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Heavy-tail phenomena: probabilistic and statistical modeling

Author: Resnick, Sidney I. Series: Springer series in operations research and financial engineering Publisher: Springer, 2007.Language: EnglishDescription: 404 p. ; 24 cm.ISBN: 0387242724Type of document: BookBibliography/Index: Includes bibliographical references and index
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Book Europe Campus
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Print QA273 .R47 2007
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001213168
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Includes bibliographical references and index

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Heavy-Tail Phenomena Probabilistic and Statistical Modeling Contents Preface ......................................................................................................................... v Acknowledgments..................................................................................................... vii 1 Introduction ............................................................................................................. 1 1.1 Welcome .......................................................................................................... 1 1.2 Survey .............................................................................................................. 1 1.3 Context and examples ...................................................................................... 3 1.3.1 Data networks ........................................................................................ 3 1.3.2 Finance................................................................................................... 5 Value-at-risk........................................................................................ 9 1.3.3 Insurance and reinsurance ................................................................... 13 Part I Crash Courses 2 Crash Course I: Regular Variation ..................................................................... 2.1 Preliminaries from analysis ........................................................................... 2.1.1 Uniform convergence ......................................................................... 2.1.2 Inverses of monotone functions........................................................... 2.1.3 Convergence of monotone functions................................................... 2.1.4 Cauchy's functional equation............................................................... 2.2 Regular variation: Definition and first properties ......................................... 17 17 17 18 19 20 20 2.2.1 A maximal domain of attraction.......................................................... 23 2.3 Regular variation: deeper results; Karamata's theorem ................................. 24 2.3.1 Uniform convergence.......................................................................... 24 2.3.2 Integration and Karamata's theorem................................................... 25 2.3.3 Karamata's representation ................................................................. 29 2.3.4 Differentiation.................................................................................... 30 2.4 Regular variation: Further properties ............................................................ 32 2.5 Problems ......................................................................................................... 35 3 Crash Course II: Weak Convergence; Implications for Heavy-Tail Analysis................................................................................................................ 39 3.1 Definitions ...................................................................................................... 39 3.2 Basic properties of weak convergence ........................................................... 40 3.2.1 Portmanteau theorem ......................................................................... 40 3.2.2 Skorohod's theorem ........................................................................... 41 3.2.3 Continuous mapping theorem ............................................................ 42 3.2.4 Subsequences and Prohorov's theorem.............................................. 43 3.3 Some useful metric spaces............................................................................ 44 3.3.1 Rd, finite-dimensional Euclidean space ............................................. 44 3.3.2 R " , sequence space .......................................................................... 45 3.3.3 C[0, 1] and C[0, oo), continuous functions ....................................... 45 3.3.4 D[O, 1] and D[O, oo) ......................................................................... 46 3.3.5 Radon measures and point measures; vague convergence ................ 48 Spaces of measures........................................................................... 48 Convergence concept ....................................................................... 49 The vague topology; more on M± (E) (and hence, more on Mp(E))............................................................................ 51 3.4 How to prove weak convergence .................................................................. 53 3.4.1 Methods in spaces useful for heavy-tail analysis............................... 53 3.4.2 Donsker's theorem ............................................................................. 54 3.5 New convergences from old ......................................................................... 55 3.5.1 Slutsky approximations ...................................................................... 55 3.5.2 Combining convergences.................................................................... 57 3.5.3 Inversion techniques........................................................................... 58 Inverses ............................................................................................. 58 Vervaat's lemma ............................................................................... 59 3.6 Vague convergence and regular variation..................................................... 61 Vague convergence on (0, oo] .......................................................... 62 3.7 Problems ....................................................................................................... 64 Part II Statistics 4 Dipping a Toe in the Statistical Water................................................................. 73 4.1 Statistical inference for heavy tails: This is a song about a .......................... 73 4.2 Exceedances, thresholds, and the POT method.............................................. 74 4.2.1 Exceedances......................................................................................... 75 4.2.2 Exceedance times ................................................................................ 