Patent Document ID: 20050114098
Application ID: 10508540
Patent Flag: 0

Claim One:
1. A random number generating method for generating random numbers in accordance with multivariate non-normal distributions based on the Yuan and Bentler method I on a computer, comprising: a step for fitting n-dimensional multivariate non-normal distributions for n-dimensional empirical distributions by using a computer; and a step for generating the random numbers including pseudo-random numbers by means of methods including additive generator method, M-sequence, generalized feedback shift-register method, and Mersenne Twister, and excluding congruential method, quasi-random numbers, low discrepancy sequences, and physical random numbers, with the use of a computer; said fitting step using the following formulae (151) and (152) for an application with respect to the third and fourth order moments of said empirical distributions: E ⁡ ( vech ⁡ ( XX ′ ) ⁢ X ′ ) = γ ⁢ ⁢ D n + ⁡ ( T ⊗ T ) ⁢ ( ∑ j = 1 m ⁢ ζ j ⁢ E ii ⊗ e i ) ⁢ T ′ ⁢ ⁢ and ( 151 ) 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 ′ ) , ( 152 ) where E (•) is an expectation (and so forth), vech (•) is a vector consists of matrix elements being not duplicated in symmetrical matrix, D n is n-order duplication matrix, D n + is the Moore-Penrose generalized inverse matrix of D n , {circle over (×)} is the Kronecker product, and E ii is e i e i ′ when e i is ith column unit; wherein said Yuan and Bentler method I is as follows: independent random variables ξ 1 ,. .. , ξ m satisfy E (ξ j )=0, E (ξ j 2 )=1, E (ξ j 3 )=ζ j , E (ξ j 4 )=κ j (1≦j≦m) with respect to parameters ζ j and κ j ; a random variable ν independent from ξ j satisfies E (ν)=0, E (ν 2 )=1, E (ν 3 )=γ, 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 satisfies TT′=Σ with respect to a matrix Σ=(σ ij ) where the matrix T′ is a transposed matrix of T wherein a random vector X=(x 1 ,. .. , x n )′ given by the following expression (153) satisfies Cov (X)=Σ: 
 X=νTξ, (153) where Cov(•) is a variance covariance matrix of a vector, and ξ=(ξ 1 ,. .. , ξ m )′.