Advances in heavy tailed risk modeling a handbook of operational risk

www.ebook3000.com www.ebook3000.com Advances in Heavy Tailed Risk Modeling A Handbook of Operational Risk www.ebook3000.com www.ebook3000.com Advances in Heavy Tailed Risk Modeling A Handbook of Operational Risk Gareth W Peters Pavel V Shevchenko www.ebook3000.com Copyright © 2015 by John Wiley & Sons, Inc All rights reserved Published by John Wiley & Sons, Inc., Hoboken, New Jersey Published simultaneously in Canada No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, (978) 750-8400, fax (978) 750-4470, or on the web at www.copyright.com Requests to the 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information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) 762-2974, outside the United States at (317) 572-3993 or fax (317) 572-4002 Wiley also publishes its books in a variety of electronic formats Some content that appears in print may not be available in electronic formats For more information about Wiley products, visit our web site at www.wiley.com Library of Congress Cataloging-in-Publication Data: Peters, Gareth W., 1978Advances in heavy tailed risk modeling : a handbook of operational risk / Gareth W Peters, Department of Statistical Science, University College of London, London, United Kingdom, Pavel V Shevchenko., Division of Computational Informatics, The Commonwealth Scientific and Industrial Research Organization, Sydney, Australia pages cm Includes bibliographical references and index ISBN 978-1-118-90953-9 (hardback) Risk management Operational risk I Shevchenko, Pavel V II Title HD61.P477 2014 658.15 5–dc23 2014015418 Printed in the United States of America 10 www.ebook3000.com Gareth W Peters This is dedicated to three very inspirational women in my life: Chen Mei–Peters, my mother Laraine Peters and Youxiang Wu; your support, encouragement and patience has made this possible Mum, you instilled in me the qualities of scientific inquiry, the importance of questioning ideas and scientific rigour This is especially for my dear Chen who bore witness to all the weekends in the library, the late nights reading papers and the ups and downs of toiling with mathematical proofs across many continents over the past few years Pavel V Shevchenko To my dear wife Elena www.ebook3000.com Embarking upon writing this book has proven to be an adventure through the landscape of ideas Bringing forth feelings of adventure analogous to those that must have stimulated explorers such as Columbus to voyage to new lands In the depth of winter, I finally learned that within me there lay an invincible summer Albert Camus www.ebook3000.com Contents in Brief Motivation for Heavy-Tailed Models Fundamentals of Extreme Value Theory for OpRisk 17 Heavy-Tailed Model Class Characterizations for LDA 105 Flexible Heavy-Tailed Severity Models: α-Stable Family 139 Flexible Heavy-Tailed Severity Models: Tempered Stable and Quantile Transforms 227 Families of Closed-Form Single Risk LDA Models 279 Single Risk Closed-Form Approximations of Asymptotic Tail Behaviour 353 Single Loss Closed-Form Approximations of Risk Measures 433 Recursions for Distributions of LDA Models 517 A Miscellaneous Definitions and List of Distributions 587 vii www.ebook3000.com www.ebook3000.com 620 References Steutel, F W., & Van Harn, K 2003 Infinite Divisibility of Probability Distributions on the Real Line CRC Press Sundt, B 1998 A generalisation of the De Pril transform Scandinavian actuarial journal, 1998(1), 41–48 Sundt, B 2005 On some properties of De Pril transforms of counting distributions ASTIN Bulletin, 25(1), 19–31 Sundt, B., & Jewell, W S 1981 Further results on recursive evaluation of compound distributions ASTIN Bulletin, 12(1), 27–39 Sundt, B., & Vernic, R 2009 Recursions for Convolutions and Compound Distributions with Insurance Applications Springer: Berlin Tadikamalla, P R 1980 A look at the Burr and related distributions International Statistical Review/Revue Internationale