Positive Tempered Stable Distributions and Related Subordinators

Description

Contains methods for the simulation of positive tempered stable distributions and related subordinators. Including classical tempered stable, rapidly deceasing tempered stable, truncated stable, truncated tempered stable, generalized Dickman, truncated gamma, generalized gamma, and p-gamma. For details, see Dassios et al (2019) <doi:10.1017/jpr.2019.6>, Dassios et al (2020) <doi:10.1145/3368088>, Grabchak (2021) <doi:10.1016/j.spl.2020.109015>.

Details

The DESCRIPTION file:

Index of help topics:

SubTS-package           Positive Tempered Stable Distributions and
                        Related Subordinators
dF1                     Pdf for f_1
dF2                     Pdf for f_2
dGGa                    Pdf of the generalized gamma distribution
dSubCTS                 PDF of CTS subordinator
getk1                   Constant K_1
getk2                   Constant K_2
rDickman                Simulation from the generalized Dickman
                        distribution
rF1                     Simulation from f_1
rF2                     Simulation from f_2
rGGa                    Simulates from the generalized gamma
                        distribution
rPGamma                 Simulation from p-gamma distributions.
rPRDTS                  Simulation from p-RDTS distributions.
rSubCTS                 Simulates of CTS subordinators
rTrunGamma              Simulation from the truncated gamma
                        distribution
rTrunS                  Simulation from the truncated stable
                        distribution
rTrunTS                 Simulation from the truncated tempered stable
                        distribution.
simCondS                Simulation from a conditioned stable
                        distribution.
simTandW                Simulation of hitting time and overshoot.

Author(s)

NA

Maintainer: NA

References

A. Dassios, Y. Qu, J.W. Lim (2019). Exact simulation of generalised Vervaat perpetuities. Journal of Applied Probability, 56(1):57-75.

A. Dassios, Y. Qu, J.W. Lim (2020). Exact simulation of a truncated Levy subordinator. ACM Transactions on Modeling and Computer Simulation, 30(10), 17.

M. Grabchak (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham.

M. Grabchak (2021). An exact method for simulating rapidly decreasing tempered stable distributions. Statistics and Probability Letters, 170: Article 109015.

Examples

rPRDTS(20, 2, 1, .7, 2)
rPRDTS(20, 2, 1, 0, 2)
rPRDTS(20, 2, 1, -.7, 2)
rDickman(10, 1)
rTrunGamma(10, 2, 1)
rPGamma(20, 2, 2, 2)
rTrunS(10, 2, .6)
rTrunTS(10, 2, 2, .6)

Pdf for f_1

Description

Evaluates the pdf f_1(x) intruduced in Grabchak (2021).

Usage

dF1(x, a, p)

Arguments

Details

Evaluates the pdf

f_1(x) = exp(-x^p)*x^(-1-a)/K_1, x>1

where K_1 is a normalizing constant. This is distribution is needed to simulate p-RDTS random variables.

Value

Returns a vector of real numbers corresponding to the values of f_1(x).

Author(s)

Michael Grabchak and Lijuan Cao

References

M. Grabchak (2021). An exact method for simulating rapidly decreasing tempered stable distributions. Statistics and Probability Letters, 170: Article 109015.

Examples

  x = (10:20)/10
  dF1(x, .5, 2)

Pdf for f_2

Description

Evaluates the pdf f_2(x) intruduced in Grabchak (2021).

Usage

dF2(x, a, p)

Arguments

Details

Evaluates the pdf

f_2(x) = (exp(-x^p) - exp(-x))*x^(-1-a)/K_2, 0<x<1

where K_2 is a normalizing constant. This distribution is needed to simulate p-RDTS random variables.

Value

Returns a vector of real numbers corresponding to the values of f_2(x).

Author(s)

Michael Grabchak and Lijuan Cao

References

M. Grabchak (2021). An exact method for simulating rapidly decreasing tempered stable distributions. Statistics and Probability Letters, 170: Article 109015.

