Using Complex Integration to Compute Multivariate Normal Probabilities |
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Authors: | W C Soong Jason C Hsu |
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Institution: | 1. Department of Statistics , National Chung Hsing University , Taipei , Taiwan , 104, R. O. C.;2. Department of Statistics , Ohio State University , Columbus , OH , 43210 , USA |
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Abstract: | Abstract The execution of most multiple comparison methods involves, at least in part, the computation of the probability that a multivariate normal or multivarite t random vector is in a hyper-rectangle. In multiple comparison with a control as well as multiple comparison with the best (of normal populations or multinomial cell probabilities), the correlation matrix R of the random vector is nonsingular and of the form ![/></span>, where <b>D</b> is a diagonal matrix and <span class=](/na101/home/literatum/publisher/tandf/journals/content/ucgs20/1997/ucgs20.v006.i04/10618600.1997.10474750/production/images/medium/ucgs_397_ilf0001.gif) ![/></span> is a known vector. It is well known that, in this case, the multivariate normal rectangular probability can be expressed as a one-dimensional integral and successfully computed using Gaussian quadrature techniques. However, in multiple comparison with the mean (sometimes called analysis of means) of normal distributions, all-pairwise comparisons of three normal distributions, as well as simultaneous inference on multinomial cell probabilities themselves, the correlation matrix is singular and of the form <span class=](/na101/home/literatum/publisher/tandf/journals/content/ucgs20/1997/ucgs20.v006.i04/10618600.1997.10474750/production/images/medium/ucgs_397_ilf0002.gif) ![/></span>. It is not well known that, in this latter case, the multivariate normal rectangular probability can still be expressed as a single integral, albeit one with complex variables in its integrand. Previously published proofs of the validity of this expression either contained a gap or relied on a numerical demonstration, and this article will provide an analytic proof. Furthermore, we explain how this complex integral can be computed accurately, using Romberg integration of complex variables when the dimension is low, and using ?idák's inequality as an approximation when the dimension is at least moderate.</td>
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Keywords: | Complex integration Multiple comparisons Multivariate normal Rectangular probability |
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