Chapter 12. Approximation of Integrals

  1. André I. Khuri

Published Online: 26 MAR 2003

DOI: 10.1002/0471394882.ch12

Advanced Calculus with Applications in Statistics, Second Edition

Advanced Calculus with Applications in Statistics, Second Edition

How to Cite

Khuri, A. I. (2002) Approximation of Integrals, in Advanced Calculus with Applications in Statistics, Second Edition, John Wiley & Sons, Inc., Hoboken, NJ, USA. doi: 10.1002/0471394882.ch12

Author Information

  1. University of Florida, Gainesville, Florida, USA

Publication History

  1. Published Online: 26 MAR 2003
  2. Published Print: 1 NOV 2002

Book Series:

  1. Wiley Series in Probability and Statistics

Book Series Editors:

  1. David J. Balding,
  2. Peter Bloomfield,
  3. Noel A. C. Cressie,
  4. Nicholas I. Fisher,
  5. Iain M. Johnstone,
  6. J. B. Kadane,
  7. Louise M. Ryan,
  8. David W. Scott,
  9. Adrian F. M. Smith and
  10. Jozef L. Teugels

ISBN Information

Print ISBN: 9780471391043

Online ISBN: 9780471394884

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Keywords:

  • Bayesian statistics;
  • Gaussian;
  • Gauss–Hermite quadrature;
  • importance sampling;
  • Laplace's approximation;
  • Markov chain Monte Carlo;
  • Monte Carlo;
  • multiple integrals;
  • Newton–Cotes methods;
  • saddlepoint approximation;
  • Simpson's method;
  • trapezoidal method

Summary

This last chapter provides an exposition of methods for approximating integrals, including the trapezoidal method, Simpson's method, Newton–Cotes methods, Gaussian quadrature, and the method of Laplace. Approximation of multidimensional integrals are also included, in addition to the Monte Carlo method. The last section on applications in statistics covers approximation using the Gauss–Hermite quadrature, the mean-squared error of a quadrature estimator. Also included is a discussion concerning the moments of a ratio of quadratic forms, and the use of Laplace approximation in Bayesian statistics.