Patent ID: 7475102

Claim:
A method for generating random numbers on a computer in accordance with multivariate non-normal distributions based on the Yuan and Bentler method I, comprising: fitting, using a computer, n-dimensional multivariate non-normal distributions for n-dimensional empirical distributions; generating, using a computer, the random numbers including pseudo-random numbers by at least one of an additive generator method, an M-sequence method, a generalized feedback shift-register method, and a Mersenne Twister method, and excluding a congruential method, quasi-random numbers, low discrepancy sequences, and physical random numbers; wherein the fitting employs the following formulae (a) and (b) for an application with respect to third and fourth order moments of the empirical distributions: E ⁡ ( vech ⁡ ( XX ′ ) ⁢ X ′ ) = γ ⁢ ⁢ D n + ⁡ ( T ⊗ T ) ⁢ ( ∑ j = 1 m ⁢ ζ j ⁢ E ii ⊗ e i ) ⁢ T ′ ( a ) var ⁡ ( vech ⁡ ( XX ′ ) ) = 2 ⁢ β ⁢ ⁢ D n + ⁡ ( Σ ⊗ Σ ) ⁢ D n + ′ + ⁢ ⁢ ( β - 1 ) ⁢ vech ⁡ ( Σ ) ⁢ vech ′ ⁡ ( Σ ) + ⁢ ⁢ β ⁢ ∑ j = 1 m ⁢ ( κ j - 3 ) ⁢ vech ⁡ ( t j ⁢ t j ′ ) ⁢ vech ′ ⁡ ( t j ⁢ t j ′ ) ( b ) where E (•) is an expectation, vech (•) is a vector consisting of matrix elements not duplicated in symmetrical matrix, D n is an n-order duplication matrix, D n + is the Moore-Penrose generalized inverse matrix of D n , is the Kronecker product, and E ii is e i e i ′ when e i is the ith column unit; wherein the Yuan and Bentler method I uses: independent random variables ξ l , . . . , ξ m that satisfy E (ξ j )=0, E (ξ j 2 )=1, E (ξ j 3 )=ζ j , and E (κ j 4 )=κ j (1≦j≦m) with respect to parameters ξ j and κ j ; a random variable ν independent from ξ j that satisfies E (ν)=0, E (ν 2 )=1, E (ν 3 )=γ, and E (ν 4 )=β with respect to parameters γ and β; a non-random n×m (m≦n) matrix T=(t ij ) of n rank that satisfies TT′=Σ with respect to a matrix Σ=(σ ij ) where the matrix T′ is a transposed matrix of T and where a random vector X=(x i , . . . , x n )′ given by the following expression (C) satisfies Cov (X)=Σ; X=νTξ (c) where Cov (•) is a variance covariance matrix of a vector, and ξ=(ξ 1 , . . . , ξ m )′.