Univariate Stable Distributions

Models for Heavy Tailed Data

verfasst von: John P. Nolan

Verlag: Springer International Publishing

Buchreihe : Springer Series in Operations Research and Financial Engineering

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This textbook highlights the many practical uses of stable distributions, exploring the theory, numerical algorithms, and statistical methods used to work with stable laws. Because of the author’s accessible and comprehensive approach, readers will be able to understand and use these methods. Both mathematicians and non-mathematicians will find this a valuable resource for more accurately modelling and predicting large values in a number of real-world scenarios.
Beginning with an introductory chapter that explains key ideas about stable laws, readers will be prepared for the more advanced topics that appear later. The following chapters present the theory of stable distributions, a wide range of applications, and statistical methods, with the final chapters focusing on regression, signal processing, and related distributions. Each chapter ends with a number of carefully chosen exercises. Links to free software are included as well, where readers can put these methods into practice.
Univariate Stable Distributions is ideal for advanced undergraduate or graduate students in mathematics, as well as many other fields, such as statistics, economics, engineering, physics, and more. It will also appeal to researchers in probability theory who seek an authoritative reference on stable distributions.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Basic Properties of Univariate Stable Distributions

Abstract

Stable distributions are a rich class of probability distributions that allow skewness and heavy tails and have many intriguing mathematical properties. The class was characterized by Paul Lévy in his study of sums of independent identically distributed terms in the 1920s.

John P. Nolan

Chapter 2. Modeling with Stable Distributions

Abstract

Stable distributions have been proposed as a model for many types of physical and economic systems. There are several reasons for using a stable distribution to describe a system. The first is where there are solid theoretical reasons for expecting a non-Gaussian stable model, e.g. reflection off a rotating mirror yielding a Cauchy distribution, hitting times for a Brownian motion yielding a Lévy distribution, the gravitational field of stars yielding the Holtsmark distribution; see below for these and other examples.

John P. Nolan

Chapter 3. Technical Results for Univariate Stable Distributions

Abstract

This chapter contains proofs of the results stated in Chapter 1, as well as mathematical results about stable distributions. Both theoretical and computational expressions are derived for stable densities and distribution functions. Readers who are primarily interested in using stable distributions in applications may want to skip this chapter and return to it later for specific facts they need.

John P. Nolan

Chapter 4. Univariate Estimation

Abstract

This chapter describes various methods of estimating stable parameters: among them tail, quantile, empirical characteristic function, and maximum likelihood methods. Unless stated otherwise, we will assume that \(X_1,X_2,\ldots ,X_n\) are a stable sample, i.e. independent and identically distributed copies with a stable law. In most cases, the continuous \({{\mathbf {S}}\left( \alpha , \beta ,\gamma ,\delta ;0\right) }\) parameterization is used to avoid problems arising from the discontinuity at \(\alpha =1\). Diagnostics for assessing stability are discussed and applications to several data sets are given. The performance of the methods are compared in simulations in Section 4.9.

John P. Nolan

Chapter 5. Stable Regression

Abstract

Ordinary least squares (OLS) is a well-established and important procedure for solving regression problems. In the case of regression with normally distributed errors, the OLS solution is the same as the maximum likelihood solution. Although small departures from normality do not affect the model greatly, errors from a heavy tailed distribution will generally result in extreme observations that can greatly affect the estimated OLS regression coefficients.

John P. Nolan

Chapter 6. Signal Processing with Stable Distributions

Abstract

In many engineering problems, heavy tailed noise occurs in a variety of settings. In the engineering literature, this is referred to as impulsive or spiky noise.

John P. Nolan

Chapter 7. Related Distributions

Abstract

This chapter contains brief discussions of distributions related to stable laws. Except when there is something related to our main interests, proofs are not given.

John P. Nolan

Backmatter

Titel: Univariate Stable Distributions
verfasst von: John P. Nolan
Verlag: Springer International Publishing
Electronic ISBN: 978-3-030-52915-4
Print ISBN: 978-3-030-52914-7
DOI: https://doi.org/10.1007/978-3-030-52915-4