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Fundamentals of Heavy Tails: Properties, Emergence, and Estimation [Kõva köide]

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(California Institute of Technology), (Stichting Centrum voor Wiskunde en Informatica (CWI), Amsterdam), (Indian Institute of Technology, Bombay)
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Fundamentals of Heavy Tails: Properties, Emergence, and EstimationSuurem pilt
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Püsilink: https://www.kriso.ee/db/9781316511732.html
Märksõnad:
Heavy tails - extreme events more common than expected - are everywhere, but they are still treated as mysterious and confusing because the necessary mathematical models are not widely known. For the first time, this book provides a rigorous introduction to heavy-tailed distributions accessible to anyone who knows elementary probability.

Heavy tails –extreme events or values more common than expected –emerge everywhere: the economy, natural events, and social and information networks are just a few examples. Yet after decades of progress, they are still treated as mysterious, surprising, and even controversial, primarily because the necessary mathematical models and statistical methods are not widely known. This book, for the first time, provides a rigorous introduction to heavy-tailed distributions accessible to anyone who knows elementary probability. It tackles and tames the zoo of terminology for models and properties, demystifying topics such as the generalized central limit theorem and regular variation. It tracks the natural emergence of heavy-tailed distributions from a wide variety of general processes, building intuition. And it reveals the controversy surrounding heavy tails to be the result of flawed statistics, then equips readers to identify and estimate with confidence. Over 100 exercises complete this engaging package.

Arvustused

'Heavy tailed distributions are ubiquitous in many disciplines which use probabilistic models. The book by Nair, Wierman and Zwart is a superb introduction to the topic and presents fundamental principles in a rigorous yet accessible manner. It is a must-read for researchers interested in understanding heavy tails.' R. Srikant, University of Illinois at Urbana-Champaign 'As one of the people who keeps discovering heavy tails in computer systems, I'm thrilled to see a book that delves into the deeper foundations behind these ubiquitous distributions. This beautifully written book is both mathematically precise and also full of intuitions and examples which make it accessible to newcomers in the field.' Mor Harchol-Balter, Carnegie Mellon University 'The book provides a fresh look at heavy-tailed probability distributions on the real line and their role in applied probability. The authors show that these distributions appear via natural algebraic operations. Their approach, towards understanding properties of these distributions, combines the key mathematical ideas alongside with informal explanations. Physical intuition is also provided, for example, the 'catastrophe/big jump principle' for heavy-tailed distributions versus the 'conspiracy principle' for light-tailed ones. The book is designed to help the practitioner and includes many interesting examples and exercises that may help to the reader to adjust and enjoy its content.' Sergey Foss, Heriot-Watt University

