The characteristic function of a stable distribution

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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

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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 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).

Value

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

References

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

Examples

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# 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)