| Title: | Symmetric Tempered Stable Distributions |
|---|---|
| Description: | Contains methods for simulation and for evaluating the pdf, cdf, and quantile functions for symmetric stable, symmetric classical tempered stable, and symmetric power tempered stable distributions. |
| Authors: | Michael Grabchak <[email protected]> and Lijuan Cao <[email protected]> |
| Maintainer: | Michael Grabchak <[email protected]> |
| License: | GPL (>= 3) |
| Version: | 1.0-2 |
| Built: | 2024-09-20 05:51:04 UTC |
| Source: | CRAN |
Contains methods for simulation and for evaluating the pdf, cdf, and quantile functions for symmetric stable, symmetric classical tempered stable, and symmetric power tempered stable distributions.
The DESCRIPTION file:
| Package: | SymTS |
| Type: | Package |
| Title: | Symmetric Tempered Stable Distributions |
| Version: | 1.0-2 |
| Date: | 2023-01-14 |
| Author: | Michael Grabchak <[email protected]> and Lijuan Cao <[email protected]> |
| Maintainer: | Michael Grabchak <[email protected]> |
| Description: | Contains methods for simulation and for evaluating the pdf, cdf, and quantile functions for symmetric stable, symmetric classical tempered stable, and symmetric power tempered stable distributions. |
| License: | GPL (>= 3) |
| NeedsCompilation: | yes |
| Packaged: | 2023-01-15 00:42:22 UTC; lcao2 |
| Repository: | CRAN |
| Date/Publication: | 2023-01-15 01:00:02 UTC |
Index of help topics:
SymTS-package Symmetric Tempered Stable Distributions
dCTS PDF of CTS Distribution
dPowTS PDF of PowTS Distribution
dSaS PDF of Symmetric Stable Distribution
pCTS CDF of CTS Distribution
pPowTS PDF of PowTS Distribution
pSaS CDF of Symmetric Stable Distribution
qCTS Quantile Function of CTS Distribution
qPowTS Quantile Function of PowTS Distribution
qSaS Quantile Function of Symmetric Stable
Distribution
rCTS Simulation from CTS Distribution
rPowTS Simulation from PowTS Distribution
rSaS Simulation from Symmetric Stable Distribution
Michael Grabchak <[email protected]> and Lijuan Cao <[email protected]>
Maintainer: Michael Grabchak <[email protected]>
M. Grabchak (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham.
S. T. Rachev, Y. S. Kim, M. L. Bianchi, and F. J. Fabozzi (2011). Financial Models with Levy Processes and Volatility Clustering. Wiley, Chichester.
J. Rosinski (2007). Tempering stable processes. Stochastic Processes and Their Applications, 117(6):677-707.
G. Samorodnitsky and M. Taqqu (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, Boca Raton.
Evaluates the pdf for the symmetric classical tempered stable distribution. When alpha=0 this is the symmetric variance gamma distribution.
dCTS(x, alpha, c = 1, ell = 1, mu = 0)dCTS(x, alpha, c = 1, ell = 1, mu = 0)
x |
Vector of points. |
alpha |
Number in [0,2) |
c |
Parameter c >0 |
ell |
Parameter ell>0 |
mu |
Location parameter, any real number |
The integration is preformed using the QAWF method in the GSL library for C. For this distribution the Rosinski measure R(dx) = c*delta_ell(dx) + c*delta_(-ell)(dx), where delta is the delta function. The Levy measure is M(dx) = c*ell^(alpha) *e^(-x/ell)*x^(-1-alpha) dx. The characteristic function is, for alpha not equal 0,1:
f(t) = exp( 2*c*gamma(-alpha)*(1+ell^2 t^2)^(alpha/2)*(cos(alpha*atan(ell*t))-1)) *e^(i*t*mu),
for alpha = 1 it is
f(t) = (1+ell^2 t^2)^c*exp(-2*c*ell*t*atan(ell*t)) *e^(i*t*mu),
and for alpha=0 it is
f(t) = (1+t^2 ell^2)^(-c) *e^(i*t*mu).
When alpha=0 and c<=.5, the pdf is unbounded. It is infinite at mu and the method returns Inf in that case. This does not affect pCTS, qCTS, or rCTS.
