Random Number Generation and Monte Carlo Methods
Monte Carlo simulation has become one of the most important tools in all fields of science. Simulation methodology relies on a good source of numbers that appear to be random. These "pseudorandom" numbers must pass statistical tests just as random samples would. Methods for producing pseudorandom numbers and transforming those numbers to simulate samples from various distributions are among the most important topics in statistical computing. This book surveys techniques of random number generation and the use of random numbers in Monte Carlo simulation. The book covers basic principles, as well as newer methods such as parallel random number generation, nonlinear congruential generators, quasi Monte Carlo methods, and Markov chain Monte Carlo. The best methods for generating random variates from the standard distributions are presented, but also general techniques useful in more complicated models and in novel settings are described. The emphasis throughout the book is on practical methods that work well in current computing environments. The book includes exercises and can be used as a test or supplementary text for various courses in modern statistics. It could serve as the primary test for a specialized course in statistical computing, or as a supplementary text for a course in computational statistics and other areas of modern statistics that rely on simulation. The book, which covers recent developments in the field, could also serve as a useful reference for practitioners. Although some familiarity with probability and statistics is assumed, the book is accessible to a broad audience. The second edition is approximately 50% longer than the first edition. It includes advances in methods for parallel random number generation, universal methods for generation of nonuniform variates, perfect sampling, and software for random number generation. The material on testing of random number generators has been expanded to include a discussion of newer software for testing, as well as more discussion about the tests themselves. The second edition has more discussion of applications of Monte Carlo methods in various fields, including physics and computational finance. James Gentle is University Professor of Computational Statistics at George Mason University. During a thirteen-year hiatus from academic work before joining George Mason, he was director of research and design at the world's largest independent producer of Fortran and C general-purpose scientific software libraries. These libraries implement several random number generators, and are widely used in Monte Carlo studies. He is a Fellow of the American Statistical Association and a member of the International Statistical Institute. He has held several national offices in the American Statistical Association and has served as an associate editor for journals of the ASA as well as for other journals in statistics and computing. Recent activities include serving as program director of statistics at the National Science Foundation and as research fellow at the Bureau of Labor Statistics.
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Simulating Random Numbers from a Uniform Distribution
Quality of Random Number Generators
Simulating Random Numbers from Specific Distributions
Generation of Random Samples Permutations
Monte Carlo Methods
Software for Random Number Generation
Monte Carlo Studies in Statistics
A Notation and Definitions
B Solutions and Hints for Selected Exercises
acceptance/rejection method algorithm alias method applications approximation basic Bernoulli beta binary binomial bits Box-Muller transformation called chi-squared correlation density p(x describe DIEHARD tests dimensions discrete distribution discussed distribution function distribution with parameters efficient elements equation evaluate example Exercise exponential distribution finite Fortran gamma distribution given goodness-of-fit test Halton sequences implemented independent integers interval inverse CDF method L'Ecuyer lagged Fibonacci linear congruential majorizing density Markov chain Marsaglia mass points matrix modulus Monte Carlo methods Monte Carlo study multiple recursive multivariate distributions Niederreiter normal distribution order statistics output performed Poisson probability density probability function problem properties pseudorandom pseudorandom numbers quasirandom quasirandom sequences random deviates random number random sample random variable random walk ratio-of-uniforms method realization regions S-Plus Section seed simple simulation stochastic stream test statistic tion transformation truncated uniform deviates uniform distribution univariate values variance variates vector yield
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