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Approximating integrals via Monte Carlo and deterministic methods. (English) Zbl 0958.65009
Oxford Statistical Science Series. 20. Oxford: Oxford University Press. ix, 288 p. (2000).
This text concerning numerical approximation of integrals is a substantial outgrowth of the review paper by the authors [Statist. Sci. 10, 254-272 (1995)]. Here several methods – deterministic or not – used in practice are reviewed from both theoretical and practical point of view. The authors succeeded admirably in bringing the reader up to date with the current situation in field launching him or her on many interesting research problems. Efforts are also made to keep this text as self-contained as possible.
The authors have brought together in a single volume a considerable amount of information. The central ideas are presented and illustrated abundantly. For purposes of including as much material as possible, peripheral ideas are presented in exercises. The text is as fluid as possible so that it can be easily understood by readers having basic notions in real and complex analysis, numerical analysis, algebra, probability, and statistics. Notions of programming languages such as C and Fortran will be beneficial for implementing the methods discussed. An extensive bibliography is found at the end of the text.
The book is geared toward a wide variety of research workers particularly to physical and mathematical scientists facing integration problems as well as toward graduate students or advanced undergraduate students with reasonable mathematical background. It would also be suitable as a course text in numerical approximation of integrals or as part in scientific computing.

65C05 Monte Carlo methods
65-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to numerical analysis
62G05 Nonparametric estimation
65C40 Numerical analysis or methods applied to Markov chains
65D32 Numerical quadrature and cubature formulas
62E10 Characterization and structure theory of statistical distributions
62D05 Sampling theory, sample surveys
65C60 Computational problems in statistics (MSC2010)
60J22 Computational methods in Markov chains