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
\({C_{0}=2 (2\pi)^{-\frac{d+1}{2}}|{\Sigma}|^{-\frac{1}{2}},}\)
\(d(\boldsymbol{y})=(\boldsymbol{y}-\boldsymbol{\mu})^{'}{{\Omega}^{-1}}(\boldsymbol{y}-\boldsymbol{\mu}),\)
\({m}=\boldsymbol{\lambda}^{'}{{\Omega}}^{-1}(\boldsymbol{y}-\boldsymbol{\mu}),\)
\({\Omega}={\Sigma}+\boldsymbol{\lambda}\boldsymbol{\lambda}^{'},\)
\({\delta}=1-\boldsymbol{\lambda}^{'}{\Omega}^{-1}\boldsymbol{\lambda}\).
dssg(Y, alpha, Mu, Sigma, Lambda)simulated realizations of size \(n\) from positive \(\alpha\)-stable distribution.
a vector (or an \(n\times d\) matrix) at which the density function is approximated.
the tail thickness parameter.
a vector giving the location parameter.
a positive definite symmetric matrix specifying the dispersion matrix.
a vector giving the skewness parameter.
Mahdi Teimouri
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)
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