75 Subsequence principle ...................................................................... 76 4.2.3 Peaks over threshold............................................................................ 77 4.3 The tail empirical measure............................................................................. 78 4.4 The Hill estimator........................................................................................... 80 4.4.1 Random measures and the consistency of the Hill estimator . . . ....... 81 4.4.2 The Hill estimator in practice.............................................................. 85 4.4.3 Variants of the Hill plot ...................................................................... 89 The smooHill plot .............................................................................. 89 Changing the scale, Alt plotting........................................................ 90 4.5 Alternative estimators I: The Pickands estimator .......................................... 90 4.5.1 Extreme-value theory .......................................................................... 91 4.5.2 The Pickands estimator........................................................................ 93 4.6 Alternative estimators II: QQ plotting and the QQ estimator........................ 97 4.6.1 Quantile-quantile or QQ plots: Preliminaries ..................................... 97 4.6.2 QQ plots: The method ......................................................................... 98 4.6.3 QQ plots and location-scale families................................................... 100 4.6.4 Adaptation to the heavy-tailed case: Are the data heavy tailed? ........ 101 4.6.5 Additional remarks and related plots................................................... 102 Diagnosing deviations from the line in the QQ plot ........................ 102 A related plot: The PP plot................................................................ 104 Another variant: The tail plot for heavy tails .................................... 104 4.6.6 The QQ estimator ................................................................................ 106 Consistency of the QQ estimator...................................................... 108 4.7 How to compute value-at-risk ....................................................................... 111 4.8 Problems ........................................................................................................ 114 Part III Probability 5 The Poisson Process.............................................................................................. 119 5.1 The Poisson process as a random measure.................................................... 119 5.1.1 Definition and first properties ............................................................ 119 5.1.2 Point transformations ......................................................................... 120 5.1.3 Augmentation or marking................................................................... 122 5.2 Models for data transmission......................................................................... 123 5.2.1 Background ........................................................................................ 124 5.2.2 Probability models ............................................................................. 125 5.2.3 Long-range dependence ..................................................................... 126 Simple minded detection of long-range dependence using the sample acf plot ................................................................... 127 5.2.4 The infinite-node Poisson model........................................................ 127 5.2.5 Connection between heavy tails and long-range dependence . . . ..... 130 5.3 The Laplace functional.................................................................................. 132 5.3.1 Definition and first properties ............................................................ 132 5.3.2 The Laplace functional of the Poisson process ................................. 134 5.4 See the Laplace functional flex its muscles!................................................. 137 5.4.1 The Laplace functional and weak convergence ................................. 137 Convergence of empirical measures ............................................... 138 Preservation of weak convergence under mappings of the state space ................................................................................... 141 5.4.2 A general construction of the Poisson process .................................. 143 5.4.3 Augmentation, location-dependent marking...................................... 144 5.5 Levy processes .............................................................................................. 146 5.5.1 Ito's construction of Levy processes .................................................. 146 Levy measure................................................................................... 146 Compound Poisson representations................................................. 147 Variance calculations ...................................................................... 148 Process definition ............................................................................ 149 5.5.2 Basic properties of Levy processes ................................................... 150 The characteristic function of X(t) ................................................. 151 Independent increment property of X(t) ......................................... 151 Stationary increment property ........................................................ 152 Stochastic continuity of X·) ............................................................ 152 Subordinators................................................................................... 153 Stable Levy motion.......................................................................... 154 Symmetric a-stable Levy motion .................................................... 154 5.5.3 Basic path properties of Levy processes............................................ 155 5.6 Extremal processes ....................................................................................... 160 5.6.1 Construction....................................................................................... 161 5.6.2 Discussion .......................................................................................... 161 5.7 Problems ....................................................................................................... 162 6 Multivariate Regular Variation and the Poisson Transform .......................... 167 6.1 Multivariate regular variation: Basics .......................................................... 167 6.1.1 Multivariate regularly varying functions .......................................... 167 6.1.2 The polar coordinate transformation................................................. 168 6.1.3 The one-point uncompactification..................................................... 