Computation of the Multivariate Normal Integral over a Complex Subspace

Affiliation(s)

Abdul Salam School of Mathematical Sciences, GC Univer?sity, Lahore, Pakistan.

Air University Multan Campus, Multan, Pakistan.

Abdul Salam School of Mathematical Sciences, GC Univer?sity, Lahore, Pakistan.

Air University Multan Campus, Multan, Pakistan.

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.

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.

K. Kachiashvili and M. Hashmi, "Computation of the Multivariate Normal Integral over a Complex Subspace,"

References

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[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

[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