Handbook of Monte Carlo Methods

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1 Handbook of Monte Carlo Methods Dirk P. Kroese University of Queensland Thomas Taimre University of Queensland Zdravko I. Botev Université de Montréal WILEY A JOHN WILEY & SONS, INC., PUBLICATION

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3 Handbook of Monte Carlo Methods

4 WILEY SERIES IN PROBABILITY AND STATISTICS Established by WALTER A. SHEWHART and SAMUEL S. WILKS Editors: David J. Balding, Noel A. C. Cressie, Garrett M. Fitzmaurice, Iain M. Johnstone, Geert Molenberghs, David W. Scott, Adrian F. M. Smith, Ruey S. Tsay, Sanford Weisberg Editors Emeriti: Vic Barnett, J. Stuart Hunter, Joseph B. Kadane, JozefL. Teugels A complete list of the titles in this series appears at the end of this volume.

5 Handbook of Monte Carlo Methods Dirk P. Kroese University of Queensland Thomas Taimre University of Queensland Zdravko I. Botev Université de Montréal WILEY A JOHN WILEY & SONS, INC., PUBLICATION

6 Copyright 2011 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) , fax (978) , or on the web at Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) , fax (201) , or online at Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services or for technical support, please contact our Customer Care Department within the United States at (800) , outside the United States at (317) or fax (317) 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 Library of Congress Cataloging-in-Publication Data: Kroese, Dirk P. Handbook for Monte Carlo methods / Dirk P. Kroese, Thomas Taimre, Zdravko I. Botev. p. cm. (Wiley series in probability and statistics ; 706) Includes index. ISBN (hardback) 1. Monte Carlo method. I. Taimre, Thomas, II. Botev, Zdravko I., III. Title. QA298.K '.282 dc Printed in the United States of America

7 To Lesley DPK To Aita and Ilmar TT To my parents, Maya and Ivan zib

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9 CONTENTS Preface Acknowledgments xvii xix 1 Uniform Random Number Generation Random Numbers Properties of a Good Random Number Generator Choosing a Good Random Number Generator Generators Based on Linear Recurrences Linear Congruential Generators Multiple-Recursive Generators Matrix Congruential Generators Modulo 2 Linear Generators Combined Generators Other Generators Tests for Random Number Generators Spectral Test Empirical Tests 14 References 21

10 VIII CONTENTS 2 Quasirandom Number Generation Multidimensional Integration Van der Corput and Digital Sequences Halton Sequences Faure Sequences SoboP Sequences Lattice Methods Randomization and Scrambling 38 References 40 3 Random Variable Generation Generic Algorithms Based on Common Transformations Inverse-Transform Method Other Transformation Methods Table Lookup Method Alias Method Acceptance-Rejection Method Ratio of Uniforms Method Generation Methods for Multivariate Random Variables Copulas Generation Methods for Various Random Objects Generating Order Statistics Generating Uniform Random Vectors in a Simplex Generating Random Vectors Uniformly Distributed in a Unit Hyperball and Hypersphere Generating Random Vectors Uniformly Distributed in a Hyperellipsoid Uniform Sampling on a Curve Uniform Sampling on a Surface Generating Random Permutations Exact Sampling From a Conditional Bernoulli Distribution 80 References 83 4 Probability Distributions Discrete Distributions Bernoulli Distribution Binomial Distribution Geometric Distribution Hypergeometric Distribution Negative Binomial Distribution 94

11 CONTENTS IX Phase-Type Distribution (Discrete Case) Poisson Distribution Uniform Distribution (Discrete Case) Continuous Distributions Beta Distribution Cauchy Distribution Exponential Distribution F Distribution Fréchet Distribution Gamma Distribution Gumbel Distribution Laplace Distribution Logistic Distribution Log-Normal Distribution Normal Distribution Pareto Distribution Phase-Type Distribution (Continuous Case) Stable Distribution Student's t Distribution Uniform Distribution (Continuous Case) Wald Distribution Weibull Distribution Multivariate Distributions Dirichlet Distribution Multinomial Distribution Multivariate Normal Distribution Multivariate Student's t Distribution Wishart Distribution 148 References Random Process Generation Gaussian Processes Markovian Gaussian Processes Stationary Gaussian Processes and the FFT Markov Chains Markov Jump Processes Poisson Processes Compound Poisson Process Wiener Process and Brownian Motion Stochastic Differential Equations and Diffusion Processes Euler's Method Milstein's Method 187

