Multivariate stable densities and distribution functions: general and elliptical case. John P. Nolan. American University. 8 November 2005. 1 Introduction.
Deutsche Bundesbank’s 2005 Annual Fall Conference
Multivariate stable densities and distribution functions: general and elliptical case John P. Nolan American University 8 November 2005
1 Introduction Stable distributions are a class of probability distributions that generalize the normal law, allowing heavy tails and skewness that make them attractive in modeling financial data. While there are many attractive theoretical properties of stable laws, the use of these models in practice has been restricted by the lack of formulas for stable densities and distribution functions. The univariate stable distributions are now mostly accessible. There are reliable programs to compute stable densities, distribution functions, and quantiles. And there are fast methods to simulate stable r.v. and several methods of estimating stable parameters based on maximum likelihood, quantiles, empirical characteristic functions, and fractional moments. On the other hand, multivariate stable laws are only partially accessible. This is a function of the lack of closed form expressions for densities, and the possible complexity of the dependence structures. In the bivariate case, there are some methods of computing densities and estimating, but these are difficult to implement in higher dimensions. Building on earlier results for general multivariate stable densities, new formulas are presented for multivariate stable distribution functions. Like the density formulas, these expressions are in polar form, and involve the projection parameter functions and integrals involving a new type of special function. 1
This is described in the next section. Section 3 focuses on a computationally tractable case - elliptically contoured stable laws. It is shown that one can compute densities, approximate cumulative probabilities and fit elliptical stable distributions in higher dimensions. The existing software works for dimension d ≤ 40. The remainder of this section introduces notation and background information.
A univariate stable r.v. X with index α, skewness β, scale γ, location δ has characteristic function φ(u) = φ(u|α, β, γ, δ) = E exp(iuZ) = exp(−γ α [|u|α + iβη(u, α)] + iuδ), where 0 < α < 2, −1 ≤ β ≤ 1, γ > 0, δ ∈ R and ½ −(sign u) tan(πα/2)|u|α η(u, α) = (2/π)u ln |u|
α 6= 1 α = 1.
In the notation of Samorodnitsky and Taqqu (1994), this is a Sα (γ, β, δ) distribution. We will use the notation X ∼ S(α, β, γ, δ). Let S be the unit sphere in Rd : S = {u ∈ Rd : |u| = 1}. Feldheim (1937) showed that any stable random vector can be characterized by a spectral measure Λ (a finite measure on S) and a shift vector δ ∈ Rd . We will write X ∼ S(α, Λ, δ) if the joint characteristic function of X is µ Z ¶ α E exp(ihu, Xi) = exp − [|hu, si| + iη(hu, si, α)] Λ(ds) + ihu, δi . S
Another way to describe a stable random vector is in terms of projections. This uses the fact that for any vector u, the projection hu, Xi is univariate α-stable with some skewness β(u), some scale γ(u) and some shift δ(u). We will write X ∼ S(α, β(·), γ(·), δ(·)) if X is stable with hu, Xi ∼ S(α, β(u), γ(u), δ(u)) for every u ∈ Rd . This is called the projection parameterization. The spectral measure determines the projection parameter functions by: µZ ¶1/α α γ(u) = |hu, si| Λ(ds) S R |hu, si|α sign (hu, si)Λ(ds) β(u) = S γ(u)α ½ hu, δi α 6= 1 R δ(u) = 2 hu, δi − π S hu, si ln |hu, si|Λ(ds) α = 1. 2
Conversely, the projection functions β(·), γ(·) and δ(·) determine Λ, but in general there is no simple formula for this. Nolan et al. (2001) gives a way of numerically computing a discrete approximation to the spectral measure using the parameter functions. Both parameterizations are useful. For mathematical simplicity, it is easier to use a description of the distribution that involves a measure on the unit sphere. But it can be difficult to work with an arbitrary measure. The projection parameterization has several advantages. First, it builds on the definition of one dimensional stable distributions to give a quick way of describing all multivariate stable distributions: if φ(u|α, β, γ, δ) is the characteristic function of a one dimensional stable, then E exp(ihu, Xi) = φ(1|α, β(u), γ(u), δ(u)). Second, below it will be used in formulas for multivariate stable densities and cumulative probabilities. Third, Nolan et al. (2001) shows that it gives a useful way to estimate multivariate stable distributions by reducing to one dimensional stable projections. Finally, Nolan (2005a) shows that the projection based description is useful when quantifying the closeness of two multivariate stable laws. While the projection parameterization has some advantages, it is sometimes cumbersome to work simultaneously with the three functions β(·), γ(·) and δ(·).
