dssg {mixSSG}

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

Description

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

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

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, p_1,p_2,\cdots, p_N (for N=3000) are independent realizations following positive stable distribution that are generated using command rpstable(3000, alpha). Otherwise, if d(\boldsymbol{y})> 2L^{\frac{2}{\alpha}}, we have

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 T_{\nu}(x) is distribution function of the Student's t with \nu degrees of freedom, \Phi(x|a,b) is the cumulative density function of normal distribution wih mean a and standard deviation b, and {C_{0}=2 (2\pi)^{-\frac{d+1}{2}}|{\Sigma}|^{-\frac{1}{2}},} d(\boldsymbol{y})=(\boldsymbol{y}-\boldsymbol{\mu})^{'}{{\Omega}^{-1}}(\boldsymbol{y}-\boldsymbol{\mu}), {m}=\boldsymbol{\lambda}^{'}{{\Omega}}^{-1}(\boldsymbol{y}-\boldsymbol{\mu}), {\Omega}={\Sigma}+\boldsymbol{\lambda}\boldsymbol{\lambda}^{'}, {\delta}=1-\boldsymbol{\lambda}^{'}{\Omega}^{-1}\boldsymbol{\lambda}.

Usage

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

Arguments

Y

a vector (or an n\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 n from positive \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)