The characteristic function of a stable distribution

It computes the theoretical characteristic function of a stable distribution for two different parametrizations. It is used in the vignette to illustrate the estimation of the parameters using GMM.

charStable(theta, tau, pm = 0)

Arguments

theta

Vector of parameters of the stable distribution. See details.

tau

A vector of numbers at which the function is evaluated.

pm

The type of parametization. It takes the values 0 or 1.

Value

It returns a vector of complex numbers with the dimension equals to length(tau).

Details

The function returns the vector \(\Psi(\theta,\tau,pm)\) defined as \(E(e^{ix\tau}\), where \(\tau\) is a vector of real numbers, \(i\) is the imaginary number, \(x\) is a stable random variable with parameters \(\theta\) = \((\alpha,\beta,\gamma,\delta)\) and pm is the type of parametrization. The vector of parameters are the characteristic exponent, the skewness, the scale and the location parameters, respectively. The restrictions on the parameters are: \(\alpha \in (0,2]\), \(\beta\in [-1,1]\) and \(\gamma>0\). For mode details see Nolan(2009).

References

Nolan J. P. (2020), Univariate Stable Distributions - Models for Heavy Tailed Data. Springer Series in Operations Research and Financial Engineering. URL https://edspace.american.edu/jpnolan/stable/.

Examples

# GMM is like GLS for linear models without endogeneity problems

pm <- 0
theta <- c(1.5,.5,1,0) 
tau <- seq(-3, 3, length.out = 20)
char_fct <- charStable(theta, tau, pm)