de Statistique, 48(3), 337–344 Takács, L 1977 On the ordered partial sums of real random variables Journal of Applied Probability, 14(1), 75–88 Tang, Q 2006 Insensitivity to negative dependence of the asymptotic behavior of precise large deviations Electronic Journal of Probability, 11(4), 107–120 Tang, Q 2008 Insensitivity to negative dependence of asymptotic tail probabilities of sums and maxima of sums Stochastic Analysis and Applications, 26(3), 435–450 Tarasov, V E 2008 Fractional vector calculus and fractional Maxwell?s equations Annals of Physics, 323(11), 2756–2778 Tarov, V A 2004 Smoothly varying functions and perfect proximate orders Mathematical Notes, 76(1-2), 238–243 Tawn, J A 1990 Modelling multivariate extreme value distributions Biometrika, 77(2), 245–253 Taylor, J W 2008 Estimating value at risk and expected shortfall using expectiles Journal of Financial Econometrics, 6(2), 231–252 Teugels, J L 1975 The class of subexponential distributions Annals of Probability, 3(6), 1000–1011 Teugels, J L., & Veraverbeke, N 1973 Cramér-Type Estimates for the Probability of Ruin Research Institute Center for Statistics C.O.R.E Discussion Paper No 7316, available at: http://hdl.handle.net/1942/95 Thorin, O (1977) On the infinite divisibility of the lognormal distribution, Scandinavian Actuarial Journal, 1977 (3), 121–148 Tong, B., & Wu, C 2012 Asymptotics for operational risk quantified with a spectral risk measure Journal of Operational Risk, 7(3), 91 Trinidad, W I 1990 A proof of Pollaczek-Spitzer identity International Journal of Mathematics and Mathematical Sciences, 13(4), 737–740 Tukey, J W 1977a Exploratory Data Analysis Vol 231 Reading, MA; Addison-Wesley Tukey, J W 1977b Modern techniques in data analysis In: NSF-Sponsored Regional Research Conference at Southern Massachusetts University (North Dartmouth, MA) Tweedie, M C K 1947 Functions of a statistical variate with given means, with special reference to Laplacian distributions Proceedings of the Cambridge Philosophical Society, Vol 43 Cambridge University Press, 100 Tweedie, M C K 1984 An index which distinguishes between some important exponential families Statistics: Applications and New Directions: Proc Indian Statistical Institute Golden Jubilee International Conference, 579–604 Uchaikin, V V., & Zolotarev, V M 1999 Chance and Stability: Stable Distributions and their Applications Walter de Gruyter Usero, D 2007 Fractional Taylor Series for Caputo Fractional Derivatives Construction of Numerical Schemes http://www mat ucm es/deptos/ma/inv/prepub/new/2007-10 pdf Vinogradov, V 1994 Refined Large Deviation Limit Theorems Vol 315 CRC Press Waldmann, K.-H 1996 Modified recursions for a class of compound distributions ASTIN Bulletin, 26(2), 213–224 References 621 Walin, J F., & Paris, J 1998 On the use of equispaced discrete distributions ASTIN Bulletin, 28(2), 241–255 Waller, L A., Turnbull, B W., & Hardin, J M 1995 Obtaining distribution functions by numerical inversion of characteristic functions with applications American Statistician, 49(4), 346–350 Wang, Y., Cheng, D., & Wang, K 2005 The closure of a local subexponential distribution class under convolution roots, with applications to the compound Poisson process Journal of Applied Probability, 42(4), 1194–1203 Warde, W D., & Katti, S K 1971 Infinite divisibility of discrete distributions, II Annals of Mathematical Statistics, 42(3), 1088–1090 Wedderburn, R W M 1974 Quasi-likelihood functions, generalized linear models, and the Gauss?Newton method Biometrika, 61(3), 439–447 Weissman, I 1978 Estimation of parameters and large quantiles based on the k largest observations Journal of the American Statistical Association, 73(364), 812–815 Wendel, J G 1960 Order statistics of partial sums The Annals of Mathematical Statistics, 31(4), 1034–1044 Wendel, J G 1961 The non-absolute convergence of Gil-Pelaez’inversion integral Annals of Mathematical Statistics, 32(1), 338–339 Weron, A 1984 Stable processes and measures; a