yoichi Nishiyama

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 rn([fcirc]n−fn) converges weakly to a Borel limit in L2(ℝd), then the limit is necessarily degenerate.

Suggested Citation

  • Yoichi Nishiyama, 2011. "Impossibility of weak convergence of kernel density estimators to a non-degenerate law in (ℝ)," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(1), pages 129-135.
    • Handle: RePEc:taf:gnstxx:v:23:y:2011:i:1:p:129-135
    DOI: 10.1080/10485251003678507
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    Cited by:

    1. Abdelaati Daouia & Byeong U. Park, 2013. "On Projection-type Estimators of Multivariate Isotonic Functions," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 40(2), pages 363-386, June.
    2. Zheng Fang & Juwon Seo, 2019. "A Projection Framework for Testing Shape Restrictions That Form Convex Cones," Papers 1910.07689, arXiv.org, revised Sep 2021.

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