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)
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. |
The function returns the vector Ψ(θ,τ,pm) defined as E(e^{ixτ}, where τ is a vector of real numbers, i is the imaginary number, x is a stable random variable with parameters θ = (α,β,γ,δ) 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: α \in (0,2], β\in [-1,1] and γ>0. For mode details see Nolan(2009).
It returns a vector of complex numbers with the dimension equals to length(tau).
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/.
# 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)
Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.