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

Yoichi Nishiyama

Journal of Nonparametric Statistics, 2011, vol. 23, issue 1, 129-135

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.

Date: 2011
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Citations: View citations in EconPapers (3)

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DOI: 10.1080/10485251003678507

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