Multivariate elliptically contoured stable distributions: theory and estimation

Abstract

Stable distributions with elliptical contours are a class of distributions that are useful for modeling heavy tailed multivariate data. This paper describes the theory of such distributions, presents formulas for calculating their densities, and methods for fitting the data and assessing the fit. Efficient numerical routines are implemented and evaluated in simulations. Applications to data sets of a financial portfolio with 30 assets and to a bivariate radar clutter data set are presented.

Appendix A: Additional facts about the amplitude distribution

There are many other facts about the amplitude density and cdf. Since they are useful in finance applications, in signal processing, and astronomy, we collect them here.

Using the Bergstrom series expansions for stable densities in Eqs. (4) and (5) leads to series expansions for $f_R(r)$ and $F_R(r)$: when $0 < \alpha < 1$

$$\begin{aligned} F_R(r) \!&= \! 1 - {2 \over \pi \alpha \Gamma (d/2)} \sum _{k=1}^\infty {(-1)^{k+1} \Gamma \left(\frac{k \alpha +2}{2}\right) \Gamma \left(\frac{k\alpha +d}{2} \right) \sin \left(\frac{k \alpha \pi }{2}\right) \over k \,k! } \left(\frac{r}{2 \gamma _0}\right)^{-k \alpha }\qquad \end{aligned}$$

(18)

$$\begin{aligned} f_R(r) \!&= \! {1 \over \pi \gamma _0 \Gamma (d/2)} \sum _{k=1}^\infty {(-1)^{k+1} \Gamma \left(\frac{k \alpha +2}{2}\right) \Gamma \left(\frac{k\alpha +d}{2} \right) \sin \left(\frac{k \alpha \pi }{2}\right) \over k! } \left(\frac{r}{2 \gamma _0}\right)^{-k \alpha -1}\qquad \end{aligned}$$

(19)

When $1 < \alpha < 2$,

$$\begin{aligned} F_R(r)&= {4 \over \alpha \Gamma (d/2)} \sum _{k=0}^\infty {(-1)^k \Gamma \left({2 k + d \over \alpha } \right) \over (2k+d)\, k! \, \Gamma \left({ 2 k + d \over 2} \right)} \left( \frac{r}{2 \gamma _0} \right)^{2k+d }\end{aligned}$$

(20)

$$\begin{aligned} f_R(r)&= {2 \over \alpha \gamma _0 \Gamma (d/2)} \sum _{k=0}^\infty {(-1)^k \Gamma \left({2 k + d \over \alpha } \right) \over k! \, \Gamma \left({ 2 k + d \over 2} \right)} \left( \frac{r}{2 \gamma _0} \right)^{2k+d-1 } \end{aligned}$$

(21)

When $\alpha < 1$, (18) and (19) converges absolutely for any $r>0$; when $\alpha > 1$, they are asymptotic series as $r \rightarrow \infty$. Likewise, (20) and (21) are absolutely convergent for $\alpha > 1$ and an asymptotic series for $\alpha < 1$ for $r$ near $0$.

Let $f_d(r)=f_{R,d}(r)$ be the amplitude density and $F_d(r)=F_{R,d}(r)$ be the amplitude d.f. in $d$ dimensions. An argument using (6) and (7) shows

$$\begin{aligned} F_{d+2}(r)=F_d(r)-{r \over d} f_d(r) \quad \text{ and} \quad f_{d+2}(r) = \frac{d-1}{d} f_d(r) - \frac{r}{d} f^\prime _d(r). \end{aligned}$$

(22)

One consequence of the latter expression is that the score function for $R$ can be computed without explicitly differentiating:

$$\begin{aligned} -\frac{d}{dr} \log f_d(r) = - \frac{f_d^{\prime }(r)}{f_d(r)} = \frac{d-1}{r} - \frac{d \, f_{d+2}(r)}{ r f_d(r)}. \end{aligned}$$

When $\alpha =2$, $R^2= X_1^2 + \cdots + X_d^2 = 2 \gamma _0^2 T$, where $T$ is chi-squared with $d$ degrees of freedom. The d.f. and density are $F_R(r)= F_T(r^2/(2 \gamma _0^2))=1-\Gamma (d/2,r^2/(4 \gamma _0^2))/\Gamma (d/2)$ and $f_R(r) = (r/\gamma ^2)$ $f_T(r^2/(2 \gamma _0^2))$. In two dimensions, $R=\sqrt{2} \gamma _0 \sqrt{T}$ is a Rayleigh distribution with density and d.f.

$$\begin{aligned} f_R(r) = {1 \over 2 \gamma _0^2} r e^{-r^2/(4 \gamma _0^2)} \quad \text{ and} \quad F_R(r) =1 - e^{-r^2/(4 \gamma _0^2)}. \end{aligned}$$

(23)

(Note that this is not the customary scaling for the Rayleigh, which is based on $\mathbf{X}\sim N(0,\gamma _0^2 I)$ and has density $r/\gamma _0^2 \exp (-r^2/(2 \gamma _0^2))$ and d.f. $1 - \exp (-r^2/(2 \gamma _0^2))$.)

