yuima (version 1.15.27)

rng: Random numbers and densities

Description

simulate function can use the specific random number generators to generate Levy paths.

Usage

rGIG(x,lambda,delta,gamma)
dGIG(x,lambda,delta,gamma)
rGH(x,lambda,alpha,beta,delta,mu,Lambda)
dGH(x,lambda,alpha,beta,delta,mu,Lambda)
rIG(x,delta,gamma)
dIG(x,delta,gamma)
rNIG(x,alpha,beta,delta,mu,Lambda)
dNIG(x,alpha,beta,delta,mu,Lambda)
rvgamma(x,lambda,alpha,beta,mu,Lambda)
dvgamma(x,lambda,alpha,beta,mu,Lambda)
rbgamma(x,delta.plus,gamma.plus,delta.minus,gamma.minus)
dbgamma(x,delta.plus,gamma.plus,delta.minus,gamma.minus)
rstable(x,alpha,beta,sigma,gamma)
rpts(x,alpha,a,b)
rnts(x,alpha,a,b,beta,mu,Lambda)

Value

rXXX

Collection of of random numbers or vectors

dXXX

Density function

Arguments

x

Number of R.Ns to be geneated.

a

parameter

b

parameter

delta

parameter written as \(\delta\) below

gamma

parameter written as \(\gamma\) below

mu

parameter written as \(\mu\) below

Lambda

parameter written as \(\Lambda\) below

alpha

parameter written as \(\alpha\) below

lambda

parameter written as \(\lambda\) below

sigma

parameter written as \(\sigma\) below

beta

parameter written as \(\beta\) below

delta.plus

parameter written as \(\delta_+\) below

gamma.plus

parameter written as \(\gamma_+\) below

delta.minus

parameter written as \(\delta_-\) below

gamma.minus

parameter written as \(\gamma_-\) below

Author

The YUIMA Project Team

Contacts: Hiroki Masuda and Yuma Uehara

Details

GIG (generalized inverse Gaussian): The density function of GIG distribution is expressed as:

\(f(x)= 1/2*(\gamma/\delta)^\lambda*1/bK_\lambda(\gamma*\delta)*x^(\lambda-1)*exp(-1/2*(\delta^2/x+\gamma^2*x))\)

where \(bK_\lambda()\) is the modified Bessel function of the third kind with order lambda. The parameters \(\lambda, \delta\) and \(\gamma\) vary within the following regions:

\(\delta>=0, \gamma>0\) if \(\lambda>0\),

\(\delta>0, \gamma>0\) if \(\lambda=0\),

\(\delta>0, \gamma>=0\) if \(\lambda<0\).

The corresponding Levy measure is given in Eberlein, E., & Hammerstein, E. A. V. (2004) (it contains IG).

GH (generalized hyperbolic): Generalized hyperbolic distribution is defined by the normal mean-variance mixture of generalized inverse Gaussian distribution. The parameters \(\alpha, \beta, \delta, \mu\) express heaviness of tails, degree of asymmetry, scale and location, respectively. Here the parameter \(\Lambda\) is supposed to be symmetric and positive definite with \(det(\Lambda)=1\) and the parameters vary within the following region:

\(\delta>=0, \alpha>0, \alpha^2>\beta^T \Lambda \beta\) if \(\lambda>0\),

\(\delta>0, \alpha>0, \alpha^2>\beta^T \Lambda \beta\) if \(\lambda=0\),

\(\delta>0, \alpha>=0, \alpha^2>=\beta^T \Lambda \beta\) if \(\lambda<0\).

The corresponding Levy measure is given in Eberlein, E., & Hammerstein, E. A. V. (2004) (it contains NIG and vgamma).

IG (inverse Gaussian (the element of GIG)): \(\Delta\) and \(\gamma\) are positive (the case of \(\gamma=0\) corresponds to the positive half stable, provided by the "rstable").

