rng {yuima}

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)

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

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)).

Value

rXXX

Collection of of random numbers or vectors

dXXX

Density function

Note

Some density-plot functions are still missing: as for the non-Gaussian stable densities, one can use, e.g., stabledist package. The rejection-acceptance method is used for generating pts and nts. It should be noted that its acceptance rate decreases at exponential order as a and b become larger: specifically, the rate is given by exp(a*\Gamma(-\alpha)*b^(\alpha))

Author(s)

The YUIMA Project Team

Contacts: Hiroki Masuda [e-mail] and Yuma Uehara [e-mail]

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. 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. 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. doi:10.1098/rspa.1977.0041

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

Eberlein, E. (2001). Application of generalized hyperbolic Levy motions to finance. In Levy processes (pp. 319-336). Birkhauser Boston. 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. 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. 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. 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. 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. 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. 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. 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

## Not run: 
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)

## End(Not run)