Approximating the density function of skewed sub-Gaussian stable distribution.
Description
Suppose d-dimensional random vector Y follows a skewed sub-Gaussian stable distribution with density function
fY(y∣Θ) for Θ=(α,μ,Σ,λ) where α, μ, Σ, and λ are tail thickness, location, dispersion matrix, and skewness parameters, respectively. Herein, we give a good approximation for fY(y∣Θ). First , for N=50, define
where, p1,p2,⋯,pN (for N=3000) are independent realizations following positive stable distribution that are generated using command rpstable(3000, alpha). Otherwise, if d(y)>2Lα2, we have
where Tν(x) is distribution function of the Student's t with ν degrees of freedom, Φ(x∣a,b) is the cumulative density function of normal distribution wih mean a and standard deviation b, and
C0=2(2π)−2d+1∣Σ∣−21,d(y)=(y−μ)′Ω−1(y−μ),m=λ′Ω−1(y−μ),Ω=Σ+λλ′,δ=1−λ′Ω−1λ.
Usage
dssg(Y, alpha, Mu, Sigma, Lambda)
Arguments
Y
a vector (or an n×d matrix) at which the density function is approximated.
alpha
the tail thickness parameter.
Mu
a vector giving the location parameter.
Sigma
a positive definite symmetric matrix specifying the dispersion matrix.
Lambda
a vector giving the skewness parameter.
Value
simulated realizations of size n from positive α-stable distribution.
Author(s)
Mahdi Teimouri
Examples
n <- 4
alpha <- 1.4
Mu <- rep(0, 2)
Sigma <- diag(2)
Lambda <- rep(2, 2)
Y <- rssg(n, alpha, Mu, Sigma, Lambda)
dssg(Y, alpha, Mu, Sigma, Lambda)