survey In: Probability Theory on Vector Spaces III Springer, 306–364 Weron, R 1996a Correction To: “On the Chambers-Mallows-Stuck Method for Simulating Skewed Stable Random Variables” Tech Rept Hugo Steinhaus Center, Wroclaw University of Technology Weron, R 1996b On the Chambers-Mallows-Stuck method for simulating skewed stable random variables Statistics & Probability Letters, 28(2), 165–171 Weron, R 2006 Modeling and Forecasting Electiricty Loads and Prices: A Statistical Approach Wiley West, M 1987 On scale mixtures of normal distributions Biometrika, 74(3), 646–648 Willekens, E 1989 Asymptotic approximations of compound distributions and some applications Bulletin de la Société Mathématique de Belgique Série B, 41(1), 55–61 Willekens, E., & Teugels, J L 1992 Asymptotic expansions for waiting time probabilities in an M/G/1 queue with long-tailed service time Queueing Systems, 10(4), 295–311 Williams, D 1991 Probability with Martingales Cambridge University Press Wintner, A 1936 On a class of Fourier transforms American Journal of Mathematics, 58(1), 45–90 Wintner, A 1938 Lectures by Aurel Wintner on Asymptotic Distributions and Infinite Convolutions, 1937-1938 Edwards Brothers, Inc Wintner, A 1956 Cauchy’s stable distributions and an “explicit formula” of Mellin American Journal of Mathematics, 78(4), 819–861 Wright, E M 1935 The asymptotic expansion of the generalized Bessel function Proceedings of the London Mathematical Society, 2(1), 257–270 Yamazato, M 1978 Unimodality of infinitely divisible distribution functions of class L Annals of Probability, 6(4), 523–531 Yu, J 2004 Empirical characteristic function estimation and its applications Econometric Reviews, 23(2), 93–123 Zolotarev, V M 1983 Univariate Stable Distributions Moscow: Nauka Zolotarev, V M 1986 One-dimensional stable distributions Translations of Mathematical Monographs American Mathematical Society Zolotarev, V M 1994 On representation of densities of stable laws by special functions Theory of Probability & its Applications, 39(2), 354–362 Index α-Stable LDA models, 333 ν -Stable family, 332 Tail properties, 332 Additive Exponential Dispersion Models, 312 Ali-Mikhail-Haq copula, 397 alpha-Stable distribution asymptotic density expansion, 196 asymptotic tail behaviour, 220 Characterization 1, 149 Characterization 2, 149 Characterization 3, 153 Characterization 4, 156 density approximation FFT and Bergstrom, 189 density approximation polynomial series, 196 density approximation quadrature, 187 density approximation reparameterization, 198 density Scale Mixture of Normals (SMiN), 206 distribution approximation quadrature, 208 distribution approximation series, 209 domain of attraction, 152, 155 Fractional Lower Order Moments, 223 Levy distribution, 151 Levy LePage series, 205 Nolan’s S0 and S1-Type Parameterization, 178 parameter estimation, 215 quantile function, 210 simulation, 214 Zolotarev’s A-Type parameterization, 167, 172 Zolotarev’s B-Type density, 185 Zolotarev’s B-Type parameterization, 175 Zolotarev’s C and E-Type parameterizations, 176 Zolotarev’s W-Type parameterization, 176 alpha-Stable distribution spectral measure, 162 Archimedean copula, 397 Asymmetric Laplace distribution, 327 Maximum-Likelihood estimation, 329 Asymmetric Power distribution, 446 Asymptotic equivalence in distribution, 110 asymptotically locally constant distribution, 120 Benktander II distribution, 68 Bergstrom series expansion, 196 Beta distribution, 592 Binomial-Levy LDA Model, 345 bounds aggregate tail distributions, 111 probabilities of convolutions, 115 Box–Muller transformation, 66 Bromwich Inversion Integral for Laplace Transforms, 241 Burr distribution, 60 Burr Type III distribution, 61 Burr Type XII distribution, 60 Caputo fractional derivative, 513 Central Limit Theorem classical, 153 Lindeberg condition, 154 Lindeberg Levy, 153 Lyapunov condition, 154 Advances in Heavy Tailed Risk Modeling: A Handbook of Operational Risk, First Edition Gareth W Peters and Pavel