dCTS {SymTS}

PDF of CTS Distribution

Description

Evaluates the pdf for the symmetric classical tempered stable distribution. When alpha=0 this is the symmetric variance gamma distribution.

Usage

dCTS(x, alpha, c = 1, ell = 1, mu = 0)

Arguments

x

Vector of points.

alpha

Number in [0,2)

c

Parameter c >0

ell

Parameter ell>0

mu

Location parameter, any real number

Details

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).

Note

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.

Author(s)

Michael Grabchak and Lijuan Cao

References

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

Examples

x = (-10:10)/10
dCTS(x,.5)