NIG (normal inverse Gaussian (the element of GH)): Normal inverse Gaussian distribution is defined by the normal mean-variance mixuture of inverse Gaussian distribution. The parameters \(\alpha, \beta, \delta\) and \(\mu\) express the heaviness of tails, degree of asymmetry, scale and location, respectively. They satisfy the following conditions: \(\Lambda\) is symmetric and positive definite with \(det(\Lambda)=1; \delta>0; \alpha>0\) with \(\alpha^2-\beta^T \Lambda \beta >0\).

vgamma (variance gamma (the element of GH)): Variance gamma distribution is defined by the normal mean-variance mixture of gamma distribution. The parameters satisfy the following conditions: Lambda is symmetric and positive definite with \(det(\Lambda)=1; \lambda>0; \alpha>0\) with \(\alpha^2-\beta^T \Lambda \beta >0\). Especially in the case of \(\beta=0\) it is variance gamma distribution.

bgamma (bilateral gamma): Bilateral gamma distribution is defined by the difference of independent gamma distributions \(Gamma(\delta_+,\gamma_+) and Gamma(\delta_-,\gamma_-)\). Its Levy density \(f(z)\) is given by: \(f(z)=\delta_+/z*exp(-\gamma_+*z)*ind(z>0)+\delta_-/|z|*exp(-\gamma_-*|z|)*ind(z<0)\), where the function \(ind()\) denotes an indicator function.

stable (stable): Parameters \(\alpha, \beta, \sigma\) and \(\gamma\) express stability, degree of skewness, scale and location, respectively. They satisfy the following condition: \(0<\alpha<=2; -1<=\beta<=1; \sigma>0; \gamma\) is a real number.

pts (positive tempered stable): Positive tempered stable distribution is defined by the tilting of positive stable distribution. The parameters \(\alpha, a\) and \(b\) express stability, scale and degree of tilting, respectively. They satisfy the following condition: \(0<\alpha<1; a>0; b>0\). Its Levy density \(f(z)\) is given by: \(f(z)=az^(-1-\alpha)exp(-bz)\).

nts (normal tempered stable): Normal tempered stable distribution is defined by the normal mean-variance mixture of positive tempered stable distribution. The parameters \(\alpha, a, b, \beta, \mu\) and \(\Lambda\) express stability, scale, degree of tilting, degree of asymmemtry, location and degree of mixture, respectively. They satisfy the following condition: Lambda is symmetric and positive definite with \(det(\Lambda)=1; 0<\alpha<1; a>0; b>0\). In one-dimensional case, its Levy density \(f(z)\) is given by: \(f(z)=2a/(2\pi)^(1/2)*\exp(\beta*z)*(z^2/(2b+\beta^2))^(-\alpha/2-1/4)*bK_(\alpha+1/2)(z^2(2b+\beta^2)^(1/2))\).

References

## rGIG, dGIG, rIG, dIG

Chhikara, R. (1988). The Inverse Gaussian Distribution: Theory: Methodology, and Applications (Vol. 95). CRC Press.

Hormann, W., & Leydold, J. (2014). Generating generalized inverse Gaussian random variates. Statistics and Computing, 24(4), 547-557. tools:::Rd_expr_doi("10.1111/1467-9469.00045")

Jorgensen, B. (2012). Statistical properties of the generalized inverse Gaussian distribution (Vol. 9). Springer Science & Business Media. https://link.springer.com/book/10.1007/978-1-4612-5698-4

Michael, J. R., Schucany, W. R., & Haas, R. W. (1976). Generating random variates using transformations with multiple roots. The American Statistician, 30(2), 88-90. tools:::Rd_expr_doi("10.1080/00031305.1976.10479147")

## rGH, dGH, rNIG, dNIG, rvgamma, dvgamma

Barndorff-Nielsen, O. (1977). Exponentially decreasing distributions for the logarithm of particle size. In Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences (Vol. 353, No. 1674, pp. 401-419). The Royal Society. tools:::Rd_expr_doi("10.1098/rspa.1977.0041")

Barndorff-Nielsen, O. E. (1997). Processes of normal inverse Gaussian type. Finance and stochastics, 2(1), 41-68. tools:::Rd_expr_doi("10.1007/s007800050032")

Eberlein, E. (2001). Application of generalized hyperbolic Levy motions to finance. In Levy processes (pp. 319-336). Birkhauser Boston. tools:::Rd_expr_doi("10.1007/978-1-4612-0197-7_14")

Eberlein, E., & Hammerstein, E. A. V. (2004). Generalized hyperbolic and inverse Gaussian distributions: limiting cases and approximation of processes. In Seminar on stochastic analysis, random fields and applications IV (pp. 221-264). Birkh??user Basel. tools:::Rd_expr_doi("10.1007/978-1-4612-0197-7_14")