V Shevchenko © 2015 John Wiley & Sons, Inc Published 2015 by John Wiley & Sons, Inc 623 624 characteristic function, 140 comonotonicity and stochastic decreasing, 396 compound Poisson Levy process, 297 compound process tail decomposition, 415 consistent variation, 129 convolution, 282 convolution of distributions, 110 convolution root closure, 137 convolution semi-group, 281 continous, 290 generator, 290 Levy process, 295 truncated Poisson, 291 convolution symmetrization, 283 counting process, 75 De Bruyn conjugate, 126 De Pril’s first method, 538 De Pril’s second method, 539 Decoupage de Levy, 79 Discrete distributions D-distributions, 526 extrapolation methods and acceleration, 533 infinite divisibility, 527 Linnik laws, 532 non-degenerate, 525 Panjer class, 525 Sibuya distribution, 530 Stable, 530 discretization of distribution, 519 distributions of common type, 25 domain of attraction, 145 Stable and Tweedie convergence, 303 dominant variation, 130 doubly stochastic Binomial-Levy LDA model, 346 doubly stochastic Negative Binomial-Levy LDA model, 347 doubly stochastic Poisson-Levy LDA model, 345 Elongation transform, 255 equality in distribution, 21 Esscher transform, 229 expectiles, 478 Exponential Dispersion Family, 309, 441 scale invariance, 313 Exponential Dispersion models infinite divisibility, 319 Exponential tilting, 229 Index extended and O-Type regular variation, 135, 359 extremal domain of attraction conditions, 33, 35 Auxiliary function, 36 extremal limit distribution extremal limit distribution families, 39 Extreme Value Theory Block Maxima Approach, 40 elemental percentile method, 95 Generalized Extreme Value distribution, 26 Generalized Pareto distribution (GPD), 56 GEV distribution moments, 26, 54 Gumbel family distributions, 65 Maximum-likelihood GPD, 87 mixed estimation methods MLE and L-Moments, 43 Pareto distribution, 55 Penultimate VaR approximation, 468 Pickands estimator, 96 POT’s domain of attraction, 83 Power-Normalized EVT domain of attraction, 469 probability weighted moments GPD, 93 profile likelihood GPD, 91 simulation of GPD, 57 small sample GPD, 90 threshold exceedance, 72 Extreme value theory, 17 Extremal Limit Problem, 25 Extreme Value Theory Peaks Over Threshold (POTs), 81 Faa Di Bruno’s formula, 513 Farlie-Gumbel-Morgenstern copula, 401 Fast Fourier Transform, 189 First-Order stochastic dominance, 403 Fourier Inversion theorem, 181 fractional derivatives of composite functions, 513 Fractional Fourier Transform, 191 Fundamental Theorem of Calculus, 512 g-and-h distribution, 127, 257, 262 ABC, 274 index of regular variation, 268 moments, 265 percentile matching, 270 sample L-moments, 271 simulation, 262 slow variation, 269 625 Index g-and-h distribution L-moments, 261 Gaussian copula, 401 generalised inverse, 28, 452 properties, 28 generalized moments, 115 Gil–Pelaez inversion integral, 182 GPD small sample behaviour, 91 log-concave density, 529 LogNormal distribution, 65 Long-Tailed distribution, 131 lower and upper negative association and dependence, 394 LQ-moments distribution, 46 Lévy–Khintchine representation, 161 hazard function, 114 hazard rate, 114 higher order Panjer class, 573 Hinde–Demetrio frequency distributions, 325 Hougaard Levy domain of attraction, 307 Hougaard Levy process, 306 Hyper Geometric function, 203 Matuszewska index, 137 max-sum equivalence, 112 mixed poisson distribution, 542 monotone density theorem, 481 multivariate negative dependence, 394 Infinite Divisible, 144, 286, 293 absolute moments, 295 exponential moments, 295 fractional moments, 296 large deviations, 296 Lipschitz function, 296 Karamata’s Representation Theorem, 125, 269, 487 Kolmogorov canonical characteristic function, 147 Kolmogorov three series theorem, 362 L-Class of distribution, 158 L-moment estimator, 273 L-moment Tukey transforms, 261 L-moments, 46 distribution based, 44 GEV method of moments, 49 sample estimators, 45 L-skewness and L-kurtosis distribution, 47 Lagrange inversion formula, 