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.
Author(s): John P. Nolan
Series: Springer Series in Operations Research and Financial Engineering
Publisher: Springer
Year: 2020
Language: English
Pages: 333
City: Cham
Preface
Contents
1 Basic Properties of Univariate Stable Distributions
1.1 Definition of stable random variables
1.2 Other definitions of stability
1.3 Parameterizations of stable laws
1.4 Densities and distribution functions
1.5 Tail probabilities, moments, and quantiles
1.6 Sums of stable random variables
1.7 Simulation
1.8 Generalized Central Limit Theorem and Domains of Attraction
1.9 Multivariate stable
1.10 Problems
2 Modeling with Stable Distributions
2.1 Lighthouse problem
2.2 Distribution of masses in space
2.3 Random walks
2.4 Hitting time for Brownian motion
2.5 Differential equations and fractional diffusions
2.6 Financial applications
2.6.1 Stock returns
2.6.2 Value-at-risk and expected shortfall
2.6.3 Other financial applications
2.6.4 Multiple assets
2.7 Signal processing
2.8 Miscellaneous applications
2.8.1 Stochastic resonance
2.8.2 Network traffic and queues
2.8.3 Earth Sciences
2.8.4 Physics
2.8.5 Embedding of Banach spaces
2.8.6 Hazard function, survival analysis, and reliability
2.8.7 Biology and medicine
2.8.8 Discrepancies
2.8.9 Computer Science
2.8.10 Long tails in business, political science, and medicine
2.8.11 Extreme values models
2.9 Behavior of the sample mean and variance
2.10 Appropriateness of infinite variance models
2.11 Problems
3 Technical Results for Univariate Stable Distributions
3.1 Proofs of Basic Theorems of Chapter 1摥映數爠eflinkchap:basic.univariate11
3.1.1 Stable distributions as infinitely divisible distributions
3.2 Densities and distribution functions
3.2.1 Series expansions
3.2.2 Modes
3.2.3 Duality
3.3 Numerical algorithms
3.3.1 Computation of distribution functions and densities
3.3.2 Spline approximation of densities
3.3.3 Simulation
3.4 Functions gd, widetildegd, hd and widetildehd
3.4.1 Score functions
3.5 More on parameterizations
3.6 Tail behavior
3.7 Moments and other transforms
3.8 Convergence of stable laws in terms of (α,β,γ,δ)
3.9 Combinations of stable random variables
3.10 Distributions derived from stable distributions
3.10.1 Log-stable
3.10.2 Exponential stable
3.10.3 Amplitude of a stable random variable
3.10.4 Ratios and products of stable terms
3.10.5 Wrapped stable distribution
3.10.6 Discretized stable distributions
3.11 Stable distributions arising as functions of other distributions
3.11.1 Exponential power distributions
3.11.2 Stable mixtures of extreme value distributions
3.12 Stochastic series representations
3.13 Generalized Central Limit Theorem and Domains of Attraction
3.14 Entropy
3.15 Differential equations and stable semi-groups
3.16 Problems
4 Univariate Estimation
4.1 Order statistics
4.2 Tail-based estimation
4.2.1 Hill estimator
4.3 Extreme value estimate of α
4.4 Quantile-based estimation
4.5 Characteristic function-based estimation
4.5.1 Choosing values of u
4.6 Moment-based estimation
4.7 Maximum likelihood estimation
4.7.1 Asymptotic normality and Fisher information matrix
4.7.2 The score function for δ
4.8 Other methods of estimation
4.8.1 Log absolute value estimation
4.8.2 U statistic-based estimation
4.8.3 Miscellaneous methods
4.9 Comparisons of estimators
4.9.1 Using overlinex and s to estimate location and scale
4.9.2 Statistical efficiency and execution time
4.10 Assessing a stable fit
4.10.1 Graphical diagnostics
4.10.2 Likelihood ratio tests and goodness-of-fit tests
4.11 Applications
4.12 Estimation when in the domain of attraction
4.13 Fitting stable distributions to concentration data
4.14 Estimation for discretized stable distributions
4.15 Problems
5 Stable Regression
5.1 Maximum likelihood estimation of the regression coefficients
5.1.1 Parameter confidence intervals: linear case
5.1.2 Linear examples
5.2 Nonlinear regression
5.2.1 Parameter confidence intervals: nonlinear case
5.2.2 Nonlinear example
5.3 Problems
6 Signal Processing with Stable Distributions
6.1 Unweighted stable filters
6.2 Weighted and matched stable filters
6.3 Calibration and numerical issues
6.3.1 Calibration
6.3.2 Evaluating and minimizing the cost function
6.4 Evaluation of stable filters
6.5 Problems
7 Related Distributions
7.1 Pareto distributions
7.2 t distributions
7.3 Other types of stability
7.3.1 Max-stable and min-stable
7.3.2 Multiplication-stable
7.3.3 Geometric-stable distributions and Linnik distributions
7.3.4 Discrete stable
7.3.5 Generalized convolutions and generalized stability
7.4 Mixtures of stable distributions: scale, sum, and convolutions
7.5 Infinitely divisible distributions
7.6 Problems
A Mathematical Facts
A.1 Sums of random variables
A.2 Symmetric random variables
A.3 Moments
A.4 Characteristic functions
A.5 Laplace transforms
A.6 Mellin transforms
A.7 Gamma and related functions
B Stable Quantiles
C Stable Modes
D Asymptotic standard deviations and correlation coefficients for ML estimators
References
Index
Author Index
Symbol Index