Examples

  x = (0:10)/10
  dF2(x, .5, 1.5)

Pdf of the generalized gamma distribution

Description

Evaluates the pdf of the generalized gamma distribution.

Usage

dGGa(x, a, p, b)

Arguments

Details

Evaluates the pdf of the generalized gamma distribution with density

g(x) = exp(-b*x^p)*x^(a-1)/K_3, x>0,

where K_3 is a normalizing constant. This distribution is needed to simulate p-RDTS random variables with negative alpha values.

Value

Returns a vector of real numbers corresponding to the values of g(x).

Author(s)

Michael Grabchak and Lijuan Cao

References

M. Grabchak (2021). An exact method for simulating rapidly decreasing tempered stable distributions. Statistics and Probability Letters, 170: Article 109015.

E.W. Stacy (1962) A generalization of the gamma distribution. Annals of Mathematical Statistics, 33(3):1187-1192.

Examples

  x = (0:20)/10
  dGGa(x, 2.5, 1.5, 3.1)

PDF of CTS subordinator

Description

Evaluates the pdf of the classical tempered stable (CTS) subordinator. When alpha=0 this is the pdf of the gamma distribution.

Usage

dSubCTS(x, alpha, c, ell)

Arguments

Details

Returns the pdf of a classical tempered stable subordinator. The distribution has Laplace transform

L(z) = exp( c int_0^infty (e^(-xz)-1)e^(-x/ell) x^(-1-alpha) dx), z>0

and Levy measure

M(dx) = c e^(-x/ell) x^(-1-alpha) 1(x>0)dx.

Value

Returns a vector of real numbers corresponding to the values of pdf.

Note

Uses the method dtweedie in the Tweedie package.

Author(s)

Michael Grabchak and Lijuan Cao

References

M. Grabchak (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham.

Examples

  x = (0:20)/10
  dSubCTS(x, .5, 1, 1.5)

Constant K_1

Description

Evaluates the constant K_1, which is the normalizing constant for f_1.

Usage

getk1(alpha, p)

Arguments

Details

Evaluates

K_1 = int_1^infty exp(-x^p)*x^(-1-alpha) dx.

This is needed to simulate p-RDTS random variables.

Value

Returns a positive real number.

Author(s)

Michael Grabchak and Lijuan Cao

References

M. Grabchak (2021). An exact method for simulating rapidly decreasing tempered stable distributions. Statistics and Probability Letters, 170: Article 109015.

Examples

 getk1(1.5,2.5)

Constant K_2

Description

Evaluates the constant K_2, which is the normalizing constant for f_2.

Usage

getk2(alpha, p)

Arguments

Details

Evaluates

K_2 = int_0^1 ( exp(-x^p) - exp(-x) )*x^(-1-alpha) dx.

This is needed to simulate p-RDTS random variables.

Value

Returns a positive real number.

Author(s)

Michael Grabchak and Lijuan Cao

References

M. Grabchak (2021). An exact method for simulating rapidly decreasing tempered stable distributions. Statistics and Probability Letters, 170: Article 109015.

Examples

 getk2(0.5,2.5)

Simulation from the generalized Dickman distribution

Description

Simulates from the generalized Dickman distribution using Algorithm 3.1 in Dassios, Qu, and Lim (2019).

Usage

rDickman(n, t, b = 1)

Arguments

Details

Simulates from the generalized Dickman distribution by using Algorithm 3.1 in Dassios, Qu, and Lim (2019). This distribution has Laplace transform

L(z) = exp( t int_0^b (e^(-xz)-1) x^(-1) dx), z>0

and Levy measure

M(dx) = t x^(-1) 1(0<x<b) dx.

When b=1 and t=1, this is the Dickman distribution.

Value

Returns a vector of n random numbers.