212 Laguerre polynomials, 201 Landau notation Big Oh and Little Oh, 106 large deviations inequality, 297 large sample asymptotics, 308 Levy canonical characteristic function, 147 Levy Cramer continuity theorem, 141 Levy measure, 294 Levy triplets, 161 Levy truncation functions, 162 local sub-exponential distribution, 120 locally heavy tailed, 128 n-decomposable random variable, 143 n-divisible random variable, 144 n-fold convolution, 115 Natural Exponential Family, 305 mean and variance function, 305 Negative Binomial distribution, 103 Negative Binomial-Levy LDA model, 347 negative regression dependence, 395 Normal Exponential Family steepness and regularity, 311 Normal Inverse Gaussian semi-group, 288 O-regular variation distributions, 359 order statistics, 23 pairwise negative quadrant dependence, 395 pairwise positive quadrant dependence, 395 Panjer recursion, 567 Partial Rejection Control, 556 Pickands-Balkema-deHaan theorem, 73 Poisson summation of characteristic functions, 142 Poisson-Levy LDA model, 344 Poisson-Tweedie discrete, 324 Poisson-Tweedie models, 309 Pollaczek–Spitzer–Wendell identity, 364 Polya’s sufficient condition for characteristic functions, 141 Potter bounds, 126 probability distribution function, 21 probability generating function, 525 probability weighted moments, 94 profile likelihood, 53 deviance statistic, 53 626 quantile function, 29 GEV distribution, 31 properties, 29 tail, 29 recursions continous Panjer recursion, 581 De Pril transform, 537 higher order recursions, 575 mixed Poisson, 577 Panjer recursion, 566 partial sums, 535 random sums, 565 Waldmann’s recursion, 575 Wilmot class, 580 regular and slow variation, 121 quantile functions, 127 uniform convergence, 124 renewal process, 75 representation theorem of slowly varying functions, 431 reproductive Exponential Dispersion models, 313 return levels Generalized Extreme Value distribution, 54 Richardson extrapolation, 533 risk measures admissible risk spectrum, 492 CARA risk aversion function, 494 empirical quantile process, 450 ES for Asymmetric Power distribution, 447 ES for Exponential Dispersion Models, 443 ES for Exponential Dispersion Models Partial Sums, 444 ES for g-and-h, 446 EVT Hill estimator, 471 Extreme Value Theory penultimate approximation, 468 g-and-h model, 445 non-parametric estimators, 448 penultimate single-loss approximation POT’s, 473 second order VaR, 458 single loss approximation VaR, 452 spectral risk measure single loss approximations, 478 TCE for Exponential Dispersion Models, 443 tempered, 503 VaR first and second-order single-loss approximations, 451 VaR for Asymmetric Power distribution, 447 Index VaR for Exponential Dispersion Models, 441 VaR for g-and-h, 445 VaR single-loss approximation with mean correction, 458 VaR time scaling property single loss approximation, 457 VaR via point processes, 473 Var, ES and spectral risk, 439 risk weighted assets, 438 Rosinski measure, 231 second order regular variation, 486 self-decomposable random variables, 145 Sequential Monte Carlo samplers, 550 severity distribution splice model, 11 slowly varying distribution tail, 122 small sample asymptotics, 307 smoothly varying function, 129 splice models, 11 Stable rate of convergence Stable rate of convergence, 225 Stieltjes integral, 157 sub-exponential distribution, 113, 118 subversively varying tail, 131 tail asymptotic compound process light-tailed severity, 370 compound processes, 367 consistent variation, 373 convolution root closure, 358 convolution root closure O-regular varying, 360 dependent severity and frequency, 374 dominant frequency distribution, 372 first order single-loss approximation, 376 first order single-loss regular variation, 380 first-order single-loss sub-exponential, 377 Frechet domain of attraction, 373 generalized tail ratios, 381 Gumbel domain of attraction, 373 higher order single-loss approximation smooth variation, 432 higher order single-loss approximations, 414 higher order single-loss approximations