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Main Directions in the Theory of Probability Metrics Read chapter Read first chapter

2013 | OriginalPaper | Chapter

22. Negative Definite Kernels and Metrics: Recovering Measures from Potentials

Authors : Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi

Published in: The Methods of Distances in the Theory of Probability and Statistics

Publisher: Springer New York

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Abstract

Introduce probability metrics through strongly negative definite kernel functions and provide examples, Introduce probability metrics through m-negative definite kernels and provide examples, Introduce the notion of potential corresponding to a probability measure, Present the problem of recovering a probability measure from its potential, Consider the relation between the problems of convergence of measures and the convergence of their potentials, Characterize probability distributions using the theory of recovering probability measures from potentials.

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previous chapter Positive and Negative Definite Kernels and Their Properties
next chapter Statistical Estimates Obtained by the Minimal Distances Method
Footnotes
1
Sriperumbudur et al. [2010] discuss metrics similar to the \(\mathfrak{N}\)-distances that we cover in this chapter. However, the results they present were already reported in the literature.
 
2
See, for example, Akhiezer [1961].
 
3
See Vakhaniya et al. [1985].
 
4
See Klebanov and Zinger [1990].
 
5
See, for example, Kagan et al. [1973].
 
6
See, for example, Kakosyan et al. [1984].
 
7
See Klebanov et al. [2001].
 
8
See Klebanov et al. [2001].
 
Literature
Akhiezer NI (1961) The classical moment problem. Gosudarstv. Izdat. Fiz-Mat. Lit., Moscow (in Russian)
Braverman MSh (1987) A method for the characterization of probability distributions. Teor Veroyatnost i Primenen (in Russian) 32:552–556MathSciNetMATH
Gorin EA, Koldobskii AL (1987) Measure potentials in Banach spaces. Sibirsk Mat Zh 28(1):65–80MathSciNet
Kagan AM, Linnik YV, Rao CR (1973) Characterization problems of mathematical statistics. Wiley, New York
Kakosyan AV, Klebanov LB, Melamed IA (1984) Characterization problems of mathematical statistics. In: Lecture notes in mathematics, vol 1088. Springer, Berlin
Klebanov LB, Zinger AA (1990) Characterization of distributions: problems, methods, applications. In: Grigelionis B, et al (eds) Probability theory and mathematical Statistics VSP/TEV, vol 1, pp 611–617
Klebanov LB, Kozubowski TJ, Rachev ST, Volkovich VE (2001) Characterization of distributions symmetric with respect to a group of transformations and testing of corresponding statistical hypothesis. Statist Prob Lett 53:241–247MathSciNetMATHCrossRef
Koldobskii L (1982) Isometric operators in vector-valued \({L}^{p}\)-spaces. Zap Nauchn Sem Leningrad Otdel Mat Inst Steklov 107:198–203MathSciNet
Plotkin AI (1970) Isometric operators on \({L}^{p}\)-spaces. Dokl Akad Nauk 193(3):537–539 (in Russian)MathSciNet
Plotkin AI (1971) Extensions of \({L}^{p}\) isometries. Zap Nauchn Sem Leningrad Otdel Mat Inst Steklov (in Russian) 22:103–129MathSciNetMATH
Rudin W (1976) \({L}^{p}\) isometries and equimeasurability. Indiana Univ Math J 25(3):215–228
Sriperumbudur BA, Fukumizu GK, Schölkopf B (2010) Hilbert space embeddings and metrics on probability measures. J Mach Learn Res 11:1517–1561MathSciNetMATH
Vakhaniya NN, Tarieladze VI, Chobanyan SA (1985) Probability distributions in Banach spaces. Nauka, Moscow (in Russian)
Zinger AA, Klebanov L (1991) Characterization of distribution symmetry by moment properties. In: Stability problems of stochastic models, Moscow: VNII Sistemnykh Isledovaniii pp 70–72 (in Russian)
Metadata
Title
Negative Definite Kernels and Metrics: Recovering Measures from Potentials
Authors
Svetlozar T. Rachev
Lev B. Klebanov
Stoyan V. Stoyanov
Frank J. Fabozzi
Copyright Year
2013
Publisher
Springer New York
Book
The Methods of Distances in the Theory of Probability and Statistics

Print ISBN: 978-1-4614-4868-6

Electronic ISBN: 978-1-4614-4869-3

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-1-4614-4869-3_22