Random Number Generation and Monte Carlo Methods

Transcription

1 James E. Gentle Random Number Generation and Monte Carlo Methods With 30 Illustrations Springer

2 Contents Preface vii 1 Simulating Random Numbers from a Uniform Distribution Linear Congruential Generators Structure in the Generated Numbers Skipping Ahead in Linear Congruential Generators ShufHing the Output Stream Tests of Linear Congruential Generators Computer Implementation of Linear Congruential Generators Insuring Exact Computations Restriction that the Output Be > 0 and < Efficiency Considerations Vector Processors Other Congruential Generators Multiple Recursive Generators Lagged Fibonacci Add-with-Carry, Subtract-with-Borrow, and Multiply-with-Carry Generators Inversive Congruential Generators Other Nonlinear Congruential Generators Matrix Congruential Generators Feedback Shift Register Generators Generalized Feedback Shift Registers and Variations Skipping Ahead in GFSR Generators Other Sources of Uniform Random Numbers Generators Based on Chaotic Systems Tables of Random Numbers Portable Random Number Generators Combining Generators Wichmann/Hill Generator L'Ecuyer Combined Generators 33 XI

3 xii CONTENTS Properties of Combined Generators Independent Streams and Parallel Random Number Generation Lehmer Trees Combination Generators Monte Carlo on Parallel Processors 37 Exercises 38 2 Transformations of Uniform Deviates: General Methods Inverse CDF Method Acceptance/Rejection Methods Mixtures of Distributions Mixtures and Acceptance Methods Ratio-of-Uniforms Method Alias Method Use of Stationary Distributions of Markov Chains Weighted Resampling Methods for Distributions with Certain Special Properties General Methods for Multivariate Distributions Generating Samples from a Given Distribution 80 Exercises 80 3 Simulating Random Numbers from Specific Distributions Some Specific Univariate Distributions Standard Distributions and Folded Distributions Normal Distribution Exponential, Double Exponential, and Exponential Power Distributions Gamma Distribution Beta Distribution Student's t, Chi-Squared, and F Distributions Weibull Distribution Binomial Distribution Poisson Distribution Negative Binomial and Geometrie Distributions Hypergeometric Distribution Logarithmic Distribution Other Specific Univariate Distributions General Families of Univariate Distributions Some Specific Multivariate Distributions Multivariate Normal Distribution Multinomial Distribution 106

4 CONTENTS xiii Correlation Matrices and Variance-Covariance Matrices Points on a Sphere Two-Way Tables Other Specific Multivariate Distributions General Multivariate Distributions Distributions with Specified Correlations Data-Based Random Number Generation Geometrie Objects 117 Exercises Generation of Random Samples and Permutations Random Samples Permutations Generation of Nonindependent Samples Order Statistics Nonindependent Sequences: Nonhomogeneous Poisson Process Censored Data 127 Exercises Monte Carlo Methods Evaluating an Integral Variance of Monte Carlo Estimators Variance Reduction Analytic Reduction Antithetic Variates Importance and Stratified Sampling Common Variates Constrained Sampling Latin Hypercube Sampling Computer Experiments Computational Statistics Monte Carlo Methods for Inference Bootstrap Methods Evaluating a Posterior Distribution 145 Exercises Quality of Random Number Generators Analysis of the Algorithm Empirical Assessments Statistical Tests Anecdotal Evidence Quasirandom Numbers Haiton Sequences 160

5 xiv CONTENTS Sobol' Sequences Comparisons Variations Some Examples of Applications Computations Programming Issues 164 Exercises Software for Random Number Generation The User Interface for Random Number Generators Controlling the Seeds in Monte Carlo Studies Random Number Generation in IMSL Libraries Random Number Generation in S-Plus 172 Exercises Monte Carlo Studies in Statistics Simulation as an Experiment Reporting Simulation Experiments An Example 180 Exercises 190 A Notation and Definitions 193 B Solutions and Hints for Selected Exercises 199 Bibliography 205 Literature in Computational Statistics 206 World Wide Web, News Groups, List Servers, and Bulletin Boards 208 References 211 Author Index 237 Subject Index 243