Abstract
New formulas are derived for multivariate normal probabilities defined for hyper-rectangular probability regions. The formulas use conditioning with a sequence of bivariate normal probabilities. The result is an approximate formula for multivariate normal probabilities which uses a product of bivariate normal probabilities. The new approximation method is compared with approximation methods based on products of univariate normal probabilities, using tests with random covariance-matrix/probability-region problems for up to twenty variables. The reordering of variables is studied to improve efficiency of the new method.
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References
Connors, R.D., Hess, S., Daly, A.: Analytic approximations for computing probit choice probabilities. Transportmetrica 10(2), 119–139 (2014)
Drezner, Z., Wesolowsky, G.O.: On the computation of the bivariate normal integral. J. Stat. Comput. Simul. 3, 101–107 (1990)
Genz, A.: Numerical computation of rectangular bivariate and trivariate normal and \(t\) probabilities. Stat. Comput. 14, 151–160 (2004)
Genz, A.: MVNXPB, a MATLAB/Octave function for the approximation of multivariate Normal probabilities. (2013), at www.math.wsu.edu/faculty/genz/software/matlab
Genz, A., Bretz, F.: Computation of Multivariate Normal and t Probabilities. Lecture Notes in Statistics 195, Springer, New York (2009)
Golub, G.H., Van Loan, C.F.: Matrix Computations, 4th edn. Johns Hopkins University Press, Baltimore (2013)
Gibson, G.J., Glasbey, C.A., Elston, D.A.: Monte Carlo evaluation of multivariate normal integrals and sensitivity to variate ordering. In: Dimov, I.T., Sendov, B., Vassilevski, P.S. (eds.) Advances in Numerical Methods and Applications, pp. 120–126. World Scientific Publishing, River Edge (1994)
Ho, H.J., Lin, T.I., Chen, H.Y., Wang, W.L.: Some results on the truncated multivariate t distribution. J. Stat. Plan. Inference 142, 25–40 (2012)
Joe, H.: Approximations to multivariate normal rectangle probabilities based on conditional expectations. J. Am. Stat. Assoc. 90, 957–964 (1995)
Kamakura, W.A.: The estimation of multinomial probit models: a new calibration algorithm. Transp. Sci. 23, 253–265 (1989)
McFadden, D., Train, K.: Mixed MNL models for discrete response. J. Appl. Econom. 15, 447–470 (2000)
Mendell, N.R., Elston, R.C.: Multifactorial qualitative traits: genetic analysis and prediction of recurrence risks. Biometrics 30, 41–57 (1974)
Muthén, B.: Moments of the censored and truncated bivariate normal distribution. Br J Math Stat Psychol 43, 131–143 (1991)
Ochi, Y., Prentice, R.L.: Likelihood inference in a correlated probit regression model. Biometrika 71, 531–543 (1984)
Rosenbaum, S.: Moments of a truncated bivariate normal distribution. J. R. Stat. Soc. 23, 405–408 (1961)
Schervish, M.J.: Algorithm AS 195: multivariate normal probabilities with error bound. J. R. Stat. Soc. Ser. C 33, 81–94 (1984), correction 34, 103–104 (1985)
Stewart, G.W.: The efficient generation of random orthogonal matrices with an application to condition estimators. SIAM J. Numer. Anal. 17(3), 403–409 (1980)
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Trinh, G., Genz, A. Bivariate conditioning approximations for multivariate normal probabilities. Stat Comput 25, 989–996 (2015). https://doi.org/10.1007/s11222-014-9468-y
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DOI: https://doi.org/10.1007/s11222-014-9468-y