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Symmetrizing k-nn and Mutual k-nn Smoothers Read chapter Read first chapter

2018 | OriginalPaper | Chapter

Non-parametric Lower Bounds and Information Functions

Author : S. Y. Novak

Published in: Nonparametric Statistics

Publisher: Springer International Publishing

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Abstract

We argue that common features of non-parametric estimation appear in parametric cases as well if there is a deviation from the classical regularity condition. Namely, in many non-parametric estimation problems (as well as some parametric cases) unbiased finite-variance estimators do not exist; neither estimator converges locally uniformly with the optimal rate; there are no asymptotically unbiased with the optimal rate estimators; etc.
We argue that these features naturally arise in particular parametric subfamilies of non-parametric classes of distributions. We generalize the notion of regularity of a family of distributions and present a general regularity condition, which leads to the notions of the information index and the information function.
We argue that the typical structure of a continuity modulus explains why unbiased finite-variance estimators cannot exist if the information index is larger than two, while in typical non-parametric situations neither estimator converges locally uniformly with the optimal rate. We present a new result on impossibility of locally uniform convergence with the optimal rate.

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previous chapter Efficiency of the V -Fold Model Selection for Localized Bases
next chapter Modification of Moment-Based Tail Index Estimator: Sums Versus Maxima
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Metadata
Title
Non-parametric Lower Bounds and Information Functions
Author
S. Y. Novak
Copyright Year
2018
Book
Nonparametric Statistics

Print ISBN: 978-3-319-96940-4

Electronic ISBN: 978-3-319-96941-1

Copyright Year: 2018

DOI
https://doi.org/10.1007/978-3-319-96941-1_5

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