Muu info

An accessible yet rigorous package of probabilistic and statistical tools for anyone who must understand or model extreme events.
Preface ix
Acknowledgments xiii
1 Introduction
1(26)
1.1 Defining Heavy-Tailed Distributions
5(4)
1.2 Examples of Heavy-Tailed Distributions
9(15)
1.3 What's Next
24(1)
1.4 Exercises
24(3)
Part I Properties
27(78)
2 Scale Invariance, Power Laws, and Regular Variation
29(27)
2.1 Scale Invariance and Power Laws
30(2)
2.2 Approximate Scale Invariance and Regular Variation
32(4)
2.3 Analytic Properties of Regularly Varying Functions
36(12)
2.4 An Example: Closure Properties of Regularly Varying Distributions
48(2)
2.5 An Example: Branching Processes
50(3)
2.6 Additional Notes
53(1)
2.7 Exercises
54(2)
3 Catastrophes, Conspiracies, and Subexponential Distributions
56(29)
3.1 Conspiracies and Catastrophes
58(4)
3.2 Subexponential Distributions
62(5)
3.3 An Example: Random sums
67(5)
3.4 An Example: Conspiracies and Catastrophes in Random Walks
72(8)
3.5 Additional Notes
80(1)
3.6 Exercises
81(4)
4 Residual Lives, Hazard Rates, and Long Tails
85(20)
4.1 Residual Lives and Hazard Rates
86(4)
4.2 Heavy Tails and Residual Lives
90(3)
4.3 Long-Tailed Distributions
93(4)
4.4 An Example: Random Extrema
97(3)
4.5 Additional Notes
100(1)
4.6 Exercises
101(4)
Part II Emergence
105(70)
5 Additive Processes
107(20)
5.1 The Central Limit Theorem
108(4)
5.2 Generalizing the Central Limit Theorem
112(2)
5.3 Understanding Stable Distributions
114(4)
5.4 The Generalized Central Limit Theorem
118(2)
5.5 A Variation: The Emergence of Heavy Tails in Random Walks
120(4)
5.6 Additional Notes
124(1)
5.7 Exercises
125(2)
6 Multiplicative Processes
127(21)
6.1 The Multiplicative Central Limit Theorem
128(3)
6.2 Variations on Multiplicative Processes
131(7)
6.3 An Example: Preferential Attachment and Yule Processes
138(6)
6.4 Additional Notes
144(1)
6.5 Exercises
145(3)
7 Extremal Processes
148(27)
7.1 A Limit Theorem for Maxima
150(4)
7.2 Understanding Max-Stable Distributions
154(2)
7.3 The Extremal Central Limit Theorem
156(5)
7.4 An Example: Extremes of Random Walks
161(7)
7.5 A Variation: The Time between Record Breaking Events
168(2)
7.6 Additional Notes
170(1)
7.7 Exercises
171(4)
Part III Estimation
175(63)
8 Estimating Power-Law Distributions: Listen to the Body
177(20)
8.1 Parametric Estimation of Power-Laws Using Linear Regression
179(6)
8.2 Maximum Likelihood Estimation for Power-Law Distributions
185(2)
8.3 Properties of the Maximum Likelihood Estimator
187(2)
8.4 Visualizing the MLE via Regression
189(3)
8.5 A Recipe for Parametric Estimation of Power-Law Distributions
192(2)
8.6 Additional Notes
194(1)
8.7 Exercises
195(2)
9 Estimating Power-Law Tails: Let the Tail Do the Talking
197(41)
9.1 The Failure of Parametric Estimation
199(4)
9.2 The Hill Estimator
203(2)
9.3 Properties of the Hill Estimator
205(4)
9.4 The Hill Plot
209(6)
9.5 Beyond Hill and Regular Variation
215(11)
9.6 Where Does the Tail Begin?
226(7)
9.7 Guidelines for Estimating Heavy-Tailed Phenomena
233(2)
9.8 Additional Notes
235(1)
9.9 Exercises
236(2)
Commonly Used Notation 238(2)
References 240(9)
Index 249
Jayakrishnan Nair is Associate Professor in Electrical Engineering at IIT Bombay. His research focuses on modeling, performance evaluation, and design issues in online learning environments, communication networks, queueing systems, and smart power grids. He is the recipient of best paper awards at IFIP Performance (2010 and 2020) and ACM e-Energy (2020). Adam Wierman is Professor of Computing and Mathematical Sciences at the California Institute of Technology (Caltech). His research develops tools in machine learning, optimization, control, and economics with the goal of making the networked systems that govern our world sustainable and resilient. He is best known for his work spearheading the design of algorithms for sustainable data centers and he is the recipient of numerous awards including the ACM Sigmetrics Rising Star award, the ACM Sigmetrics Test of Time award, the IEEE Communication Society William Bennet Prize, and multiple teaching and best paper awards. Bert Zwart is group leader at CWI Amsterdam and Professor of Mathematics at Eindhoven University of Technology. He has expertise in stochastic operations research, queueing theory, and large deviations, and in the context of heavy tails, he has focused on sample path properties, designing Monte Carlo methods and applications to computer-communication and energy networks. He was area editor of Operations Research, the flagship journal of his profession, from 2009 to 2017, and was the recipient of the INFORMS Applied Probability Society Erlang prize, awarded every two years to an outstanding young applied probabilist.