Madan, D. B., Carr, P. P., & Chang, E. C. (1998). The variance gamma process and option pricing. European finance review, 2(1), 79-105. tools:::Rd_expr_doi("10.1111/1467-9469.00045")

## rbgamma, dbgamma

Kuchler, U., & Tappe, S. (2008). Bilateral Gamma distributions and processes in financial mathematics. Stochastic Processes and their Applications, 118(2), 261-283. tools:::Rd_expr_doi("10.1016/j.spa.2007.04.006")

Kuchler, U., & Tappe, S. (2008). On the shapes of bilateral Gamma densities. Statistics & Probability Letters, 78(15), 2478-2484. tools:::Rd_expr_doi("10.1016/j.spa.2007.04.006")

## rstable

Chambers, John M., Colin L. Mallows, and B. W. Stuck. (1976) A method for simulating stable random variables, Journal of the american statistical association, 71(354), 340-344. tools:::Rd_expr_doi("10.1080/01621459.1976.10480344")

Weron, Rafal. (1996) On the Chambers-Mallows-Stuck method for simulating skewed stable random variables, Statistics & probability letters, 28.2, 165-171. tools:::Rd_expr_doi("10.1016/0167-7152(95)00113-1")

Weron, Rafal. (2010) Correction to:" On the Chambers-Mallows-Stuck Method for Simulating Skewed Stable Random Variables", No. 20761, University Library of Munich, Germany. https://ideas.repec.org/p/pra/mprapa/20761.html

## rpts

Kawai, R., & Masuda, H. (2011). On simulation of tempered stable random variates. Journal of Computational and Applied Mathematics, 235(8), 2873-2887. tools:::Rd_expr_doi("10.1016/j.cam.2010.12.014")

## rnts

Barndorff-Nielsen, O. E., & Shephard, N. (2001). Normal modified stable processes. Aarhus: MaPhySto, Department of Mathematical Sciences, University of Aarhus.

Examples

if (FALSE) {
set.seed(123)

# Ex 1. (One-dimensional standard Cauchy distribution)
# The value of parameters is alpha=1,beta=0,sigma=1,gamma=0.
# Choose the values of x.
x<-10 # the number of r.n
rstable(x,1,0,1,0)

# Ex 2. (One-dimensional Levy distribution)
# Choose the values of sigma, gamma, x.
# alpha = 0.5, beta=1
x<-10 # the number of r.n
beta <- 1
sigma <- 0.1
gamma <- 0.1
rstable(x,0.5,beta,sigma,gamma)

# Ex 3. (Symmetric bilateral gamma)
# delta=delta.plus=delta.minus, gamma=gamma.plus=gamma.minus.
# Choose the values of delta and gamma and x.
x<-10 # the number of r.n
rbgamma(x,1,1,1,1)

# Ex 4. ((Possibly skewed) variance gamma)
# lambda, alpha, beta, mu
# Choose the values of lambda, alpha, beta, mu and x.
x<-10 # the number of r.n
rvgamma(x,2,1,-0.5,0)

# Ex 5. (One-dimensional normal inverse Gaussian distribution)
# Lambda=1.
# Choose the parameter values and x.
x<-10 # the number of r.n
rNIG(x,1,1,1,1)

# Ex 6. (Multi-dimensional normal inverse Gaussian distribution)
# Choose the parameter values and x.
beta<-c(.5,.5)
mu<-c(0,0)
Lambda<-matrix(c(1,0,0,1),2,2)
x<-10 # the number of r.n
rNIG(x,1,beta,1,mu,Lambda)

# Ex 7. (Positive tempered stable)
# Choose the parameter values and x.
alpha<-0.7
a<-0.2
b<-1
x<-10 # the number of r.n
rpts(x,alpha,a,b)

# Ex 8. (Generarized inverse Gaussian)
# Choose the parameter values and x.
lambda<-0.3
delta<-1
gamma<-0.5
x<-10 # the number of r.n
rGIG(x,lambda,delta,gamma)

# Ex 9. (Multi-variate generalized hyperbolic)
# Choose the parameter values and x.
lambda<-0.4
alpha<-1
beta<-c(0,0.5)
delta<-1
mu<-c(0,0)
Lambda<-matrix(c(1,0,0,1),2,2)
x<-10 # the number of r.n
rGH(x,lambda,alpha,beta,delta,mu,Lambda)
}