Author(s)

Michael Grabchak and Lijuan Cao

References

A. Dassios, Y. Qu, J.W. Lim (2019). Exact simulation of generalised Vervaat perpetuities. Journal of Applied Probability, 56(1):57-75.

M. Penrose and A. Wade (2004). Random minimal directed spanning trees and Dickman-type distributions. Advances in Applied Probability, 36(3):691-714.

Examples

rDickman(10, 1)

Simulation from f_1

Description

Simulates from the pdf f_1(x) intruduced in Grabchak (2021).

Usage

rF1(n, a, p)

Arguments

Details

Uses Algorithm 1 in Grabchak (2021) to simulate from the pdf

f_1(x) = exp(-x^p)*x^(-1-a)/K_1, x>1,

where K_1 is a normalizing constant. This is needed to simulate p-RDTS random variables.

Value

Returns a vector of n random numbers.

Author(s)

Michael Grabchak and Lijuan Cao

References

M. Grabchak (2021). An exact method for simulating rapidly decreasing tempered stable distributions. Statistics and Probability Letters, 170: Article 109015.

Examples

rF1(10, .7, 2.5)

Simulation from f_2

Description

Simulates from the pdf f_2(x) intruduced in Grabchak (2021).

Usage

rF2(n, a, p)

Arguments

Details

Uses Algorithm 2 in Grabchak (2021) to simulate from the pdf

f_2(x) = (exp(-x^p) - exp(-x))*x^(-1-a)/K_2, 0<x<1

where K_2 is a normalizing constant. This is needed to simulate p-RDTS random variables.

Value

Returns a vector of n random numbers.

Author(s)

Michael Grabchak and Lijuan Cao

References

M. Grabchak (2021). An exact method for simulating rapidly decreasing tempered stable distributions. Statistics and Probability Letters, 170: Article 109015.

Examples

rF2(10, .7, 2.5)

Simulates from the generalized gamma distribution

Description

Simulates from the generalized gamma distribution.

Usage

rGGa(n, a, p, b)

Arguments

Details

Simulates from the generalized gamma distribution with density

g(x) = exp(-b*x^p)*x^(a-1)/K_3, x>0,

where K_3 is a normalizing constant. The mathodology is explained in Section 4 of Grabchak (2021). This distribution is needed to simulate p-RDTS random variables with negative alpha values.

Value

Returns a vector of n random numbers.

Author(s)

Michael Grabchak and Lijuan Cao

References

M. Grabchak (2021). An exact method for simulating rapidly decreasing tempered stable distributions. Statistics and Probability Letters, 170: Article 109015.

E.W. Stacy (1962) A generalization of the gamma distribution. Annals of Mathematical Statistics, 33(3):1187-1192.

Examples

rGGa(20, .5, 2, 2)

Simulation from p-gamma distributions.

Description

Simulates from p-gamma distributions. These are p-RDTS distributions with alpha=0.

Usage

rPGamma(n, t, mu, p, step = 1)

Arguments

Details

Uses Theorem 1 in Grabchak (2021) to simulate from a p-Gamma distribution. This distribution has Laplace transform

L(z) = exp( t int_0^infty (e^(-xz)-1)e^(-(mu*x)^p) x^(-1) dx ), z>0

and Levy measure

M(dx) = t e^(-(mu*x)^p) x^(-1) 1(x>0)dx.

Value

Returns a vector of n random numbers.

Author(s)

Michael Grabchak and Lijuan Cao

References

M. Grabchak (2019). Rejection sampling for tempered Levy processes. Statistics and Computing, 29(3):549-558

M. Grabchak (2021). An exact method for simulating rapidly decreasing tempered stable distributions. Statistics and Probability Letters, 170: Article 109015.

M. Hofert (2011). Sampling exponentially tilted stable distributions. ACM Transactions on Modeling and Computer Simulation, 22(1), 3.

Examples

rPGamma(20, 2, 2, 2)

Simulation from p-RDTS distributions.

