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Biometrika 1994 81(3):624-629; doi:10.1093/biomet/81.3.624
© 1994 by Biometrika Trust
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MISCELLANEA

A note on Gauss—Hermite quadrature

QING LIU1 and DONALD A. PIERCE2

1 Department of Biostatistics, St. Jude Children's Research Hospital 332 North Lauderdale, P. O. Box 318, Memphis, Tennessee 38101, U. S.A.
2 Department of Statistics, Oregon State University Corvallis, Oregon 97331, U.S.A.

Received for publication 1 August 1993.
  Abstract

For Gauss—Hermite quadrature, we consider a systematic method for transforming the variable of integration so that the integrand is sampled in an appropriate region. The effectiveness of the quadrature then depends on the ratio of the integrand to some Gaussian density being a smooth function, well approximated by a low-order polynomial. It is pointed out that, in this approach, order one Gauss-Hermite quadrature becomes the Laplace approximationmxHermite quadrature becomes the Laplace approximationmxHermite quadrature becomes the Laplace approximationmxHermite quadrature becomes the Laplace approximation. Thus the quadrature as implemented here can be thought of as a higher-order Laplace approximation.

Key Words: Asymptotic approximation • Bayesian inference • Generalized linear mixed models • Integrated likelihood • Measurement errors in covariables • Numerical integration


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