Both the classical Gauss-Hermite quadrature for dx and the littleknown Gaussian quadrature for given by Steen-Byrne-Gelbard (1969)given by Steen-Byrne-Gelbard (1969)can be used to evaluate the multivariate normal integrals. In the present paper, we compare the above quadratures for the multivariate normal integrals. The simulated results show that the efficiencies of two formulas have not the significant difference if the condition of integral is very good, however, when the dimension of integral is high or the condition of correlation matrix of the multivariate normal distribution is not good, Steen et.al. formula is more efficient. In appendis, an expanded table of Gaussian quadrature for Steen et.al. is given by the present author.
In the present paper, we give a review of pseudo-random number generators. The new methods and theory appearing in 1990’s will be focused. This paper concerns with almost all kinds of generators such as the linear, nonlinear and in- versive congruential methods, Fibonacci and Tausworthe (or feedback shift regis- ter) sequences, add-with-carry and subtract-with-borrow methods, multiple prime generator and chaotic mapping, as well as the theory of combination of generators.
In the present paper, the authors suggest an algorithm to evaluate the multivariate normal integrals under the supposition that the correlation matrix R is quasi-decomposable, in which we have rij = aiaj for most i, j, and rij = aiaj + bij for the others, where bij’s are the nonzero deviations. The algorithm makes the high-dimensional normal distribution reduce to a 2-dimensional or 3-dimensional integral which can be evaluated by the numerical method with a high precision.Our supposition is close to what we encounter in practice. When correlation matrix is arbitrary, we suggest an approximate algorithm with a medium precision, it is, in general, better than some approximate algorithms. The simulation results of about 20000 high-dimensional integrals showed that the present algorithms were very efficient.
