Stochastic simulation.

*(English)*Zbl 0613.65006
Wiley Series in Probability and Mathematical Statistics. Applied Probability and Statistics. New York etc.: John Wiley & Sons. XI, 237 p.; Ł 31.85 (1987).

Due to the increased versatility of computers and decreased cost of computer time, (mathematical) simulation methods have become useful and affordable tools for statisticians, applied mathematicians and operations researchers. Consequently during the last decade a number of books have appeared in the field of digital simulation - general descriptions of the theory, e.g. S. J. Yakowitz: Computational probability and simulation (1977; Zbl 0387.60001), G. S. Fishman: Principles of discrete event simulation (1978; Zbl 0537.68104), P. Bratley, B. L. Fox, and L. E. Schrage: A guide to simulation (1983; Zbl 0515.68070) and B. Morgan: Elements of simulation (1984; Zbl 0575.65009) and monographs on special topics, e.g. D. L. Iglehart and G. S. Shedler: Regenerative simulation of response times in networks of queues (1980; Zbl 0424.90016), E. J. Dudewicz and T. G. Ralley: The handbook of random number generation and testing with TESTRAND computer code (1981; Zbl 0478.65003 and L. Devroye: Non- uniform random variate generation (1986; Zbl 0593.65005). The book under review belongs to the first category; the table of contents gives an impression of the topics:

1. Aims of simulation: The tools, models, simulation as experimentation, simulation in inference, examples. 2. Pseudo-random numbers: History and philosophy, congruential generators, shift-register generators, lattice structure, shuffling and testing, conclusions, proofs. 3. Random variables: Simple examples, general principles, discrete distributions, continuous distributions, recommendations. 4. Stochastic models: Order statistics, multivariate distributions, Poisson processes and lifetimes, Markov processes, Gaussian processes, point processes, Metropolis’ method and random fields. 5. Variance reduction: Monte-Carlo integration, importance sampling, control and antithetic variates, conditioning, experimental design. 6. Output analysis: The initial transient, batching, time-series methods, regenerative simulation, a case study. 7. Uses of simulation: Statistical inference, stochastic methods in optimization, systems of linear equations, quasi-Monte-Carlo integration, sharpening Buffon’s needle. Appendix A: Computer systems, Appendix B: Computer programs.

This description shows that all ”classical” parts of computer simulation are covered. Several aspects seem to be presented here for the first time in a textbook - in particular the analyses of shift register generators, of the effects of discrete approximations of transformed distributions, of Monte Carlo confidence intervals and of simulation outputs should be mentioned as highly wellcome examples.

Though the style of this book is especially suited for mathematically oriented readers special emphasis is given on explicit recommendations of methods and algorithms - Appendix B contains a series of computer programs. Altogether, this work is an excellent comprehensive guide to simulation methods, written by a very competent author. It is especially recommended for those users of simulation methods who want more than a ”cook book”.

1. Aims of simulation: The tools, models, simulation as experimentation, simulation in inference, examples. 2. Pseudo-random numbers: History and philosophy, congruential generators, shift-register generators, lattice structure, shuffling and testing, conclusions, proofs. 3. Random variables: Simple examples, general principles, discrete distributions, continuous distributions, recommendations. 4. Stochastic models: Order statistics, multivariate distributions, Poisson processes and lifetimes, Markov processes, Gaussian processes, point processes, Metropolis’ method and random fields. 5. Variance reduction: Monte-Carlo integration, importance sampling, control and antithetic variates, conditioning, experimental design. 6. Output analysis: The initial transient, batching, time-series methods, regenerative simulation, a case study. 7. Uses of simulation: Statistical inference, stochastic methods in optimization, systems of linear equations, quasi-Monte-Carlo integration, sharpening Buffon’s needle. Appendix A: Computer systems, Appendix B: Computer programs.

This description shows that all ”classical” parts of computer simulation are covered. Several aspects seem to be presented here for the first time in a textbook - in particular the analyses of shift register generators, of the effects of discrete approximations of transformed distributions, of Monte Carlo confidence intervals and of simulation outputs should be mentioned as highly wellcome examples.

Though the style of this book is especially suited for mathematically oriented readers special emphasis is given on explicit recommendations of methods and algorithms - Appendix B contains a series of computer programs. Altogether, this work is an excellent comprehensive guide to simulation methods, written by a very competent author. It is especially recommended for those users of simulation methods who want more than a ”cook book”.

Reviewer: N.Schmitz

##### MSC:

65Cxx | Probabilistic methods, stochastic differential equations |

65-02 | Research exposition (monographs, survey articles) pertaining to numerical analysis |

65C99 | Probabilistic methods, stochastic differential equations |

60G55 | Point processes (e.g., Poisson, Cox, Hawkes processes) |