charStable: 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.
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. (2009), Stable Disttributions.
Math/Stat Department, American University.
URL http://academic2.american.edu/~jpnolan/stable/stable.html.
# 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)