12 X CONTENTS Implicit Euler Exact Methods Error and Accuracy Brownian Bridge Geometric Brownian Motion Ornstein-Uhlenbeck Process Reflected Brownian Motion Fractional Brownian Motion Random Fields Levy Processes Increasing Levy Processes Generating Levy Processes Time Series 219 References 222 Markov Chain Monte Carlo Metropolis-Hastings Algorithm Independence Sampler Random Walk Sampler Gibbs Sampler Specialized Samplers Hit-and-Run Sampler Shake-and-Bake Sampler Metropolis-Gibbs Hybrids Multiple-Try Metropolis-Hastings Auxiliary Variable Methods Reversible Jump Sampler Implementation Issues Perfect Sampling 274 References 276 Discrete Event Simulation Simulation Models Discrete Event Systems Event-Oriented Approach More Examples of Discrete Event Simulation Inventory System Tandem Queue Repairman Problem 296 References 300

13 CONTENTS XI Statistical Analysis of Simulation Data Simulation Data Data Visualization Data Summarization Estimation of Performance Measures for Independent Data Delta Method Estimation of Steady-State Performance Measures Covariance Method Batch Means Method Regenerative Method Empirical Cdf Kernel Density Estimation Least Squares Cross Validation Plug-in Bandwidth Selection Resampling and the Bootstrap Method Goodness of Fit Graphical Procedures Kolmogorov-Smirnov Test Anderson-Darling Test x 2 Tests 340 References 343 Variance Reduction Variance Reduction Example Antithetic Random Variables Control Variables Conditional Monte Carlo Stratified Sampling Latin Hypercube Sampling Importance Sampling Minimum-Variance Density Variance Minimization Method Cross-Entropy Method Weighted Importance Sampling Sequential Importance Sampling Response Surface Estimation via Importance Sampling Quasi Monte Carlo 376 References 379

14 XII CONTENTS 10 Rare-Event Simulation Efficiency of Estimators Importance Sampling Methods for Light Tails Estimation of Stopping Time Probabilities Estimation of Overflow Probabilities Estimation For Compound Poisson Sums Conditioning Methods for Heavy Tails Estimation for Compound Sums Sum of Nonidentically Distributed Random Variables State-Dependent Importance Sampling Cross-Entropy Method for Rare-Event Simulation Splitting Method 409 References Estimation of Derivatives Gradient Estimation Finite Difference Method Infinitesimal Perturbation Analysis Score Function Method Score Function Method With Importance Sampling Weak Derivatives Sensitivity Analysis for Regenerative Processes 435 References Randomized Optimization Stochastic Approximation Stochastic Counterpart Method Simulated Annealing Evolutionary Algorithms Genetic Algorithms Differential Evolution Estimation of Distribution Algorithms Cross-Entropy Method for Optimization Other Randomized Optimization Techniques 460 References Cross-Entropy Method Cross-Entropy Method Cross-Entropy Method for Estimation Cross-Entropy Method for Optimization Combinatorial Optimization 469

15 CONTENTS XIII Continuous Optimization Constrained Optimization Noisy Optimization 476 References Particle Methods Sequential Monte Carlo Particle Splitting Splitting for Static Rare-Event Probability Estimation Adaptive Splitting Algorithm Estimation of Multidimensional Integrals Combinatorial Optimization via Splitting Knapsack Problem Traveling Salesman Problem Quadratic Assignment Problem Markov Chain Monte Carlo With Splitting 509 References Applications to Finance Standard Model Pricing via Monte Carlo Simulation Sensitivities Pathwise Derivative Estimation Score Function Method 542 References Applications to Network Reliability Network Reliability Evolution Model for a Static Network Conditional Monte Carlo Leap-Evolve Algorithm Importance Sampling for Network Reliability Importance Sampling Using Bounds Importance Sampling With Conditional Monte Carlo Splitting Method Acceleration Using Bounds 573 References Applications to Differential Equations Connections Between Stochastic and Partial Differential Equations 577

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