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Erschienen in:
Order statistics and nearest neighbors Kapitel lesen Erstes Kapitel lesen

2015 | OriginalPaper | Buchkapitel

6. Local behavior

verfasst von : Gérard Biau, Luc Devroye

Erschienen in: Lectures on the Nearest Neighbor Method

Verlag: Springer International Publishing

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Abstract

No study of a density estimate is complete without a discussion of the local behavior of it. That is, given a certain amount of smoothness at x, how fast does f n (x) tend to f(x)? It is clear that for any sequence of density estimates, and any sequence \(a_{n} \downarrow 0\), however slow, there exists a density f with x a Lebesgue point of f, such that
$$\displaystyle{\limsup _{n\rightarrow \infty }\frac{\mathbb{E}\left \vert f_{n}(\mathbf{x}) - f(\mathbf{x})\right \vert } {a_{n}} \geq 1.}$$
We will not show this, but just point to a similar theorem for the total variation error in density estimation (Devroye, 1987). However, under smoothness conditions, there is hope to get useful rates of convergence.

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Vorheriges Kapitel Weighted k-nearest neighbor density estimates
Nächstes Kapitel Entropy estimation
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Metadaten
Titel
Local behavior
verfasst von
Gérard Biau
Luc Devroye
Copyright-Jahr
2015
Buch
Lectures on the Nearest Neighbor Method

Print ISBN: 978-3-319-25386-2

Electronic ISBN: 978-3-319-25388-6

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-3-319-25388-6_6