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2017 | OriginalPaper | Chapter

3. Descriptive Statistics

Author : Johann Pfanzagl

Published in: Mathematical Statistics

Publisher: Springer Berlin Heidelberg

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Abstract

This essay discusses different structural descriptions of families of distributions, in particular the distinction between parameters and functionals, and between estimators and estimands. We also interpret and compare different stochastic orders between distributions on the real line, in particular spread orders and orders of concentration about a given center like the peak order and the Löwner order. We describe Anderson’s theorem, which says that the convolution of a symmetric and unimodal distribution on a Euclidean space with another distribution reduces the concentration about the origin in symmetric and convex sets. We discuss convolutions and spread orders, the interpretation of convolution products, and the relations between risks and measures of concentration for distributions of estimators. We also study the concept Pitman closeness between estimators of real-valued functionals.

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previous chapter Sufficiency
next chapter Optimality of Unbiased Estimators: Nonasymptotic Theory
Footnotes
1
According to Pfanzagl (1994, p. 86, Corollary 2.4.10), the function \(y\rightarrow \int f(x,y)\lambda ^k(dx)\) is concave if f is unimodal. This stronger assertion is obviously wrong: It would imply that any subconvex function is concave, since \((x,y)\rightarrow 1_{[0,1]^k}(x)\ell (y)\) is subconvex if \(\ell \) is subconvex, and \(\ell (y)=\int 1_{[0,1]^k}(x)\ell (y)\lambda ^k(dx)\). The proof uses that \(y\rightarrow \lambda ^k(C_y)\) is concave, which is true only on \(\{y\in \mathbb R^m:\lambda ^k(C_y)>0\}\), not on \(\mathbb R^m\). None of the reviewers mentioned this blunder.
 
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Metadata
Title
Descriptive Statistics
Author
Johann Pfanzagl
Copyright Year
2017
Publisher
Springer Berlin Heidelberg
Book
Mathematical Statistics

Print ISBN: 978-3-642-31083-6

Electronic ISBN: 978-3-642-31084-3

Copyright Year: 2017

DOI
https://doi.org/10.1007/978-3-642-31084-3_3

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