When $\alpha =1$, the amplitude density and d.f. have explicit formula in all dimensions. The expressions in dimensions 1, 2 and 3 are:

$$\begin{aligned} \begin{array}{lll} d=1&\quad f_R(r) = {2 \over \pi } \gamma _0 / (\gamma _0^2 + r^2)&\quad F_R(r)={2 \over \pi } \arctan (r/\gamma _0) \\ d=2&\quad f_R(r)= \gamma _0 r / (\gamma _0^2+r^2)^{3/2}&\quad F_R(r)=1- \gamma _0 / \sqrt{\gamma _0^2+r^2} \\ d=3&\quad f_R(r)=\displaystyle {{4 \gamma _0 \over 3 \pi } {\gamma _0^2 +2 r^2 \over (\gamma _0^2 + r^2)^2}}&\quad F_R(r)= \displaystyle {{2 \over \pi }\left[ \arctan (r/\gamma _0)- {\gamma _0 r \over 3 ( \gamma _0^2 + r^2)}\right]} \\ \end{array} \end{aligned}$$

Expressions for higher dimensions can be found using the recursion relations (22).

The fractional moments of $R$ can be found using (3): if $-d < p < \alpha$,

$$\begin{aligned} E (R^p)&= E |\mathbf{X}|^p = E(AT)^{p/2} = (EA^{p/2})(ET^{p/2}) \nonumber \\&= (2 \gamma _0)^{p} {\Gamma (1-p/\alpha ) \over \Gamma (1-p/2) } {\Gamma ((d+p)/2) \over \Gamma (d/2)}, \end{aligned}$$

(24)

where the first expectation (which is finite for all for all $p < \alpha$) is from Section 2.1 of Zolotarev (1986); a short calculation is used for the second expectation (which is finite for all $p > -d$). This expression holds for complex $p$ in the strip $-d < \mathfrak R p < \alpha$, giving the Mellin transform of $R$.

The above expression for moments combined with Markov’s inequality gives a uniform upper bound on tail probabilities of $R$ and isotropic $\mathbf{X}$:

$$\begin{aligned} \sup _{r > 0} \, r^p(1-F_R(r)) = \sup _{r >0} \, r^p P(|\mathbf{X}| > r) \le E (R^p), \quad 0 < p < \alpha \end{aligned}$$

(25)

Let $X$ be univariate strictly stable, e.g. $X \sim {{\mathbf{S}(\alpha ,\beta ,\gamma ,0)}}$ with $\alpha \ne 1$ or $X \sim {{\mathbf{S}(1,0 ,\gamma ,0)}}$. Section 3.6 of Zolotarev (1986) shows $\log |X|$ has mean and variance

$$\begin{aligned} E(\log |X|)&= \gamma _{Euler}\left(\frac{1}{\alpha }- 1 \right) + \log \left({\gamma \over (\cos \alpha \theta _0)^{1/\alpha }} \right) \\ \mathrm Var (\log |X|)&= {\pi ^2 (1 + 2/\alpha ^2) \over 12} -\theta _0^2 \end{aligned}$$

where $\gamma _{Euler}\approx 0.57721$ is Euler’s constant and $\theta _0 = \arctan (\beta \tan (\pi \alpha /2))/\alpha$. (Note the constant $\theta _0$ arises in our expression because Zolotarev uses a different parameterization.) The following is a multivariate generalization of this result, it uses the digamma function $\psi (z) = \Gamma \,^\prime (z)/\Gamma (z)$.

Lemma 1

$\log R$ has moment generating function $E \exp ( u \log R ) = E \, R^u$ given by (24) for $-d < u < \alpha$. The mean and variance of $\log R$ are

$$\begin{aligned} E (\log R)&= \log (2 \gamma _0) + \gamma _{Euler}\left(\frac{1}{\alpha }- \frac{1}{2} \right) + \frac{1}{2} \psi (d/2) \\ \mathrm Var (\log R)&= \frac{\pi ^2}{6} \left(\frac{1}{\alpha ^2} - \frac{1}{4} \right) + {1 \over 4} \psi \, ^\prime (d/2). \end{aligned}$$

We will not pursue it here, but there are several ways of estimating $\gamma _0$ and $\alpha$ from amplitude data: (a) maximum likelihood estimation using $f_R(r)$, (b) fractional moment methods using (24), and (c) using the first and second sample moments of $\log R$ and Lemma 1.