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2013 | Buch

The Methods of Distances in the Theory of Probability and Statistics

verfasst von: Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank Fabozzi

Verlag: Springer New York

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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)

Inhaltsverzeichnis

Frontmatter
Chapter 1. Main Directions in the Theory of Probability Metrics
Abstract
Increasingly, the demands of various real-world applications in the sciences, engineering, and business have resulted in the creation of new, more complicated probability models. In the construction and evaluation of these models, model builders have drawn on well-developed limit theorems in probability theory and the theory of random processes. The study of limit theorems in general spaces and a number of other questions in probability theory make it necessary to introduce functionals – defined on either classes of probability distributions or classes of random elements – and to evaluate their nearness in one or another probabilistic sense. Thus various metrics have appeared including the well-known Kolmogorov (uniform) metric, L p metrics, the Prokhorov metric, and the metric of convergence in probability (Ky Fan metric). We discuss these measures and others in the chapters that follow.
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi

General topics in the theory of probability metrics

Frontmatter
Chapter 2. Probability Distances and Probability Metrics: Definitions
Abstract
The goals of this chapter are to: Provide examples of metrics in probability theory; Introduce formally the notions of a probability metric and a probability distance;
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi
Chapter 3. Primary, Simple, and Compound Probability Distances and Minimal and Maximal Distances and Norms
Abstract
The goals of this chapter are to: Formally introduce primary, simple, and compound probability distances; Provide examples of and study the relationship between primary, simple, and compound distances; Introduce the notions of minimal probability distance, minimal norms, cominimal functionals, and moment functions, which are needed in the study of primary, simple, and compound probability distances.
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi
Chapter 4. A Structural Classification of Probability Distances
Abstract
The goals of this chapter are to: Introduce and motivate three classifications of probability metrics according to their metric structure, Provide examples of probability metrics belonging to a particular structural group, Discuss the generic properties of the structural groups and the links between them.
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi

Relations between compound, simple and primary distances

Frontmatter
Chapter 5. Monge–Kantorovich Mass Transference Problem, Minimal Distances and Minimal Norms
Abstract
The goals of this chapter are to: Introduce the Kantorovich and Kantorovich–Rubinstein problems in one-dimensional and multidimensional settings; Provide examples illustrating applications of the abstract problems; Provide examples illustrating applications of the abstract problems; Discuss the multivariate Kantorovich and Kantorovich–Rubinstein theorems, which provide dual representations of certain types of minimal distances and norms; Discuss a particular application leading to an explicit representation for a class of minimal norms.
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi
Chapter 6. Quantitative Relationships Between Minimal Distances and Minimal Norms
Abstract
The goals of this chapter are to: Explore the conditions under which there is equality between the Kantorovich and the Kantorovich–Rubinstein functionals; Provide inequalities between the Kantorovich and Kantorovich–Rubinstein functionals; Provide criteria for convergence, compactness, and completeness of probability measures in probability spaces involving the Kantorovich and Kantorovich–Rubinstein functionals; Analyze the problem of uniformity between the two functionals.
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi
Chapter 7. K -Minimal Metrics
Abstract
The goals of this chapter are to: Define the notion of K-minimal metrics and describe their general properties; Provide representations of the K-minimal metrics with respect to several particular metrics such as the Lévy metric, Kolmogorov metric, and p-average metric; Consider K-minimal metrics when probability measures are defined on a general separable metric space; Provide relations between the multidimensional Kantorovich and Strassen theorems.
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi
Chapter 8. Relations Between Minimal and Maximal Distances
Abstract
The goals of this chapter are to: Discuss dual representations of the maximal distances \(\check{\mu }_{c}\) and \(\mu ^{(s)}_{c}\) and to compare them with the corresponding dual representations of the minimal metric \(\widehat{\mu }\) and minimal norm \(\mu ^{ \circ }_{c}\), Provide closed-form expressions for \(\check{\mu }_{c}\) and \(\mu ^{(s)}_{c}\) in some special cases, Study the topological structure of minimal distances and minimal norms.
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi
Chapter 9. Moment Problems Related to the Theory of Probability Metrics: Relations Between Compound and Primary Distances
Abstract
Explore the general relations between compound and primary probability distances that are similar to the relations between compound and simple probability distances
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi

