dCTS {TempStable}

Density function of the classical tempered stable (CTS) distribution

Description

The probability density function (PDF) of the classical tempered stable distributions is not available in closed form. Relies on fast Fourier transform (FFT) applied to the characteristic function.

Usage

dCTS(
  x,
  alpha = NULL,
  deltap = NULL,
  deltam = NULL,
  lambdap = NULL,
  lambdam = NULL,
  mu = NULL,
  theta = NULL,
  dens_method = "FFT",
  a = -20,
  b = 20,
  nf = 2048,
  ...
)

Arguments

x

A numeric vector of quantiles.

alpha

Stability parameter. A real number between 0 and 2.

deltap

Scale parameter for the right tail. A real number > 0.

deltam

Scale parameter for the left tail. A real number > 0.

lambdap

Tempering parameter for the right tail. A real number > 0.

lambdam

Tempering parameter for the left tail. A real number > 0.

mu

A location parameter, any real number.

theta

Parameters stacked as a vector.

dens_method

Algorithm for numerical evaluation. Choose between "FFT" (default) and "Conv".

a

Starting point of FFT, if dens_method == "FFT". -20 by default.

b

Ending point of FFT, if dens_method == "FFT". 20 by default.

nf

Pieces the transformation is divided in. Limited to power-of-two size. 2048 by default.

...

Possibility to modify charCTS().

Details

theta denotes the parameter vector (alpha, deltap, deltam, lambdap, lambdam, mu). Either provide the parameters individually OR provide theta. Methods include the FFT or alternatively by convolving two totally positively skewed tempered stable distributions, see Massing (2022).

The "FFT" method is automatically selected for Mac users, as the "Conv" method causes problems.

Value

As x is a numeric vector, the return value is also a numeric vector of densities.

References

Massing, T. (2023), 'Parametric Estimation of Tempered Stable Laws'

Examples

x <- seq(0,15,0.25)
y <- dCTS(x,0.6,1,1,1,1,1,NULL,"FFT",-20,20,2048)
plot(x,y)