slow variation, 431 independent frequency and severity, 375 Inverse Laplace Characters, 424 Laplace Characters, 423 large deviations for partial sums, 367 627 Index large number of losses, 362 long tailed distribution, 361 partial sum, 356 partial sum distributional lower bound, 357 partial sum long tailed, 362 partial sums o-regularly varying, 359 remainder analysis regular variation, 386 remainer analysis for sub-exponential, 385 Saddlepoint conditions, 370 second-order single-loss approximations, 389 second-order single-loss approximation dependent risks, 413 stochastic bounds for distributions of partial sums, 403 tail balance condition, 126 tail dependence measure, 398 tail distribution function, 21 tail equivalence, 21 tempered α-Stable LDA model, 231, 349 tempered Levy measure, 230 tempered Stable model closure under convolution, 243 cummulants, 244, 245 density, 237, 241 Fractional Lower Order Moments, 243 Levy measure MTS, 239 parameter estimation, 246 representation 1, 232 representation (CTS), 232 representation (RDTS), 233 representation (GTS), 234 representation , 236 representation (MTS), 239 representation (NMTS), 240 representation CGMY, 235 shot noise representation, 249 simulation, 248 standardized, 245 tail behaviour, 252 Tier I Capital, 437 Tier I Capital Ratio, 438 Tier II Capital, 438 trimmed L-moments distribution, 46 estimator variance, 48 sample estimators, 48 Tukey distribution, 268 Tukey transformation, 253 h-transform, 254 j-transform, 254 k-transform, 254 Tweedie convergence, 307 Tweedie loss model cumulant function, 314 Tweedie models parameter estimation, 317 underflow, 572 uniform convergence, 20 unimodality of distribution, 158 Vinogradov notation, 108 Volterra integral equations second kind, 581 von Mises condition von Mises condition, 40 weak convergence in distribution Helly–Bray theorem, 22 weak convergence in distribution, 21 weak convergence Levy process to Stable Levy process, 304 Weibull Stretched Exponential distribution, 119 40 Loss amounts not exceeding threshold Xi ∈ Bc 35 Threshold level u defining set B = (u,∞) Loss amounts exceeding threshold Xi ∈ B τ3 = 30 τ7 = 23 τ13 = 43 τ8 = 34 Loss amounts 25 20 15 10 0 50 100 150 200 250 300 350 Time (days) FIGURE 2.18 An example realization of a single risk process {(Xi , ti )}i=1:n with marked losses that exceeded a threshold loss amount u The times on the x-axis at which the losses occurred correspond to the days ti , the amounts correspond to the values Xi and the secondary process comprised of a subset of the losses, denoted by τj ’s as illustrated on the figure, corresponds to the indexes of the j th exceedance Advances in Heavy Tailed Risk Modeling: A Handbook of Operational Risk, First Edition Gareth W Peters and Pavel V Shevchenko © 2015 John Wiley & Sons, Inc Published 2015 by John Wiley & Sons, Inc Loss amounts 40 Loss amounts not exceeding threshold Xi ∈ Bc Threshold level u defining set B = (u,∞) Loss amounts exceeding threshold Xi ∈ B 35 30 τ3 = τ7 = 23 τ8 = 34 A = (y,∞)×(t1,t2) τ13 = 43 25 Loss amount y 20 Threshold u 15 10 0 50 100 200 300 250 350 Time t2 Time (days) Time t1 set (t1,t2) ⊂ [0,T] 150 FIGURE 2.19 An example realization of a single risk process {(Xi , ti )}i=1:n with marked losses that exceeded a threshold loss amount u The times on the x-axis at which the losses occurred correspond to the days ti , the amounts correspond to the values Xi and the secondary process comprised of a subset of the losses, denoted by τj ’s as illustrated on the figure, correspond to the indexes of the j th exceedance In addition, an example region A = (t1 , t2 ) × (y, ∞) is marked for points exceeding threshold u in time interval (t1 , t2 ) ⊂ [0, T ] 90 10 120 150 α = 0.5 α = 0.75 α = 1.1 α = 1.5 α = 1.9 60 30 180 210 330 240 300 270 (a) 90 10 120 150 λ = 0.1 λ = 0.25 λ = 0.5 λ = 0.75 λ=1 60 30 180 330 210 300 240 270 (b) 90 10 120 150 β = −0.8 β = −0.4 β=0 β = 0.25 β = 0.9 60 30 180 330 210 300 240 (c) 270 FIGURE 4.2 (a) Example of