dssg {mixSSG}R Documentation

Approximating the density function of skewed sub-Gaussian stable distribution.

Description

Suppose dd-dimensional random vector Y\boldsymbol{Y} follows a skewed sub-Gaussian stable distribution with density function fY(yΘ)f_{\boldsymbol{Y}}(\boldsymbol{y} | \boldsymbol{\Theta}) for Θ=(α,μ,Σ,λ){\boldsymbol{\Theta}}=(\alpha,\boldsymbol{\mu},\Sigma, \boldsymbol{\lambda}) where α\alpha, μ\boldsymbol{\mu}, Σ\Sigma, and λ\boldsymbol{\lambda} are tail thickness, location, dispersion matrix, and skewness parameters, respectively. Herein, we give a good approximation for fY(yΘ)f_{\boldsymbol{Y}}(\boldsymbol{y} | \boldsymbol{\Theta}). First , for N=50{\cal{N}}=50, define

L=Γ(Nα2+1+α2)Γ(d+Nα2+α2)Γ(Nα2+1)Γ(d+Nα2)(N+1). L=\frac{\Gamma(\frac{{\cal{N}}\alpha}{2}+1+\frac{\alpha}{2})\Gamma\bigl(\frac{d+{\cal{N}}\alpha}{2}+\frac{\alpha}{2}\bigr)}{ \Gamma(\frac{{\cal{N}}\alpha}{2}+1)\Gamma\bigl(\frac{d+{\cal{N}}\alpha}{2}\bigr)({\cal{N}}+1)}.

If d(y)2L2αd(\boldsymbol{y})\leq 2L^{\frac{2}{\alpha}}, then

fY(yΘ)C02πδNi=1Nexp{d(y)2pi}Φ(m0,δpi)pid2, f_{\boldsymbol{Y}}(\boldsymbol{y} | {\boldsymbol{\Theta}}) \simeq \frac{{C}_{0}\sqrt{2\pi \delta }}{N} \sum_{i=1}^{N} \exp\Bigl\{-\frac{d(\boldsymbol{y})}{2p_{i}}\Bigr\}\Phi \bigl( m| 0, \sqrt{\delta p_{i}} \bigr)p_{i}^{-\frac{d}{2}},

where, p1,p2,,pNp_1,p_2,\cdots, p_N (for N=3000N=3000) are independent realizations following positive stable distribution that are generated using command rpstable(3000, alpha). Otherwise, if d(y)>2L2αd(\boldsymbol{y})> 2L^{\frac{2}{\alpha}}, we have

fY(yΘ)C0d(y)δπj=1N(1)j1Γ(jα2+1)sin(jπα2)Γ(j+1)[d(y)2]d+1+jα2Γ(d+jα2)Td+jα(md+jαd(y)δ), f_{\boldsymbol{Y}}(\boldsymbol{y} | \boldsymbol{\Theta})\simeq \frac{{C}_{0}\sqrt{d(\boldsymbol{y})\delta}}{\sqrt{\pi}} \sum_{j=1}^{{\cal{N}}}\frac{ (-1)^{j-1}\Gamma(\frac{j\alpha}{2}+1)\sin \bigl(\frac{j\pi \alpha}{2}\bigr)} {\Gamma(j+1)\bigl[\frac{d(\boldsymbol{y})}{2}\bigr]^{\frac{d+1+j\alpha}{2}}}\Gamma\Bigl(\frac{d+j\alpha}{2}\Bigr) T_{d+j\alpha}\biggl(m\sqrt{\frac{d+j\alpha}{d(\boldsymbol{y})\delta}}\biggr),

where Tν(x)T_{\nu}(x) is distribution function of the Student's tt with ν\nu degrees of freedom, Φ(xa,b)\Phi(x|a,b) is the cumulative density function of normal distribution wih mean aa and standard deviation bb, and C0=2(2π)d+12Σ12,{C_{0}=2 (2\pi)^{-\frac{d+1}{2}}|{\Sigma}|^{-\frac{1}{2}},} d(y)=(yμ)Ω1(yμ),d(\boldsymbol{y})=(\boldsymbol{y}-\boldsymbol{\mu})^{'}{{\Omega}^{-1}}(\boldsymbol{y}-\boldsymbol{\mu}), m=λΩ1(yμ),{m}=\boldsymbol{\lambda}^{'}{{\Omega}}^{-1}(\boldsymbol{y}-\boldsymbol{\mu}), Ω=Σ+λλ,{\Omega}={\Sigma}+\boldsymbol{\lambda}\boldsymbol{\lambda}^{'}, δ=1λΩ1λ{\delta}=1-\boldsymbol{\lambda}^{'}{\Omega}^{-1}\boldsymbol{\lambda}.

Usage

dssg(Y, alpha, Mu, Sigma, Lambda)

Arguments

Y

a vector (or an n×dn\times 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 nn from positive α\alpha-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)

[Package mixSSG version 2.1.1 Index]