Applications of minimal probability distances

Frontmatter
Chapter 10. Moment Distances
Abstract
In this chapter we show that in some cases the investigation of the convergence of a sequence of distributions {F n } to a prescribed distribution function (DF) F (or to a prescribed class \(\mathcal{K}\) of DFs) can be replaced by studying the convergence of certain characteristics of F n to the corresponding characteristics of F (or characteristics of \(\mathcal{K}\)).
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi
Chapter 11. Uniformity in Weak and Vague Convergence
Abstract
The goals of this chapter are to: • Extend the notion of uniformity, • Study the metrization of weak convergence, • Describe the notion of vague convergence, • Consider the question of its metrization.
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi
Chapter 12. Glivenko–Cantelli Theorem and Bernstein–Kantorovich Invariance Principle
Abstract
This chapter begins with an application of the theory of probability metrics to the problem of convergence of the empirical probability measure.
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi
Chapter 13. Stability of Queueing Systems
Abstract
The subject of this chapter is the fundamental problem of the stability of a sequence of stochastic models that can be interpreted as approximations or perturbations of a given initial model.
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi
Chapter 14. Optimal Quality Usage
Abstract
In this chapter, we discuss the problem of optimal quality usage as a multidimensional Monge–Kantorovich problem. We begin by stating and interpreting the one-dimensional and the multidimensional problems. We provide conditions for optimality and weak optimality in the multivariate case for particular choices of the cost function. Finally, we derive an upper bound for the minimal total losses for a special choice of the cost function and compare it to the upper bound involving the first difference pseudomoment.
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi

Ideal metrics

Frontmatter
Chapter 15. Ideal Metrics with Respect to Summation Scheme for i.i.d. Random Variables
Abstract
The subject of this chapter is the application of the theory of probability metrics to limit theorems arising from summing independent and identically distributed (i.i.d.) random variables (RVs).
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi
Chapter 16. Ideal Metrics and Rate of Convergence in the CLT for Random Motions
Abstract
The ideas developed in Chap. 15 are discussed in this chapter in the context of random motions defined on \({\mathbb{R}}^{d}\).
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi
Chapter 17. Applications of Ideal Metrics for Sums of i.i.d. Random Variables to the Problems of Stability and Approximation in Risk Theory
Abstract
In this chapter, we present applications of ideal probability metrics to insurance risk theory. First, we describe and analyze themathematical framework.
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi
Chapter 18. How Close Are the Individual and Collective Models in Risk Theory?
Abstract
The subject of this chapter is individual and collective models in insurance risk theory and how ideal probability metrics can be employed to calculate the distance between them.
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi
Chapter 19. Ideal Metric with Respect to Maxima Scheme of i.i.d. Random Elements
Abstract
In this chapter, we discuss the problem of estimating the rate of convergence in limit theorems arising from the maxima scheme of independent and identically distributed (i.i.d.) random elements.
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi
Chapter 20. Ideal Metrics and Stability of Characterizations of Probability Distributions
Abstract
No probability distribution is a true representation of the probabilistic law of a given random phenomenon: assumptions such as normality, exponentiality, and the like are seldom if ever satisfied in practice.
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi

Euclidean-like distances and their applications

Frontmatter
Chapter 21. Positive and Negative Definite Kernels and Their Properties
Abstract
The goals of this chapter are to: Formally introduce positive and negative definite kernels, Describe the properties of positive and negative definite kernels, Provide examples of positive and negative definite kernels and to characterize coarse embeddings in a Hilbert space, Introduce strictly and strongly positive and negative definite kernels.
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi
Chapter 22. Negative Definite Kernels and Metrics: Recovering Measures from Potentials
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.
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi
Chapter 23. Statistical Estimates Obtained by the Minimal Distances Method
Abstract
The goals of this chapter are to: Consider the problem of parameter estimation by the method of minimal distances, Study the properties of the estimators.
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi
Chapter 24. Some Statistical Tests Based on $$\mathfrak{N}$$ -Distances
Abstract
In this chapter, we construct statistical tests based on the theory of \(\mathfrak{N}\)-distances. We consider a multivariate two-sample test, a test to determine if two distributions belong to the same additive type, and tests for multivariate normality with unknown mean and covariance matrix.
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi
Chapter 25. Distances Defined by Zonoids
Abstract
The goals of this chapter are to: • Introduce \(\mathfrak{N}\)-distances defined by zonoids, • Explain the connections between \(\mathfrak{N}\)-distances and zonoids.
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi
Chapter 26. $$\mathfrak{N}$$ -Distance Tests of Uniformity on the Hypersphere
Abstract
The goals of this chapter are to: Discuss statistical tests of uniformity based on the \(\mathfrak{N}\)-distance theory, Calculate the asymptotic distribution of the test statistic.
Svetlozar T. Rachev, Lev B. Klebanov, Stoyan V. Stoyanov, Frank J. Fabozzi
Backmatter
Metadaten
Titel
The Methods of Distances in the Theory of Probability and Statistics
verfasst von
Svetlozar T. Rachev
Lev B. Klebanov
Stoyan V. Stoyanov
Frank Fabozzi
Copyright-Jahr
2013
Verlag
Springer New York
Electronic ISBN
978-1-4614-4869-3
Print ISBN
978-1-4614-4868-6
DOI
https://doi.org/10.1007/978-1-4614-4869-3