%0 Journal Article %1 plackett1954reduction %A Plackett, R. L. %D 1954 %J Biometrika %K Gaussian_processes integrals multivariate_Gaussian numerical_methods useful_identity %P 351--360 %T A reduction formula for normal multivariate integrals %U http://www.jstor.org/stable/2332716 %V 41 %X MR Let (X1,X2,⋯,Xn) be a vector of chance variables with a nonsingular multivariate normal distribution. The problem is to evaluate P(X1>a1,⋯,Xn>an). The author obtains a reduction formula for this probability, involving integrals of partial derivatives of the probability with respect to the elements of the covariance matrix of (X1,⋯,Xn). For n=3 and n=4, the reduction formula enables the author to express the probability as a finite sum of single integrals of tabulated functions. These integrals have to be evaluated by numerical quadrature, but for certain cases simple approximations to them are given.

@article{plackett1954reduction, abstract = {[MR] Let (X1,X2,⋯,Xn) be a vector of chance variables with a nonsingular multivariate normal distribution. The problem is to evaluate P(X1>a1,⋯,Xn>an). The author obtains a reduction formula for this probability, involving integrals of partial derivatives of the probability with respect to the elements of the covariance matrix of (X1,⋯,Xn). For n=3 and n=4, the reduction formula enables the author to express the probability as a finite sum of single integrals of tabulated functions. These integrals have to be evaluated by numerical quadrature, but for certain cases simple approximations to them are given. }, added-at = {2012-10-18T06:54:51.000+0200}, author = {Plackett, R. L.}, biburl = {https://www.bibsonomy.org/bibtex/216aeb1bde598e2e51a73f8acd03edbb0/peter.ralph}, description = {The case of n=2 of the basic identity appears in Pearson 1901, http://archive.org/details/philtrans01501516}, fjournal = {Biometrika}, interhash = {8ed6e6d313cd674ab4b63d818328fe93}, intrahash = {16aeb1bde598e2e51a73f8acd03edbb0}, issn = {0006-3444}, journal = {Biometrika}, keywords = {Gaussian_processes integrals multivariate_Gaussian numerical_methods useful_identity}, mrclass = {60.0X}, mrnumber = {0065047 (16,377c)}, mrreviewer = {L. Weiss}, pages = {351--360}, timestamp = {2015-02-20T02:54:32.000+0100}, title = {A reduction formula for normal multivariate integrals}, url = {http://www.jstor.org/stable/2332716}, volume = 41, year = 1954 }