The multivariate stable distribution is a multivariate probability distribution that is a multivariate generalisation of the univariate stable distribution. The multivariate stable distribution defines linear relations between stable distribution marginals. In the same way as for the univariate case, the distribution is defined in terms of its characteristic function.
The multivariate stable distribution can also be thought as an extension of the multivariate normal distribution. It has parameter, α, which is defined over the range 0 < α ≤ 2, and where the case α = 2 is equivalent to the multivariate normal distribution. It has an additional skew parameter that allows for non-symmetric distributions, where the multivariate normal distribution is symmetric.
Let
S
Rd\colonS=\{u\inRd\colon|u|=1\}
X
X\simS(\alpha,Λ,\delta)
X
\operatorname{E}\exp(iuTX)=\exp\left\{-\int\limitss\left\{|uTs|\alpha+i\nu(uTs,\alpha)\right\}Λ(ds)+iuT\delta\right\}
where 0 < α < 2, and for
y\inR
\nu(y,\alpha)=\begin{cases}-sign(y)\tan(\pi\alpha/2)|y|\alpha&\alpha\ne1,\\ (2/\pi)yln|y|&\alpha=1.\end{cases}
This is essentially the result of Feldheim,[2] that any stable random vector can be characterized by a spectral measure
Λ
S
\delta\inRd
Another way to describe a stable random vector is in terms of projections. For any vector
u
uTX
\alpha-
\beta(u)
\gamma(u)
\delta(u)
X\simS(\alpha,\beta( ⋅ ),\gamma( ⋅ ),\delta( ⋅ ))
uTX\sims(\alpha,\beta( ⋅ ),\gamma( ⋅ ),\delta( ⋅ ))
u\inRd
The spectral measure determines the projection parameter functions by:
\gamma(u)= l(\ints|uTs|\alphaΛ(ds)r)1/\alpha
\beta(u)=\ints|uTs|\alphasign(uTs)Λ(ds)/\gamma(u)\alpha
\delta(u)=\begin{cases}uT\delta&\alpha\ne1\\uT\delta-\ints\tfrac{\pi}{2}uTsln|uTs|Λ(ds)&\alpha=1\end{cases}
There are special cases where the multivariate characteristic function takes a simpler form. Define the characteristic function of a stable marginal as
\omega(y|\alpha,\beta)=\begin{cases}|y|\alpha\left[1-i\beta(\tan\tfrac{\pi\alpha}{2})sign(y)\right]&\alpha\ne1\\ |y|\left[1+i\beta\tfrac{2}{\pi}sign(y)ln|y|\right]&\alpha=1\end{cases}
The characteristic function is
E\exp(iuT
\alpha|u| | |
X)=\exp\{-\gamma | |
0 |
\alpha+iuT\delta)\}
\alpha=2
\alpha<2
\Sigma
E\exp(iuTX)=\exp\{-(uT\Sigmau)\alpha/2+iuT\delta)\}
\delta\inRd
\Sigma
E\exp(iuTX)=\exp\{-(uT\Sigmau)+iuT\delta)\}
The marginals are independent with
Xj\simS(\alpha,\betaj,\gammaj,\deltaj)
E\exp(iuTX)=
m | |
\exp\left\{-\sum | |
j=1 |
\omega(uj|\alpha,\betaj)\gamma
\alpha | |
j |
+iuT\delta)\right\}
If the spectral measure is discrete with mass
λj
sj\inS,j=1,\ldots,m
E\exp(iuTX)=
m | |
\exp\left\{-\sum | |
j=1 |
Ts | |
\omega(u | |
j|\alpha,1)λ |
\alpha | |
j |
+iuT\delta)\right\}
If
X\simS(\alpha,\beta( ⋅ ),\gamma( ⋅ ),\delta( ⋅ ))
b\inRm,
\alpha
\gamma(AT ⋅ ),
\beta(AT ⋅ ),
\delta(AT ⋅ )+bT.
Recently[4] it was shown how to compute inference in closed-form in a linear model (or equivalently a factor analysis model), involving independent component models.
More specifically, let
Xi\simS(\alpha,
\beta | |
xi |
,
\gamma | |
xi |
,
\delta | |
xi |
),i=1,\ldots,n
n x n
Yi=
n | |
\sum | |
i=1 |
AijXj
Xi
Yi=S(\alpha,
\beta | |
yi |
,
\gamma | |
yi |
,
\delta | |
yi |
)
Xi
Yi
An application for this construction is multiuser detection with stable, non-Gaussian noise.