Michael Grabchak and Lijuan Cao
M. Grabchak (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham.
x = (-10:10)/10 dCTS(x,.5)x = (-10:10)/10 dCTS(x,.5)
Evaluates the pdf for the symmetric power tempered stable distribution.
dPowTS(x, alpha, c = 1, ell = 1, mu = 0)dPowTS(x, alpha, c = 1, ell = 1, mu = 0)
x |
Vector of points |
alpha |
Number in [0,2) |
c |
Parameter c >0 |
ell |
Parameter ell>0 |
mu |
Location parameter, any real number |
The integration is preformed using the QAWF method in the GSL library for C. For this distribution the Rosinski measure R(dx) = c*(alpha+ell+1)*(alpha+ell)*(1+|x|)^(-2-alpha-ell)(dx).
We do not allow for the case alpha=0 and c<=.5*(1+ell), as, in this case, the pdf is unbounded. This does not affect pPowTS, qPowTS, or rPowTS.
Michael Grabchak and Lijuan Cao
M. Grabchak (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham.
x = (-10:10)/10 dPowTS(x,.5)x = (-10:10)/10 dPowTS(x,.5)
Evaluates the pdf for the symmetric alpha stable distribution. For alpha=1 this is the Cauchy distribution.
dSaS(x, alpha, c = 1, mu = 0)dSaS(x, alpha, c = 1, mu = 0)
x |
Vector of points. |
alpha |
Index of stability; Number in (0,2) |
c |
Scale parameter, c>0 |
mu |
Location parameter, any real number |
The integration is preformed using the QAWF method in the GSL library for C. The characteristic function is
f(t) = e^(-c |t|^alpha) *e^(i*t*mu).
Michael Grabchak and Lijuan Cao
G. Samorodnitsky and M. Taqqu (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, Boca Raton.
x = (-10:10)/10 dSaS(x,.5)x = (-10:10)/10 dSaS(x,.5)
Evaluates the cdf for the symmetric classical tempered stable distribution. When alpha=0 this is the symmetric variance gamma distribution.
pCTS(x, alpha, c = 1, ell = 1, mu = 0)pCTS(x, alpha, c = 1, ell = 1, mu = 0)
x |
Vector of probabilities. |
alpha |
Number in [0,2) |
c |
Parameter c >0 |
ell |
Parameter ell>0 |
mu |
Location parameter, any real number |
For details about this distribution see the the describtion of dCTS.
Michael Grabchak and Lijuan Cao
M. Grabchak (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham.
x = (-10:10)/10 pCTS(x,.5)x = (-10:10)/10 pCTS(x,.5)
Evaluates the cdf for the symmetric power tempered stable distribution.
pPowTS(x, alpha, c = 1, ell = 1, mu = 0)pPowTS(x, alpha, c = 1, ell = 1, mu = 0)
x |
Vector of probabilities. |
alpha |
Number in [0,2) |
c |
Parameter c >0 |
ell |
Parameter ell>0 |
mu |
Location parameter, any real number |
The integration is preformed using the QAWF method in the GSL library for C. For this distribution the Rosinski measure R(dx) = c*(alpha+ell+1)*(alpha+ell)*(1+|x|)^(-2-alpha-ell)(dx).
Michael Grabchak and Lijuan Cao
M. Grabchak (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham.
x = (-10:10)/10 pPowTS(x,.5)x = (-10:10)/10 pPowTS(x,.5)
Evaluates the cdf for the symmetric alpha stable distribution. For alpha=1 this is the Cauchy distribution.
pSaS(x, alpha, c = 1, mu = 0)pSaS(x, alpha, c = 1, mu = 0)
x |
Vector of probabilities. |
alpha |
Index of stability; Number in (0,2) |
c |
Scale parameter, c>0 |
mu |
Location parameter, any real number |
The integration is preformed using the QAWF method in the GSL library for C. The characteristic function is
f(t) = e^(-c |t|^alpha) *e^(i*t*mu).