Description

Simulates from p-rapidly decreasing tempered stable (p-RDTS) distributions.

Usage

rPRDTS(n, t, mu, alpha, p, step = 1)

Arguments

Details

Simulates from a p-RDTS distribution. When alpha >=0, uses Theorem 1 in Grabchak (2021) and when alpha<0 uses the method in Section 4 of Grabchak (2021). This distribution has Laplace transform

L(z) = exp( t int_0^infty (e^(-xz)-1)e^(-(mu*x)^p) x^(-1-alpha) dx ), z>0

and Levy measure

M(dx) = t e^(-(mu*x)^p) x^(-1-alpha) 1(x>0)dx.

Value

Returns a vector of n random numbers.

Author(s)

Michael Grabchak and Lijuan Cao

References

M. Grabchak (2019). Rejection sampling for tempered Levy processes. Statistics and Computing, 29(3):549-558

M. Grabchak (2021). An exact method for simulating rapidly decreasing tempered stable distributions. Statistics and Probability Letters, 170: Article 109015.

M. Hofert (2011). Sampling exponentially tilted stable distributions. ACM Transactions on Modeling and Computer Simulation, 22(1), 3.

Examples

rPRDTS(20, 2, 1, .7, 2)
rPRDTS(20, 2, 1, 0, 2)
rPRDTS(20, 2, 1, -.7, 2)

Simulates of CTS subordinators

Description

Simulates from classical tempered stable (CTS) distributions. When alpha=0 this is the gamma distribution.

Usage

rSubCTS(n, alpha, c, ell, method = NULL)

Arguments

Details

Simulates a CTS subordinator. The distribution has Laplace transform

L(z) = exp( c int_0^infty (e^(-xz)-1)e^(-x/ell) x^(-1-alpha) dx), z>0

and Levy measure

M(dx) = c e^(-x/ell) x^(-1-alpha) 1(x>0)dx.

Value

Returns a vector of n random numbers.

Note

Uses the method retstable in the copula package.

Author(s)

Michael Grabchak and Lijuan Cao

References

M. Grabchak (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham.

Examples

rSubCTS(20, .7, 1, 1)

Simulation from the truncated gamma distribution

Description

Simulates from the truncated gamma distribution.

Usage

rTrunGamma(n, t, mu, b = 1, step = 1)

Arguments

Details

Simulates from the truncated gamma distribution. This distribution has Laplace transform

L(z) = exp( t int_0^b (e^(-xz)-1) x^(-1)e^(-mu*x) dx), z>0

and Levy measure

M(dx) = t x^(-1) e^(-mu*x) 1(0<x<b) dx.

The simulation is performed by applying rejection sampling (Algorithm 4.4 in Dassios, Qu, Lim (2020)) to the generalized Dickman distribution. We simulate from the latter using Algorithm 3.1 in Dassios, Qu, Lim (2019).

Value

Returns a vector of n random numbers.

Author(s)

Michael Grabchak and Lijuan Cao

References

A. Dassios, Y. Qu, J.W. Lim (2019). Exact simulation of generalised Vervaat perpetuities. Journal of Applied Probability, 56(1):57-75.

A. Dassios, Y. Qu, J.W. Lim (2020). Exact simulation of a truncated Levy subordinator. ACM Transactions on Modeling and Computer Simulation, 30(10), 17.

M. Grabchak (2019). Rejection sampling for tempered Levy processes. Statistics and Computing, 29(3):549-558

M. Hofert (2011). Sampling exponentially tilted stable distributions. ACM Transactions on Modeling and Computer Simulation, 22(1), 3.

Examples

rTrunGamma(10, 2, 1)

Simulation from the truncated stable distribution

Description

Simulates from the truncated stable distribution.

