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9780387001784

Random Number Generation and Monte Carlo Methods

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  • ISBN13:

    9780387001784

  • ISBN10:

    0387001786

  • Edition: 2nd
  • Format: Hardcover
  • Copyright: 2003-07-01
  • Publisher: Springer Nature
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Summary

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.

Author Biography

James E. Gentle is University Professor of Computational Statistics at George Mason University.

Table of Contents

Preface vii
Simulating Random Numbers from a Uniform Distribution
1(60)
Uniform Integers and an Approximate Uniform Density
5(6)
Simple Linear Congruential Generators
11(16)
Structure in the Generated Numbers
14(6)
Tests of Simple Linear Congruential Generators
20(1)
Shuffling the Output Stream
21(2)
Generation of Substreams in Simple Linear Congruential Generators
23(4)
Computer Implementation of Simple Linear Congruential Generators
27(4)
Ensuring Exact Computations
28(1)
Restriction that the Output Be in the Open Interval (0,1)
29(1)
Efficiency Considerations
30(1)
Vector Processors
30(1)
Other Linear Congruential Generators
31(5)
Multiple Recursive Generators
32(2)
Matrix Congruential Generators
34(1)
Add-with-Carry, Subtract-with-Borrow, and Multiply-with-Carry Generators
35(1)
Nonlinear Congruential Generators
36(2)
Inversive Congruential Generators
36(1)
Other Nonlinear Congruential Generators
37(1)
Feedback Shift Register Generators
38(5)
Generalized Feedback Shift Registers and Variations
40(3)
Skipping Ahead in GFSR Generators
43(1)
Other Sources of Uniform Random Numbers
43(3)
Generators Based on Cellular Automata
44(1)
Generators Based on Chaotic Systems
45(1)
Other Recursive Generators
45(1)
Tables of Random Numbers
46(1)
Combining Generators
46(2)
Properties of Combined Generators
48(3)
Independent Streams and Parallel Random Number Generation
51(3)
Skipping Ahead with Combination Generators
52(1)
Different Generators for Different Streams
52(1)
Quality of Parallel Random Number Streams
53(1)
Portability of Random Number Generators
54(1)
Summary
55(6)
Exercises
56(5)
Quality of Random Number Generators
61(32)
Properties of Random Numbers
62(2)
Measures of Lack of Fit
64(7)
Measures Based on the Lattice Structure
64(3)
Differences in Frequencies and Probabilities
67(3)
Independence
70(1)
Empirical Assessments
71(16)
Statistical Goodness-of-Fit Tests
71(15)
Comparisons of Simulated Results with Statistical Models in Physics
86(1)
Anecdotal Evidence
86(1)
Tests of Random Number Generators Used in Parallel
87(1)
Programming Issues
87(1)
Summary
87(6)
Exercises
88(5)
Quasirandom Numbers
93(8)
Low Discrepancy
93(1)
Types of Sequences
94(4)
Halton Sequences
94(2)
Sobol' Sequences
96(1)
Comparisons
97(1)
Variations
97(1)
Computations
98(1)
Further Comments
98(3)
Exercises
100(1)
Transformations of Uniform Deviates: General Methods
101(64)
Inverse CDF Method
102(7)
Decompositions of Distributions
109(2)
Transformations that Use More than One Uniform Deviate
111(1)
Multivariate Uniform Distributions with Nonuniform Marginals
112(1)
Acceptance/Rejection Methods
113(12)
Mixtures and Acceptance Methods
125(4)
Ratio-of-Uniforms Method
129(4)
Alias Method
133(3)
Use of the Characteristic Function
136(1)
Use of Stationary Distributions of Markov Chains
137(12)
Use of Conditional Distributions
149(1)
Weighted Resampling
149(1)
Methods for Distributions with Certain Special Properties
150(5)
General Methods for Multivariate Distributions
155(4)
Generating Samples from a Given Distribution
159(6)
Exercises
159(6)
Simulating Random Numbers from Specific Distributions
165(52)
Modifications of Standard Distributions
167(3)
Some Specific Univariate Distributions
170(27)
Normal Distribution
171(5)
Exponential, Double Exponential, and Exponential Power Distributions
176(2)
Gamma Distribution
178(5)
Beta Distribution
183(1)
Chi-Squared, Student's t, and F Distributions
184(2)
Weibull Distribution
186(1)
Binomial Distribution
187(1)
Poisson Distribution
188(1)
Negative Binomial and Geometric Distributions
188(1)
Hypergeometric Distribution
189(1)
Logarithmic Distribution
190(1)
Other Specific Univariate Distributions
191(2)
General Families of Univariate Distributions
193(4)
Some Specific Multivariate Distributions
197(13)
Multivariate Normal Distribution
197(1)
Multinomial Distribution
198(1)
Correlation Matrices and Variance-Covariance Matrices
198(3)
Points on a Sphere
201(1)
Two-Way Tables
202(1)
Other Specific Multivariate Distributions
203(5)
Families of Multivariate Distributions
208(2)
Data-Based Random Number Generation
210(2)
Geometric Objects
212(5)
Exercises
213(4)
Generation of Random Samples, Permutations, and Stochastic Processes
217(12)
Random Samples
217(3)
Permutations
220(1)
Limitations of Random Number Generators
220(1)
Generation of Nonindependent Samples
221(3)
Order Statistics
221(2)
Censored Data
223(1)
Generation of Nonindependent Sequences
224(5)
Markov Process
224(1)
Nonhomogeneous Poisson Process
225(1)
Other Time Series Models
226(1)
Exercises
227(2)
Monte Carlo Methods
229(54)
Evaluating an Integral
230(3)
Sequential Monte Carlo Methods
233(2)
Experimental Error in Monte Carlo Methods
235(1)
Variance of Monte Carlo Estimators
236(3)
Variance Reduction
239(10)
Analytic Reduction
240(1)
Stratified Sampling and Importance Sampling
241(4)
Use of Covariates
245(3)
Constrained Sampling
248(1)
Stratification in Higher Dimensions: Latin Hypercube Sampling
248(1)
The Distribution of a Simulated Statistic
249(1)
Computational Statistics
250(6)
Monte Carlo Methods for Inference
251(1)
Bootstrap Methods
252(3)
Evaluating a Posterior Distribution
255(1)
Computer Experiments
256(1)
Computational Physics
257(4)
Computational Finance
261(22)
Exercises
271(12)
Software for Random Number Generation
283(14)
The User Interface for Random Number Generators
285(1)
Controlling the Seeds in Monte Carlo Studies
286(1)
Random Number Generation in Programming Languages
286(2)
Random Number Generation in IMSL Libraries
288(3)
Random Number Generation in S-Plus and R
291(6)
Exercises
295(2)
Monte Carlo Studies in Statistics
297(16)
Simulation as an Experiment
298(2)
Reporting Simulation Experiments
300(1)
An Example
301(12)
Exercises
310(3)
A Notation and Definitions
313(10)
B Solutions and Hints for Selected Exercises
323(8)
Bibliography
331(40)
Literature in Computational Statistics
332(2)
World Wide Web, News Groups, List Servers, and Bulletin Boards
334(2)
References for Software Packages
336(1)
References to the Literature
336(35)
Author Index 371(6)
Subject Index 377

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