A Numerical Procedure to Generate a Sample Covariance Matrix

U SE of random numbers, especially in Monte Carlo procedure, is an established practice in most large computing centers. T. E. Hull and A. R. Dobell 1 in their paper Random Number Generators, give a relative large number of references to early as well as recent (up to 1962) work on random numbers. J. M. Hammersley and D. C. Handscomb 2 have written a book entitled Monte Carlo Methods, that contains not only a discussion on generating random numbers, but also several applications and an extensive bibliography. References 3, 4, 5, 6, and 7 contain techniques for generating correlated random numbers. In references 6 and 7, techniques are presented for generating, with already available means for generating independent standardized normal ranldom variables, a random (p X 1) vector X, which is distributed according to a multivariate normal distribution with given mean, u and covariance matrix R. In 6 use is made of the Crout factorization, R = CCT, of the covariance matrix R in order to generate a normal vector; while in 7 a method based on conditional distributions is formulated. Reference 6 gives techniques for generating time series from stationary as well as non-stationary normal stochastic processes. Let S = A/N be the maximum likelihood estiinator of a p X p covariance matrix R from a normally distributed sample of N independent (p>X 1) random vectors xi; i==1, 2, * *, N. Thus