Usage

rTrunS(n, t, alpha, b = 1, step = 1)

Arguments

Details

Simulates from the truncated stable distribution using Algorithm 4.3 in Dassios, Qu, and Lim (2020). This distribution has Laplace transform

L(z) = exp( t * (alpha/Gamma(1-alpha)) * int_0^b (e^(-xz)-1) x^(-1-alpha) dx), z>0

and Levy measure

M(dx) = t * (alpha/Gamma(1-alpha)) * x^(-1-alpha) 1(0<x<b) dx.

Here Gamma() is the gamma function.

Value

Returns a vector of n random numbers.

Author(s)

Michael Grabchak and Lijuan Cao

References

A. Dassios, Y. Qu, J.W. Lim (2020). Exact simulation of a truncated Levy subordinator. ACM Transactions on Modeling and Computer Simulation, 30(10), 17.

M. Grabchak (2019). Rejection sampling for tempered Levy processes. Statistics and Computing, 29(3):549-558

M. Hofert (2011). Sampling exponentially tilted stable distributions. ACM Transactions on Modeling and Computer Simulation, 22(1), 3.

Examples

rTrunS(10, 2, .6)

Simulation from the truncated tempered stable distribution.

Description

Simulates from the truncated tempered stable distribution.

Usage

rTrunTS(n, t, mu, alpha, b = 1, step = 1)

Arguments

Details

Simulates from the truncated stable distribution using Algorithm 4.3 in Dassios, Qu, and Lim (2020). This distribution has Laplace transform

L(z) = exp( t * (alpha/Gamma(1-alpha)) * int_0^b (e^(-xz)-1) x^(-1-alpha) e^(-mu*x) dx), z>0

and Levy measure

M(dx) = t * (alpha/Gamma(1-alpha)) * x^(-1-alpha) e^(-mu*x) 1(0<x<b) dx.

Here Gamma() is the gamma function.

Value

Returns a vector of n random numbers.

Author(s)

Michael Grabchak and Lijuan Cao

References

A. Dassios, Y. Qu, J.W. Lim (2020). Exact simulation of a truncated Levy subordinator. ACM Transactions on Modeling and Computer Simulation, 30(10), 17.

M. Grabchak (2019). Rejection sampling for tempered Levy processes. Statistics and Computing, 29(3):549-558

M. Hofert (2011). Sampling exponentially tilted stable distributions. ACM Transactions on Modeling and Computer Simulation, 22(1), 3.

Examples

rTrunTS(10, 2, 2, .6)

Simulation from a conditioned stable distribution.

Description

Implements Algorithm 4.2 in Dassios, Qu, and Lim (2020) to simulate from a stable distribution conditioned on an appropriate event.

Usage

simCondS(t, alpha)

Arguments

Details

Implements Algorithm 4.2 in Dassios, Qu, and Lim (2020) to simulate from a stable distribution conditioned on an appropriate event. There are some typos in this algorithm, which are corrected in Grabchak (2021). These random variables are needed to simulate truncated stable, truncated tempered stable, and p-RDTS random variables.

Value

Returns one random number.

Author(s)

Michael Grabchak and Lijuan Cao

References

A. Dassios, Y. Qu, J.W. Lim (2020). Exact simulation of a truncated Levy subordinator. ACM Transactions on Modeling and Computer Simulation, 30(10), 17.

M. Grabchak (2021). An exact method for simulating rapidly decreasing tempered stable distributions. Statistics and Probability Letters, 170: Article 109015.

Examples

simCondS(2, .7)

Simulation of hitting time and overshoot.

Description

Simulates the hitting time T and the overshoot W of a stable process by implimenting Algorithm 4.1 in Dassios, Qu, and Lim (2020). This is important for simulating other distribution.

Usage

simTandW(alpha)

Arguments

Value

Returns one pair of random numbers. The first is T and the second is W.

Author(s)

Michael Grabchak and Lijuan Cao

References

A. Dassios, Y. Qu, J.W. Lim (2020). Exact simulation of a truncated Levy subordinator. ACM Transactions on Modeling and Computer Simulation, 30(10), 17.

Examples

simTandW(.6)