Simulating Copulas

Stochastic Models, Sampling Algorithms, And Applications

By: Jan-frederik Mai, Matthias Scherer

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Hardcover | 28 June 2012

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Hardcover
312 Pages

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23.5 x 15.3 x 2.2

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This book provides the reader with a background on simulating copulas and multivariate distributions in general. It unifies the scattered literature on the simulation of various families of copulas (elliptical, Archimedean, Marshall-Olkin type, etc.) as well as on different construction principles (factor models, pair-copula construction, etc.). The book is self-contained and unified in presentation and can be used as a textbook for advanced undergraduate or graduate students with a firm background in stochastics. Alongside the theoretical foundation, ready-to-implement algorithms and many examples make this book a valuable tool for anyone who is applying the methodology.

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Non-Fiction Mathematics Probability & Statistics Applied Mathematics Stochastics Calculus & Mathematical Analysis Complex Analysis Booktopia Publisher Services World Scientific Publishing

Preface	p. vii
Introduction	p. 1
Copulas	p. 4
Analytical Properties	p. 7
Sklar's Theorem and Survival Copulas	p. 14
General Sampling Methodology in Low Dimensions	p. 22
Graphical Visualization	p. 26
Concordance Measures	p. 28
Measures of Extremal Dependence	p. 33
General Classifications of Copulas	p. 36
Radial Symmetry	p. 36
Exchangeability	p. 39
Homogeneous Mixture Models	p. 41
Heterogeneous Mixture Models/Hierarchical Models	p. 48
Extreme-Value Copulas	p. 52
Archimedean Copulas	p. 57
Motivation	p. 58
Extendible Archimedean Copulas	p. 61
Kimberling's Result and Bernstein's Theorem	p. 62
Properties of Extendible Archimedean Copulas	p. 65
Constructing Multi-Parametric Families	p. 69
Parametric Families	p. 69
Exchangeable Archimedean Copulas	p. 76
Constructing Exchangeable Archimedean Copulas	p. 82
Sampling Exchangeable Archimedean Copulas	p. 85
Properties of Exchangeable Archimedean Copulas	p. 87
Hierarchical (H-Extendible) Archimedean Copulas	p. 89
Compatibility of Generators	p. 90
Probabilistic Construction and Sampling	p. 91
Properties	p. 93
Examples	p. 95
Other Topics Related to Archimedean Copulas	p. 97
Simulating from the Generator	p. 97
Asymmetrizing Archimedean Copulas	p. 99
Marshall-Olkin Copulas	p. 101
The General Marshall-Olkin Copula	p. 102
Canonical Construction of the MO Distribution	p. 104
Alternative Construction of the MO Distribution	p. 110
Properties of Marshall-Olkin Copulas	p. 118
The Exchangeable Case	p. 122
Reparameterizing Marshall-Olkin Copulas	p. 126
The Inverse Pascal Triangle	p. 129
Efficiently Sampling eMO	p. 131
Hierarchical Extensions	p. 138
The Extendible Case	p. 140
Precise Formulation and Proof of Theorem 3.1	p. 141
Proof of Theorem 3.2	p. 146
Efficient Simulation of Lévy-Frailty Copulas	p. 150
Hierarchical (H-Extendible) Lévy-Frailty Copulas	p. 153
Elliptical Copulas	p. 159
Spherical Distributions	p. 161
Elliptical Distributions	p. 166
Parametric Families of Elliptical Distributions	p. 170
Elliptical Copulas	p. 174
Parametric Families of Elliptical Copulas	p. 175
Sampling Algorithms	p. 179
A Generic Sampling Scheme	p. 179
Sampling Important Parametric Families	p. 181
Pair Copula Constructions	p. 185
Introduction to Pair Copula Constructions	p. 186
Copula Construction by Regular Vine Trees	p. 191
Regular Vines	p. 191
Regular Vine Matrices	p. 196
Simulation from Regular Vine Distributions	p. 203
h-Functions for Bivariate Copulas and Then Rotated Versions	p. 204
The Sampling Algorithms	p. 208
Dependence Properties	p. 218
Application	p. 223
Time Series Model for Each Margin	p. 224
Parameter Estimation	p. 224
Forecasting Value at Risk	p. 226
Backtesting Value at Risk	p. 227
Backtest Results	p. 228
Sampling Univariate Random Variables	p. 231
General Aspects of Generating Random Variables	p. 231
Generating Uniformly Distributed Random Variables	p. 232
Quality Criteria for RNG	p. 233
Common Causes of Trouble	p. 234
The Inversion Method	p. 234
Generating Exponentially Distributed Random Numbers	p. 235
Acceptance-Rejection Method	p. 235
Generating Normally Distributed Random Numbers	p. 238
Calculating the Cumulative Normal	p. 238
Generating Normally Distributed Random Numbers via Inversion	p. 238
Generating Normal Random Numbers with Polar Methods	p. 239
Generating Lognormal Random Numbers	p. 240
Generating Gamma-Distributed Random Numbers	p. 240
Generating Gamma-Distributed RNs with ß > 1	p. 241
Generating Gamma-Distributed RNs with ß < 1	p. 242
Relations to Other Distributions	p. 243
Generating Chi-Square-Distributed RNs	p. 243
Generating t-Distributed Random Numbers	p. 244
Generating Pareto-Distributed Random Numbers	p. 245
Generating Inverse Gaussian-Distributed Random Numbers	p. 245
Generating Stable-Distributed Random Numbers	p. 246
Generating Discretely Distributed Random Numbers	p. 247
Generating Random Numbers with Geometric and Binomial Distribution	p. 248
Generating Poisson-Distributed Random Numbers	p. 248
The Monte Carlo Method	p. 251
First Aspects of the Monte Carlo Method	p. 251
Variance Reduction Methods	p. 254
Antithetic Variates	p. 255
Antithetic Variates for Radially Symmetric Copulas	p. 257
Control Variates	p. 258
Approximation via a Simpler Dependence Structure	p. 260
Importance Sampling	p. 262
Importance Sampling via Increasing the Dependence	p. 263
Further Comments on Variance Reduction Methods	p. 265
Supplemental Material	p. 267
Validating a Sampling Algorithm	p. 267
Introduction to Lévy Subordinators	p. 268
Compound Poisson Subordinator	p. 272
Gamma Subordinator	p. 274
Inverse Gaussian Subordinator	p. 275
Stable Subordinator	p. 276
Scale Mixtures of Marshall-Olkin Copulas	p. 277
Further Reading	p. 281
Bibliography	p. 283
Index	p. 293
Table of Contents provided by Ingram. All Rights Reserved.

Simulating Copulas

Stochastic Models, Sampling Algorithms, And Applications

At a Glance

New Edition

Hardcover

Shipping

How to return your order

Defective items

You Can Find This Book In

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