the A-type parameterization of the α-stable characteristic function for a range of values of α ∈ {0.5, 0.75, 1.1, 1.5, 1.9} with β = 0, γ = 0.1 and δ = (b) Example of the A-type parameterization of the α-stable characteristic function for a range of values of γ ∈ {0.1, 0.25, 0.5, 0.75, 1} with α = 0.5, β = and δ = (c) Example of the A-type parameterization of the α-stable characteristic function for a range of values of β ∈ {−0.8, −0.4, 0, 0.25, 0.9} with α = 1.2, γ = 0.1 and δ = 90 10 120 60 150 α = 0.7 α = 0.95 α = 0.99 α = 1.05 α = 1.5 30 180 210 330 240 300 270 FIGURE 4.3 Example of A-type paramerization for a range of stability index values 90 10 120 α = 0.7 α = 0.95 α = 0.99 α = 1.05 data5 60 150 30 180 210 330 240 300 270 FIGURE 4.4 Example of B-type paramerization for a range of stability index values 0.3 0.25 30 0.3 0.3 α = 1.9 α = 1.6 α = 1.4 0.25 0.25 0.2 0.2 0.2 0.15 0.15 0.1 0.1 0.1 0.05 0.05 0.05 0 −0.05 (a) 10 −0.05 20 (b) 10 0.15 −10 −0.05 −20 −0.1 10 −0.15 20 (c) 20 10 15 −30 20 (d) 10 15 20 FIGURE 4.9 Study of first 20 summand terms of the stable density series expansion for a range of parameter values at three different x locations (a) x = 1, (b) x = 0.5, (c) x = and (d) x = 0.3 0.3 0.3 0.25 0.25 0.25 0.2 0.2 0.15 0.15 2.5 β=0 β = 0.5 β = 0.9 1.5 0.2 0.15 0.5 0.1 0.1 0.1 0.05 0.05 0.05 0 −0.05 10 20 (a) −0.05 10 20 (b) −0.5 −1 −0.05 −1.5 −0.1 10 20 (c) −2 10 15 20 (d) FIGURE 4.10 Study of first 20 summand terms of the stable density series expansion for a range of parameter values at three different x locations (a) x = 1, (b) x = 0.5, (c) x = and (d) x = g = 0.1 g = 0.5 g = 0.75 g=1 Gaussian 0.6 0.5 0.4 0.3 0.2 0.1 (a) 10 0.4 h = 0.01 h=1 h=5 Gaussian 0.3 0.2 0.1 (b) 10 FIGURE 5.7 (a) This plot shows the effect of the skewness parameter g on the elongation-transformed severity distribution versus the base Gaussian distribution with g ∈ {0.1, 0.5, 0.75, 1} In this case, the other parameters were set to a = 3, b = and h = 0.001 (b) This plot shows the effect of the kurtosis parameter h on the elongation-transformed severity distribution versus the base Gaussian distribution with h ∈ {0.01, 1, 5} In this case, the other parameters were set to a = 0, b = and g = 0.8 0.6 0.6 0.4 0.4 0.2 0.2 (a) 10 (b) 10 10 0.8 0.5 0.6 0.4 0.4 0.3 0.2 0.2 0.1 (c) 10 (d) FIGURE 5.8 (a) This plot shows the effect of the skewness parameter g on the elongation-transformed severity distribution versus the base Gaussian distribution with g ∈ {0.1, 0.5, 0.75, 1} In this case, the other parameters were set to a = 3, b = and h = 0.001 (b) This plot shows the effect of the kurtosis parameter h on the elongation-transformed severity distribution versus the base Gaussian distribution with h ∈ {0.01, 1, 5} In this case, the other parameters were set to a = 0, b = and g = (c) This plot shows the effect of the skewness parameter g on the elongation-transformed severity distribution versus the base LogNormal(0,1) distribution with g ∈ {0.1, 0.5, 0.75, 1} In this case, the other parameters were set to a = 3, b = and h = 0.001 (d) This plot shows the effect of the kurtosis parameter h on the elongation-transformed severity distribution versus the base LogNormal(0,1) distribution with h ∈ {0.01, 1, 5} In this case, the other parameters were set to a = 0, b = and g = WILEY END USER LICENSE AGREEMENT Go to www.wiley.com/go/eula to access Wiley’s ebook EULA ...www.ebook3000.com Advances in Heavy Tailed Risk Modeling A Handbook of Operational Risk www.ebook3000.com www.ebook3000.com Advances in Heavy Tailed Risk Modeling A Handbook of Operational Risk Gareth W... relevant to practice This is where the Advances in Heavy- Tailed Risk Modeling: A Handbook of Operational Risk can add value to the industry In particular, by providing a clear and detailed coverage... Library of Congress Cataloging -in- Publication Data: Peters, Gareth W., 197 8Advances in heavy tailed risk modeling : a handbook of operational risk / Gareth W Peters, Department of Statistical