| charStable {gmm} | R Documentation |
The characteristic function of a stable distribution
Description
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.
Usage
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. |
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).
Value
It returns a vector of complex numbers with the dimension equals to length(tau).
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)