Michael Grabchak and Lijuan Cao
G. Samorodnitsky and M. Taqqu (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, Boca Raton.
x = (-10:10)/10 pSaS(x,.5)x = (-10:10)/10 pSaS(x,.5)
Evaluates the quantile function for the symmetric classical tempered stable distribution. When alpha=0 this is the symmetric variance gamma distribution.
qCTS(x, alpha, c = 1, ell = 1, mu = 0)qCTS(x, alpha, c = 1, ell = 1, mu = 0)
x |
Vector of quantiles. |
alpha |
Number in [0,2) |
c |
Parameter c >0 |
ell |
Parameter ell>0 |
mu |
Location parameter, any real number |
For details about this distribution see the the describtion of dCTS.
Michael Grabchak and Lijuan Cao
M. Grabchak (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham.
x = (1:9)/10 qCTS(x,.5)x = (1:9)/10 qCTS(x,.5)
Evaluates the quantile function for the symmetric power tempered stable distribution.
qPowTS(x, alpha, c = 1, ell = 1, mu = 0)qPowTS(x, alpha, c = 1, ell = 1, mu = 0)
x |
Vector of quantiles. |
alpha |
Number in [0,2) |
c |
Parameter c >0 |
ell |
Parameter ell>0 |
mu |
Location parameter, any real number |
Michael Grabchak and Lijuan Cao
M. Grabchak (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham.
x = (1:9)/10 qPowTS(x,.5)x = (1:9)/10 qPowTS(x,.5)
Evaluates the quantile function for the symmetric alpha stable distribution. For alpha=1 this is the Cauchy distribution.
qSaS(x, alpha, c = 1, mu = 0)qSaS(x, alpha, c = 1, mu = 0)
x |
Vector of points. |
alpha |
Index of stability; Number in (0,2) |
c |
Scale parameter, c>0 |
mu |
Location parameter, any real number |
The characteristic function is
f(t) = e^(-c |t|^alpha) *e^(i*t*mu).
Michael Grabchak and Lijuan Cao
G. Samorodnitsky and M. Taqqu (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, Boca Raton.
x = (1:9)/10 qSaS(x,.5)x = (1:9)/10 qSaS(x,.5)
Simulates from the symmetric classical tempered stable distribution. When alpha=0 this is the symmetric variance gamma distribution. The simulation is performed by numerically evaluating the quantile function.
rCTS(r, alpha, c = 1, ell = 1, mu = 0)rCTS(r, alpha, c = 1, ell = 1, mu = 0)
r |
Number of observations. |
alpha |
Number in [0,2) |
c |
Parameter c >0 |
ell |
Parameter ell>0 |
mu |
Location parameter, any real number |
For details about this distribution see the the describtion of dCTS.
Michael Grabchak and Lijuan Cao
M. Grabchak (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham.
rCTS(10,.5)rCTS(10,.5)
Simulates from the symmetric power tempered stable distribution. The simulation is performed by numerically evaluating the quantile function.
rPowTS(r, alpha, c = 1, ell = 1, mu = 0)rPowTS(r, alpha, c = 1, ell = 1, mu = 0)
r |
Number of observations. |
alpha |
Number in [0,2) |
c |
Parameter c >0 |
ell |
Parameter ell>0 |
mu |
Location parameter, any real number |
For this distribution the Rosinski measure R(dx) = c*(alpha+ell+1)*(alpha+ell)*(1+|x|)^(-2-alpha-ell)(dx).
Michael Grabchak and Lijuan Cao
M. Grabchak (2016). Tempered Stable Distributions: Stochastic Models for Multiscale Processes. Springer, Cham.
pPowTS(10,.5)pPowTS(10,.5)
Simulates from the symmetric alpha stable distribution. When alpha=1 this is the Cauchy distribution. The simulation is performed using a well-known approah. See for instance Proposition 1.7.1 in Samorodnitsky and Taqqu (1994).
rSaS(r, alpha, c = 1, mu = 0)rSaS(r, alpha, c = 1, mu = 0)
r |
Number of observations. |
alpha |
Index of stability; Number in (0,2) |
c |
Scale parameter, c>0 |
mu |
Location parameter, any real number |
The characteristic function is
f(t) = e^(-c |t|^alpha)*e^(i*t*mu).
Michael Grabchak and Lijuan Cao
G. Samorodnitsky and M. Taqqu (1994). Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance. Chapman & Hall, Boca Raton.
rSaS(10,.5)rSaS(10,.5)