170 6.1.4 Multivariate regular variation of measures........................................ 172 6.2 The Poisson transform .................................................................................. 179 6.3 Multivariate peaks over threshhold............................................................... 183 6.4 Why bootstrapping heavy-tailed phenomena is difficult.............................. 184 6.4.1 An example to fix ideas ..................................................................... 184 6.4.2 Why the bootstrap sample size must be carefully chosen................ 186 The bootstrap procedure.................................................................. 186 What exactly is the bootstrap procedure?....................................... 187 When bootstrap asymptotics work ................................................. 188 When bootstrap asymptotics do not work....................................... 189 6.5 Multivariate regular variation: Examples, comments, amplification . . . ..... 191 6.5.1 Two examples .................................................................................. 192 Independence and asymptotic independence ................................. 192 Repeated components and asymptotic full dependence ................. 195 6.5.2 A general representation for the limiting measure v ....................... 196 6.5.3 A general construction of a multivariate regularly varying distribution....................................................................................... 197 6.5.4 Regularly varying densities .............................................................. 199 6.5.5 Beyond the nonnegative orthant ...................................................... 201 6.5.6 Standard vs. nonstandard regular variation....................................... 203 6.6 Problems ....................................................................................................... 206 7 Weak Convergence and the Poisson Process ..................................................... 211 7.1 Extremes ....................................................................................................... 211 7.1.1 Weak convergence of multivariate extremes: The timeless result................................................................................................ 211 7.1.2 Weak convergence of multivariate extremes: Functional convergence to extremal processes ................................................ 212 7.2 Partial sums................................................................................................... 214 7.2.1 Weak onvergence of partial sum processes to Levy processes . . .... 214 7.2.2 Weak convergence to stable Levy motion ........................................ 218 7.2.3 Continuity of the summation functional ........................................... 221 7.3 Transformations ........................................................................................... 226 7.3.1 Addition ............................................................................................ 227 Linear combinations of components of a random vector .............. 227 Adding independent vectors........................................................... 228 7.3.2 Products ............................................................................................ 231 Breiman's theorem: A factor has a relatively thin tail.................... 231 Products of heavy-tailed random variables which are jointly regularly varying ............................................................... 236 Internet data.................................................................................... 238 7.3.3 Laplace transforms ........................................................................... 239 Special case for d = 1: Karamata's Tauberian theorem................. 245 Renewal theory .............................................................................. 245 7.4 Problems ...................................................................................................... 247 8 Applied Probability Models and Heavy Tails ................................................... 253 8.1 A network model for cumulative traffic on large time scales ..................... 253 8.1.1 Model review..................................................................................... 253 8.1.2 The critical input rate ........................................................................ 255 8.1.3 Why stable Levy motion can approximate cumulative input under slow growth........................................................................... 257 The basic decomposition ................................................................ 257 One-dimensional convergence........................................................ 259 Finite-dimensional convergence .................................................... 263 8.2 A model for network activity rates .............................................................. 264 8.2.1 Mean value analysis when x, ß < 1 ................................................. 265 8.2.2 Behavior of N (t), the renewal counting function when 0 < x < 1 ........................................................................................ 265 8.2.3 Activity rates when a, ß < 1 and tails are comparable..................... 266 Counting function of {(S k, T k), k > 0} ......................................... 266 Number of active sources when tails are comparable ................... 267 8.2.4 Activity rates when 0 < a, ß < 1, and For, has a heavier tail . . ...... 269 Number of active sources when For, is heavier .............................. 271 8.3 Heavy traffic and heavy tails ....................................................................... 272 8.3.1 Crash course on waiting-time processes........................................... 273 8.3.2 Heavy-traffic approximation for queues with heavy-tailed services............................................................................................ 275 8.3.3 Approximation to a negative-drift random walk .............................. 279 8.3.4 Approximation to the supremum of a negative-drift random walk ................................................................................................ 281 8.3.5 Proof of the heavy-traffic approximation ......................................... 283 8.4 Problems ...................................................................................................... 286 Part IV More Statistics 9 Additional Statistics Topics.................................................................................291 9.1 Asymptotic normality................................................................................... 291 9.1.1 Asymptotic normality of the tail empirical measure ........................ 291 9.1.2 Asymptotic normality of the Hill estimator ..................................... 