The Methods of Distances in the Theory of Probability and Statistics

This book covers the method of metric distances and its application in probability theory and other fields. The method is fundamental in the study of limit theorems and generally in assessing the quality of approximations to a given probabilistic model. The method of metric distances is developed to study stability problems and reduces to the selection of an ideal or the most appropriate metric for the problem under consideration and a comparison of probability metrics.
After describing the basic structure of probability metrics and providing an analysis of the topologies in the space of probability measures generated by different types of probability metrics, the authors study stability problems by providing a characterization of the ideal metrics for a given problem and investigating the main relationships between different types of probability metrics. The presentation is provided in a general form, although specific cases are considered as they arise in the process of finding supplementary bounds or in applications to important special cases.
Svetlozar T. Rachev is the Frey Family Foundation Chair of Quantitative Finance, Department of Applied Mathematics and Statistics, SUNY-Stony Brook and Chief Scientist of Finanlytica, USA. Lev B. Klebanov is a Professor in the Department of Probability and Mathematical Statistics, Charles University, Prague, Czech Republic. Stoyan V. Stoyanov is a Professor at EDHEC Business School and Head of Research, EDHEC-Risk Institute-Asia (Singapore). Frank J. Fabozzi is a Professor at EDHEC Business School. (USA)

List of contents

Main directions in the theory of probability metrics.- Probability distances and probability metrics: Definitions.- Primary, simple and compound probability distances, and minimal and maximal distances and norms.- A structural classification of probability distances.-Monge-Kantorovich mass transference problem, minimal distances and minimal norms.- Quantitative relationships between minimal distances and minimal norms.- K-Minimal metrics.- Relations between minimal and maximal distances.- Moment problems related to the theory of probability metrics: Relations between compound and primary distances.- Moment distances.- Uniformity in weak and vague convergence.- Glivenko-Cantelli theorem and Bernstein-Kantorovich invariance principle.- Stability of queueing systems.-Optimal quality usage.- Ideal metrics with respect to summation scheme for i.i.d. random variables.- Ideal metrics and rate of convergence in the CLT for random motions.- Applications of ideal metrics for sums of i.i.d. random variables to the problems of stability and approximation in risk theory.- How close are the individual and collective models in risk theory?- Ideal metric with respect to maxima scheme of i.i.d. random elements.- Ideal metrics and stability of characterizations of probability distributions.- Positive and negative de nite kernels and their properties.- Negative definite kernels and metrics: Recovering measures from potential.- Statistical estimates obtained by the minimal distances method.-Some statistical tests based on N-distances.- Distances defined by zonoids.- N-distance tests of uniformity on the hypersphere.-

About the author

Svetlozar T. Rachev is a Professorin Department of Applied Mathematics and Statistics, SUNY-Stony Brook. Lev B. Klebanov is a Professor in the Department of Probability and Mathematical Statistics, MFF, Charles University, Prague, Czech Republic. Stoyan V. Stoyanov is a Professor of Finance, EDHEC Business School, Head of Research, EDHEC-Risk Institute. Frank J. Fabozzi is a Professor of Finance, EDHEC Business School

Summary

This book covers the method of metric distances and its application in probability theory and other fields. The method is fundamental in the study of limit theorems and generally in assessing the quality of approximations to a given probabilistic model. The method of metric distances is developed to study stability problems and reduces to  the selection of an ideal or the most appropriate metric for the problem under consideration and a comparison of probability metrics.
After describing the basic structure  of probability metrics and providing an analysis of the topologies in the space of probability measures generated by different types of probability metrics, the authors study stability problems by providing a characterization of the ideal metrics for a given problem and investigating the main relationships between different types of probability metrics. The presentation is provided in a general form, although specific cases are considered as they arise in the process of finding supplementary bounds or in applications to important special cases.
      Svetlozar T.  Rachev is the Frey Family Foundation Chair of Quantitative Finance, Department of Applied Mathematics and Statistics, SUNY-Stony Brook  and Chief Scientist of Finanlytica, USA. Lev B. Klebanov is a Professor in the Department of Probability and Mathematical Statistics, Charles University, Prague, Czech Republic. Stoyan V. Stoyanov is a Professor at EDHEC Business School and Head of Research, EDHEC-Risk Institute—Asia (Singapore).  Frank J. Fabozzi is a Professor at EDHEC Business School. (USA)

Additional text

From the book reviews:
 “This textbook gives a comprehensive overview of the method of metric distances and its applications in probability theory. … The text is mainly self-contained and should be accessible for readers with basic knowledge in probability theory. The exposition is well structured and covers an impressive range of topics around the central theme of probability metrics.” (Hilmar Mai, zbMATH, Vol. 1280, 2014)
“The reviewed book is divided into five parts. … The target audience is graduate students in the areas of functional analysis, geometry, mathematical programming, probability, statistics, stochastic analytics, and measure theory. The book can also be used for students in probability and statistics. The theory of probability metrics presented here can be applied to engineering, physics, chemistry, information theory, economics, and finance. Specialists from the aforementioned areas might find the book useful.” (Adriana Horníková, Technometrics, Vol. 55 (4), November, 2013)

Report

From the book reviews:
"This textbook gives a comprehensive overview of the method of metric distances and its applications in probability theory. ... The text is mainly self-contained and should be accessible for readers with basic knowledge in probability theory. The exposition is well structured and covers an impressive range of topics around the central theme of probability metrics." (Hilmar Mai, zbMATH, Vol. 1280, 2014)
"The reviewed book is divided into five parts. ... The target audience is graduate students in the areas of functional analysis, geometry, mathematical programming, probability, statistics, stochastic analytics, and measure theory. The book can also be used for students in probability and statistics. The theory of probability metrics presented here can be applied to engineering, physics, chemistry, information theory, economics, and finance. Specialists from the aforementioned areas might find the book useful." (Adriana Horníková, Technometrics, Vol. 55 (4), November, 2013)

Product details

• Authors: Frank Fabozzi, Le Klebanov, Lev Klebanov, Svetlozar Rachev, Svetlozar T Rachev, Svetlozar T. Rachev, Stoyan Stoyanov, Stoyan V. Stoyanov
• Publisher: Springer, Berlin
• Languages: English
• Product format: Paperback / Softback
• Released: 01.01.2012
• EAN: 9781489995698
• ISBN: 978-1-4899-9569-8
• No. of pages: 619
• Dimensions: 156 mm x 35 mm x 236 mm
• Weight: 961 g
• Illustrations: XVI, 619 p.
• Subjects: Natural sciences, medicine, IT, technology > Mathematics > Probability theory, stochastic theory, mathematical statistics; B, Statistics, Mathematics and Statistics, Statistical Theory and Methods, Probability Theory and Stochastic Processes, Probability & statistics, Differential calculus & equations, Probabilities, Stochastics, Probability Theory, Approximations and Expansions, Approximation theory