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 AM  Vol.3 No.5 , May 2012
Computation of the Multivariate Normal Integral over a Complex Subspace
Abstract: The computation of the multivariate normal integral over a Complex Subspace is a challenge, especially when the inte-gration region is of a complex nature. Such integrals are met with, for example, in the generalized Neyman-Pearson criterion, conditional Bayesian problems of testing many hypotheses and so on. The Monte-Carlo methods could be used for their computation, but at increasing dimensionality of the integral the computation time increases unjustifiedly. Therefore a method of computation of such integrals by series after reduction of dimensionality to one without information loss is offered below. The calculation results are given.
Cite this paper: K. Kachiashvili and M. Hashmi, "Computation of the Multivariate Normal Integral over a Complex Subspace," Applied Mathematics, Vol. 3 No. 5, 2012, pp. 489-498. doi: 10.4236/am.2012.35074.
References

[1]   S. Thompson, “On the Distribution of Type II Errors in Hypothesis Testing,” Applied Mathematics, Vol. 2, No. 2, 2011, pp. 189-195. doi:10.4236/am.2011.22021

[2]   C. R. Rao, “Linear Statistical Inference and Its Application,” 2nd Edition, John Wiley & Sons Ltd, New York, 2006.

[3]   K. J. Kachiashvili, “Generalization of Bayesian Rule of Many Simple Hypotheses Testing,” International Journal of Information Technology & Decision Making, Vol. 2, No. 1, 2003, pp. 41-70. doi:10.1142/S0219622003000525

[4]   A. V. Primak, V. V. Kafarov and K. J. Kachiashvili, “System Analysis of Air and Water Quality Con?trol,” Naukova Dumka, Kiev, 1991.

[5]   A. I. Potapov, A. G. Vinogradov, I. A. Goritskyi and E. E. Pertsov, “About Decision-Making of Presence of Objects at Group Measurements,” Questions of Radio-Electronics, Vol. 6, 1975, pp. 69-76.

[6]   P. J. David and P. Rabinovitz, “Methods of Numerical Integration. Computer Science and Applied Mathematics,” 2nd Edition, Academic Press Inc., Orlando, 1984.

[7]   A. Genz, “Numerical Computation of Multivariate Normal Probabilities,” Journal of Computational and Graphical Statistics, Vol. 1, 1992, pp. 141-149.

[8]   A. Genz, “Comparison of Methods for the Computation of Multivariate Normal Probabilities,” Computing Science and Statistics, Vol. 25, 1993, pp. 400-405.

[9]   A. Genz and F. Bretz, “Numerical Computation of Multivariate t-Probabilities with Application to Power Calculation of Multiple Contrasts,” Journal of Statistical Computation and Simulation, Vol. 63, No. 4, 1999, pp. 361378. doi:10.1080/00949659908811962

[10]   S. Joe, “Approximations to Multivariate Normal Rectangle Probabilities Based on Conditional Expectations,” Journal of the American Statistical Association, Vol. 90, 1995, pp. 957-964.

[11]   I. H. Sloan and S. Joe, “Lattice Methods for Multiple Integration,” Clarendon Press, Oxford, 1994.

[12]   V. Hajivassiliou, D. McFadden and P. Ruud, “Simulation of Multivariate Normal Rectangle Probabilities and Their Derivatives: Theoretical and Computational Results,” Journal of Econometrics, Vol. 72, No. 1-2, 1996, pp. 85134. doi:10.1016/0304-4076(94)01716-6

[13]   J. O. Berger, “Statistical Decision Theory and Bayesian Analysis,” Springer, New York, 1985.

[14]   K. J. Kachiashvili, “Bayesian Algorithms of Many Hypothesis Testing,” Ganatleba, Tbilisi, 1989.

[15]   D. V. Lindley, “The Use of Prior Probability Distributions in Statistical Inference and Decisions,” Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, 1961, pp. 453-468.

[16]   L. Tierney and J. B. Kadane, “Accurate Approximations for Posterior Moments and Marginal Densities,” Journal of the American Sta?tis?tical Association, Vol. 81, 1986, pp. 82-86.

[17]   A. Stuart, J. K. Ord and S. Arnols, “Kendall’s Advanced Theory of Statistics. Classical Inference and the Linear Model,” Vol. 2A, 6th Edition, Oxford University Press Inc., New York, 1999.

[18]   T. W. Anderson, “An introduction to Multivariate Statistical Analysis,” 3rd Edition, Wiley & Sons, Inc., New Jersey, 2003.

[19]   A. Stuart, J. K. Ord and S. Arnols, “Kendall’s Advanced Theory of Statistics. Distribution Theory,” Vol. 1, 6th Edition, Oxford University Press Inc., New York, 1994.

[20]   H. Cramer, “Mathematical Methods of Statistics,” Princeton University Press, Princeton, 1999.

[21]   M. Kendall and A. Stuart, “Distribution Theory,” Vol. 1, Charles Griffit & Company Limited, London, 1966.

[22]   G. D. Shel?lard, “Estimating the Product of Several Random Variables,” Journal of the American Sta?tis-tical Association, Vol. 47, 1952, pp. 216-221.

[23]   H. A. R. Barnett, “The Variance of the Product of Two independent Variables and Its Application to an In?ves?ti?ga?tion Based on Sample Data,” Journal of the Institute of Actuaries, Vol. 81, 1955, pp. 190-198.

[24]   L. A. Goodman, “On the Exact Variance of Products,” Journal of the American Statistical Association, Vol. 55, 1960, pp. 708-713.

[25]   L. A. Goodman, “The Variance of the Product of K Random Variables,” Journal of the American Statistical Association, Vol. 57, No. 297, 1962, pp. 54-60.

[26]   S. N. Nath, “On Product Moments from a Finite Universe,” Journal of the American Sta?tis?ti?cal Association, Vol. 63, No. 322, 1968, pp. 535-541.

[27]   S. N. Nath, “More results on Pro??duct Moments from a Finite Universe,” Journal of the American Sta?tis?ti?cal Association, Vol. 64, No. 327, 1969, pp. 864-869.

[28]   S. Nadarajah and K. Mitov, “Product Moments of Multivariate Random Vectors,” Communications in Statistics. Theory and Methods, Vol. 32, No. 1, 2003, pp. 47-60. doi:10.1081/STA-120017799

[29]   S. Kotz, N. Balakrishnan and N. L. Johnson, “Continuous Multivariate Distributions. Models and Applications,” Vol. 1, 2nd Edition, John Wiley & Sons Ltd, New York, 2000. doi:10.1002/0471722065

[30]   G. Szego, “Orthogonal Polynomials,” American Mathematical Society, New York, 1959.

[31]   K. J. Kachiashvili and D. I. Melikdzhanian, “SDpro—The Software Package for Statistical Processing of Experimental Information,” International Journal Information Technology & Decision Making, Vol. 9, No 1, 2010, pp. 1-30. doi:10.1142/S0219622010003634

[32]   K. J. Kachiashvili and A. Mueed “Conditional Bayesian Task of Testing Many Hypotheses,” Statistics, 2011, pp. 1-20. doi:10.1080/02331888.2011.602681

 
 
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