Summary
A review and comparison is presented for the use of Monte Carlo and Quasi-Monte Carlo methods for multivariate Normal and multivariate t distribution computation problems. Spherical-radial transformations, and separation-of-variables transformations for these problems are considered. The use of various Monte Carlo methods, Quasi-Monte Carlo methods and randomized Quasi-Monte Carlo methods are discussed for the different problem formulations and test results are summarized.
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References
M. Beckers and A. Haegemans. ‘Comparison of Numerical Integration Techniques for Multivariate Normal Integrals’, Computer Science Depart ment Preprint, Catholic University of Leuven, Belgium, 1992.
E. A. Cornish. ‘The Multivariate t-Distribution Associated with a Set of Normal Sample Deviates’, Australian Journal of Physics 7, pp. 531-542, 1954.
I. Deák. ‘Three Digit Accurate Multiple Normal Probabilities’ Numer. Math. 35, pp. 369-380, 1980.
I. Deák. ‘Computing Probabilities of Rectangles in Case of Multinormal Distribution’ J. Statist. Comput. Simul. 26, pp. 101-114, 1986.
I. Deák. Random Number Generation and Simulation, Akadémiai Kiadó, Budapest, Chapter 7, 1990.
L. Devroye. ‘Non-Uniform Random Variate Generation’ Springer-Verlag, Berlin, 1986.
K.-T. Fang and Y. Wang. Number-Theoretic Methods in Statistics, Chap-man and Hall, London, pp. 167-170, 1994.
H. I. Gassmann, I. Deák, and T. Szántai. ‘Computing Multivariate Normal Probabilities: a New Look’, J. Comp. Graph. Stat. 11, pp. 920-949, 2002.
A. Genz. ‘Numerical Computation of the Multivariate Normal Probabilities’, J. Comput. Graph. Stat. 1, pp. 141-150, 1992.
A. Genz. ‘A Comparison of Methods for Numerical Computation of Multivariate Normal Probabilities’, Computing Science and Statistics 25, pp. 400-405, 1993.
A. Genz and F. Bretz. Numerical Computation of Multivariate t Proba-bilities with Application to Power Calculation of Multiple Contrasts, J. Stat. Comp. Simul. 63, pp. 361-378, 1999.
A. Genz and F. Bretz. ‘Numerical Computation of Critical Values for Multiple Comparison Problems’, in Proceedings of the Statistical Com-puting Section, American Statistical Association, Alexandria, VA, pp. 84-87, 2000.
A. Genz and F. Bretz. ‘Critical Point and Power Calculations for the Studentised Range Test’, J. Stat. Comp. Simul. 71, pp. 85-97, 2001.
A. Genz and F. Bretz. ‘Comparison of Methods for the Computation of Multivariate t Probabilities’, J. Comp. Graph. Stat. 11, pp. 950-971.
A. Genz and K. S. Kwong. ‘Numerical Evaluation of Singular Multivariate Normal Distributions’, J. Stat. Comp. Simul. 68, pp. 1-21, 1999.
J. Geweke. ‘Efficient Simulation from the Multivariate Normal and Student-t Distributions Subject to Linear Constraints’, Computing Sci ence and Statistics 23, pp. 571-578, 1991.
G. J. Gibson, C. A. Glasbey, and D. A. Elston. ‘Monte-Carlo Evaluation of Multivariate Normal Integrals and Sensitivity to Variate Ordering’, in Proceedings of the Third International Conference in Numerical Methods and Applications, World Scientific, Singapore, pp. 120-126, 1994.
V. Hajivassiliou. ‘Simulating Normal Rectangle Probabilities and Their Derivatives: The effects of Vectorization’, The International Journal of Supercomputer Applications 7, pp. 231-253, 1993.
V. Hajivassiliou, D. McFadden, and O. Rudd. ‘Simulation of Multivariate Normal Rectangle Probabilities and Their Derivatives: Theoretical and Computational Results’, Journal of Econometrics, 72, pp. 85-134, 1996
F. J. Hickernell and H. S. Hong. ‘Computing Multivariate Normal Proba-bilities Using Rank-1 Lattice Sequences’, in Proceedings of the Workshop on Scientific Computing (Hong Kong), (G. H. Golub, S. H. Lui, F. T. Luk, and R. J. Plemmons, eds.), Springer-Verlag, Singapore, pp. 209-215, 1997.
