A Study On The NORTA Approach With Finite Difference Stochastic Approximation Algorithm

The NORTA(NORmal To Anything) approach is a very important method on generating n-dimensional random vectors with given marginal distributions and correlation matrix. Suppose given cumulative distribution functions Fi, where i = 1, · · ·, n,of random vector-→X =(X1, X2, · · ·, Xn) from which observations are expected and correlation matrix(ρij)n i,j=1, the key problem of the NORTA approach is how to ?nd a multidimensional standard normal random vector correlation matrix(ρ?ij)ni,j=1to ensure the generated random vector has the given correlation matrix(ρij)n i,j=1. The current solutions are fall into three categories:analytical, numerical and simulation. Compared with complexity of the analytical method and defective of the current numerical methods, simulation solutions may be more effective. From the simulation opinions,the problem is a SRFP(Stochastic Root Finding Problem), so the paper studied the FDSA(Finite Difference Stochastic Approximation) algorithm in SA(Stochastic Approximation) algorithm family which are used to solve SRFP. Considering SRFP in the NORTA approach, this paper uses the mean square error of the estimators of ρij where 1 ≤ i ≤ j ≤ n as the loss function L(θ), studied the parameters’ selection problem. Simulation experiments veri?ed the effectiveness of the algorithm while a computer program for the algorithm is also given in the Appendix. In addition, the paper brie?y described some background knowledge for generating random vectors and random vectors’ correlations, gave an current research overview on generating correlated random vectors, and also analyzed other algorithms for the NORTA approach and other methods inherited from it.