rng | R Documentation |
simulate
function can use the specific random number generators to generate Levy paths.
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)
x |
Number of R.Ns to be geneated. |
a |
parameter |
b |
parameter |
delta |
parameter written as δ below |
gamma |
parameter written as γ below |
mu |
parameter written as μ below |
Lambda |
parameter written as Λ below |
alpha |
parameter written as α below |
lambda |
parameter written as λ below |
sigma |
parameter written as σ below |
beta |
parameter written as β below |
delta.plus |
parameter written as δ_+ below |
gamma.plus |
parameter written as γ_+ below |
delta.minus |
parameter written as δ_- below |
gamma.minus |
parameter written as γ_- below |
GIG
(generalized inverse Gaussian):
The density function of GIG distribution is expressed as:
f(x)= 1/2*(γ/δ)^λ*1/bK_lambda(γ*δ)*x^(λ-1)*exp(-1/2*(δ^2/x+γ^2*x))
where bK_λ() is the modified Bessel function of the third kind with order lambda. The parameters λ, δ and γ vary within the following regions:
δ>=0, γ>0 if λ>0,
δ>0, γ>0 if λ=0,
δ>0, γ>=0 if λ<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 α, β, δ, μ express heaviness of tails, degree of asymmetry, scale and location, respectively. Here the parameter Λ is supposed to be symmetric and positive definite with det(Λ)=1 and the parameters vary within the following region:
δ>=0, α>0, α^2>β^T Λ β if λ>0,
δ>0, α>0, α^2>β^T Λ β if λ=0,
δ>0, α>=0, α^2>=β^T Λ β if λ<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)): Δ and γ are positive (the case of γ=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 α, β, δ and μ express the heaviness of tails, degree of asymmetry, scale and location, respectively. They satisfy the following conditions:
Λ is symmetric and positive definite with det(Λ)=1; δ>0; α>0 with α^2-β^T Λ β >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(Λ)=1; λ>0; α>0 with α^2-β^T Λ β >0. Especially in the case of β=0 it is variance gamma distribution.
bgamma
(bilateral gamma): Bilateral gamma distribution is defined by the difference of independent gamma distributions Gamma(δ_+,γ_+) and Gamma(δ_-,γ_-). Its Levy density f(z) is given by:
f(z)=δ_+/z*exp(-γ_+*z)*ind(z>0)+δ_-/|z|*exp(-γ_-*|z|)*ind(z<0), where the function ind() denotes an indicator function.
stable
(stable): Parameters α, β, σ and γ express stability, degree of skewness, scale and location, respectively. They satisfy the following condition: 0<α<=2; -1<=β<=1; σ>0; γ is a real number.
pts
(positive tempered stable): Positive tempered stable distribution is defined by the tilting of positive stable distribution. The parameters α, a and b express stability, scale and degree of tilting, respectively. They satisfy the following condition: 0<α<1; a>0; b>0. Its Levy density f(z) is given by: f(z)=az^(-1-α)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 α, a, b, β, μ and Λ 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(Λ)=1; 0<α<1; a>0; b>0.
In one-dimensional case, its Levy density f(z) is given by:
f(z)=2a/(2π)^(1/2)*\exp(β*z)*(z^2/(2b+β^2))^(-α/2-1/4)*bK_(α+1/2)(z^2(2b+β^2)^(1/2)).
rXXX |
Collection of of random numbers or vectors |
dXXX |
Density function |
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*Γ(-α)*b^(α))
The YUIMA Project Team
Contacts: Hiroki Masuda hmasuda@ms.u-tokyo.ac.jp and Yuma Uehara y-uehara@kansai-u.ac.jp
## 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.
## 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)
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