F. J. Hickernell, H. S. Hong, P. L’Ecuyer, and C. Lemieux. ‘Extensible Lattice Sequences for QMC Quadrature’ SIAM Journal of Scientific and Statistical Computing 22, pp. 1117-1138, 2000.
C. Lemieux and P. L’Ecuyer. ‘A Comparison of Monte Carlo, Lattice Rules and Other Low-Discrepancy Point Sets’, in Monte Carlo and Quasi-Monte Carlo methods 1998, (H. Niederreiter and J. Spanier Eds.), Springer, Berlin, pp. 326-340, 2000.
D. Nuyens and R. Cools. ‘Fast Component-By-Component Construction of Rank-1 Lattice Rules in Shift-Invariant Reproducing Kernel Hilbert Spaces’ Math. Comp. 75, pp. 903-920, 2006.
Z. Sandor and P. Andras. ‘Alternative Sampling Methods for Estimating Multivariate Normal Probabilities’, Journal of Econometrics, 120, pp. 207-234, 2004.
M. Schervish. ‘Multivariate Normal Probabilities with Error Bound’, Applied Statistics 33, pp. 81-87, 1984.
I. H. Sloan. ‘QMC Integration - Beating Intractability by Weighting the Co-ordinate Directions’ in Monte Carlo and Quasi-Monte Carlo Methods 2000 (K. T. Fang, F. J. Hickernell, and H. Niederreiter, eds.), Springer-Verlag, Berlin, pp. 103-123, 2002.
I. H. Sloan and S. Joe. Lattice Methods for Multiple Integration, Oxford University Press, Oxford, 1994.
P. N. Somerville. ‘Multiple Testing and Simultaneous Confidence Intervals: Calculation of Constants’ Comp. Stat. & Data Analysis 25, pp. 217-223, 1997.
P. N. Somerville. ‘Numerical Computation of Multivariate Normal and Multivariate-t Probabilities Over Convex Regions’, J. Comput. Graph. Stat. 7, pp. 529-545, 1998.
P. N. Somerville. ‘A Fortran 90 Program for Evaluation of Multivariate Normal and Multivariate t Integral over Convex Regions’, Journal of Statistical Software 3, 1998, available at http://www.jstatsoft.org.
P. N. Somerville. ‘Critical Values for Multiple Testing and Comparisons: One Step and Step Down Procedures’ J. Stat. Plan. & Inf., 82, pp. 129-138,1999.
P. N. Somerville. ‘Numerical Computation of Multivariate Normal and Multivariate t Probabilities over Ellipsoidal regions. Journal of Statistical Software 6, 2001, available at http://www.jstatsoft.org.
G. W. Stewart. ‘The Efficient Generation of Random Orthogonal Matrices with An Application to Condition Estimation’, SIAM J. Numer. Anal. 17, pp. 403-409, 1980.
Y. L. Tong. The Multivariate Normal Distribution, Springer-Verlag, New York, 1990.
P. M. Vijverberg. ‘Monte Carlo Evaluation of Multivariate Student’s t Probabilities’, Economics Letters 52, pp. 1-6, 1996.
W. P. M. Vijverberg. ‘Monte Carlo Evaluation of Multivariate Normal Probabilities’,Journal of Econometrics 76, pp. 281-307, 1997.
W. P. M. Vijverberg. ‘Rectangular and Wedge-shaped Multivariate Nor-mal Probabilities’, Economics Letters 68, pp. 13-20, 2000.
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Genz, A. (2008). MCQMC Methods for Multivariate Statistical Distributions. In: Keller, A., Heinrich, S., Niederreiter, H. (eds) Monte Carlo and Quasi-Monte Carlo Methods 2006. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74496-2_3
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DOI: https://doi.org/10.1007/978-3-540-74496-2_3
Publisher Name: Springer, Berlin, Heidelberg
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