296 Blood and guts................................................................................ 298 Removing the random centering .................................................... 299 Centering by 1 l x ........................................................................... 302 Conclusions .................................................................................... 303 9.2 Estimation for multivariate heavy-tailed variables ..................................... 304 9.2.1 Dependence among extreme events ................................................. 304 Example: Modeling of exchange rates .......................................... 305 9.2.2 Estimation in the standard case ........................................................ 307 9.2.3 Estimation in the nonstandard case................................................... 309 Live with diversity ......................................................................... 309 Be crude! ........................................................................................ 310 The ranks method ........................................................................... 310 Estimation of the angular measure ................................................ 313 9.2.4 How to choose k; the Starica plot ..................................................... 314 9.3 Examples ..................................................................................................... 316 9.3.1 Internet data....................................................................................... 316 Boston University data................................................................... 316 Internet HTTP response data ......................................................... 317 9.3.2 Exchange rates................................................................................. 318 9.3.3 Insurance ......................................................................................... 319 9.4 The coefficient of tail dependence and hidden regular variation ................ 322 9.4.1 Hidden regular variation.................................................................. 323 Definition of hidden regular variation............................................ 324 Topology is destiny......................................................................... 324 9.4.2 A simple characterization ................................................................. 325 9.4.3 Two examples ................................................................................... 330 9.4.4 Detection of hidden regular variation ............................................... 332 A first step ...................................................................................... 332 But wait! Why does the rank transform preserve hidden regular variation?............................................................................ 332 Estimating the hidden angular measure ......................................... 337 9.5 The sample correlation function .................................................................. 340 9.5.1 Overview ........................................................................................... 340 9.5.2 Limit theory....................................................................................... 342 Preliminaries .................................................................................. 342 Point process limits......................................................................... 343 Summing the points ........................................................................ 346 9.5.3 The heavy-tailed sample acf; x < 1 .................................................. 346 9.5.4 The classical sample acf: 1 < x < 2 ................................................... 347 9.5.5 Suggestions to use.............................................................................. 349 9.6 Problems ...................................................................................................... 350 Part V Appendices 10 Notation and Conventions ................................................................................ 359 10.1 Vector notation .......................................................................................... 359 10.2 Symbol shock ............................................................................................. 360 11 Software ..............................................................................................................363 11.1 One dimension ........................................................................................... 364 11.1.1 Hill estimation................................................................................ 364 Hillalpha.......................................................................................... 364 altHillalpha ..................................................................................... 364 smooHillalpha ................................................................................ 365 11.1.2 QQ plotting .................................................................................... 366 pppareto .......................................................................................... 366 parfit................................................................................................ 366 QQ estimator plot ........................................................................... 367 11.1.3 Estimators from extreme-value theory .......................................... 368 The Pickands estimator................................................................... 368 The moment estimator ................................................................... 369 11.2 Multivariate heavy tails ............................................................................. 370 11.2.1 Estimation of the angular distribution............................................ 370 Rank transform ............................................................................... 371 Estimate the angular density using ranks ........................................ 371 Estimate the angular density using power transforms..................... 371 Estimate the angular distribution using the rank transform ............ 372 11.2.2 The Starica plot .............................................................................. 372 Norms............................................................................................... 373 Starica plot using the power transform............................................ 373 StArica plot using the rank transform ............................................. 374 Allowing the Starica plot to choose k ............................................. 374 References ............................................................................................................. 377 Index........................................................................................................................ 397

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