2 Polar representation for densities and probabilities Expressions for univariate and multivariate stable densities and cumulative probability functions use the following special functions: for integer d > 0 define the functions for v ∈ R, Z ∞ α gd (v|α, β) = cos(vr + βη(r, α))rd−1 e−r dr 0
For a standardized univariate stable law, standard Fourier inversion of the characteristic function shows that the density is f (x|α, β) =
1 g1 (x|α, β). π
The gd functions are used in a similar way to give multivariate stable densities. The following formula for multivariate stable densities from Theorem 1(b) (when α = 1) and Theorem 2 (when α 6= 1) of Abdul-Hamid and Nolan (1998). 3
Theorem 1 Let X ∼ S(α, β(·)γ(·), δ(·)) be a nonsingular stable d-dimensional random vector with density f (x). (a) When α 6= 1, ¯ ¶ µ Z hx, si − δ(s) ¯¯ −d −d f (x) = (2π) gd ¯ α, β(s) γ (s)ds. γ(s) S (b) When α = 1, µ
Z −d
f (x) = (2π)
gd S
¯ ¶ hx, si − δ(s) + π2 β(s)γ(s) ln γ(s) ¯¯ −d ¯ 1, β(s) γ (s)ds. γ(s)
For the multivariate cumulative distribution function, define the additional functions: Gd (v|α, β) = (−1)m (d − 1)!(2π)−d × (1) ( Pm−1 (−1)j+1 P j+1 (−1) 2j+1 v 2j g2j (v|α, β) + m−1 g˜2j+1 (v|α, β) d = 2m j=0 (2j+1)! v j=0 (2j)! Pm−1 (−1)j 2j+1 Pm (−1)j+1 2j g2j+1 (v|α, β) + j=0 (2j)! v g˜2j (v|α, β) d = 2m + 1 j=0 (2j+1)! v where Z
∞
g0 (v|α, β) = Z0 ∞ g˜0 (v|α, β) = Z0 ∞ g˜d (v|α, β) =
cos(vr + βη(r, α)) − cos(βη(r, α)) −rα e dr r sin(vr + βη(r, α)) − sin(βη(r, α)) −rα e dr r α
sin(vr + βη(r, α))rd−1 e−r dr,
d > 0.
0
In the univariate case, the standardized d.f. is given by Z x 1 F (x|α, β) − F (0|α, β) = f (x|α, β) = g˜0 (x|α, β), π 0
x > 0.
We note that there are explicit formulas for F (0|α, β) and that the d.f. formula is naturally in “polar form”, giving the probability of being in the interval (0, x]. 4
Let A be a set in Rd with a polar decomposition, i.e. A ↔ (B, R(·)) where B ⊂ S and R(u) is a nonnegative function defined on B: A = {ru : u ∈ B, 0 ≤ r ≤ R(u)}.
(2)
In words, A is the cone generated by base B and bounded by R(·). The next result is proved in Nolan (2005b). Theorem 2 Let X ∼ S(α, β(·), γ(·), δ(·)) be a nonsingular strictly stable ddimensional random vector and let A be a bounded set given by (2). When (α 6= 1 and δ = 0) or (α = 1 and β(u) = 0 for all u) ¯ ¶ µ Z Z R(u)hu, si ¯¯ α, β(s) hu, si−d ds du. P (X ∈ A) = Gd ¯ γ(s) B S
The above formulas for the density and d.f. depend on the special functions gd and g˜d . It is possible to express all these new functions in terms of the complex valued function S(v|α, β) = g0 (v|α, β) + i˜ g0 (v|α, β). Derivatives of this function can be used to express d dimensional densities because of the recursive relations dgd (v|α, β) = −˜ gd+1 (v|α, β) dv
and
d˜ gd (v|α, β) = gd+1 (v|α, β). dv
Combinations of derivatives are used in expressions for stable cumulative probability functions. We note that these functions are discontinuous in α near α = 1. It is possible to use a continuous parameterization of stable laws to avoid this problem. We have a program to compute g2 (v|α, 0) reliably. The computation of other terms is possible, but involved numerically. Detailed information is given on these functions in Nolan (2005b): recursive relations, values at the origin, tail behavior, series expansions, and Zolotarev type integrals. These results offer more tractable computational expressions for these new functions. It is likely that the general results above will only be practical for a low dimensions. Even if the functions gd (v|α, β) and g˜d (v|α, β) are computed accurately and quickly, the numerical computation of general stable densities and probabilities is 5
challenging for higher dimensions. Theorem 1 requires a (d − 1) dimensional integral for each density value and Theorem 2 requires a 2(d − 1) integral for each set. While this can be done for d = 2, it gets challenging for d = 3, and very difficult in higher dimensions. For portfolio problems involving dozens or hundreds of assets, the general case is likely to remain inaccessible for years to come. In the next section, we focus on a special case that is accessible now.
3 Isotropic and elliptical stable distributions If X is α-stable and radially symmetric or isotropic, then it has characteristic function E exp(ihu, Xi) = exp(−γ0α |u|α + ihu, δi) (3) and projection parameter functions γ(u) = γ0 |u|,
β(u) = 0,
δ(u) = hu, δi.
The spectral measure in this case is a uniform distribution on S. If X is α-stable and elliptically contoured, then it has joint characteristic function E exp(ihu, Xi) = exp(−(uT Σu)α/2 + ihu, δi) (4) and projection parameter functions γ(u) = (uT Σu)1/2 ,
β(u) = 0,
δ(u) = hu, δi
for some positive definite matrix Σ and shift vector δ ∈ Rd . The spectral measure in this case is complicated; see Proposition 2.5.8 of Samorodnitsky and Taqqu (1994). We will assume that X is nonsingular, which is equivalent to Σ being strictly positive definite, i.e. for every u 6= 0, uT Σu > 0. All elliptically contoured stable distributions are scale mixtures of multivariate normal distributions, see Proposition 2.5.2 of Samorodnitsky and Taqqu (1994). Let G ∼ N (0, Σ) be a d-dimensional multivariate normal r. vector and A ∼ S(α/2, 1, γ, 0) be an independent univariate positive (α/2)-stable r. v. with 0 < α < 2. Then X = A1/2 G is α-stable elliptically contoured with joint characteristic function exp(−(γ/2)α/2 (sec πα/4)(uT Σu)α/2 ). For this reason, elliptically contoured stable distributions are called sub-Gaussian stable. 6
This gives a formula for simulating elliptical stable distributions. In particular, if 0 < α < 2, A ∼ S(α/2, 1, 2γ02 (cos πα/4)2/α , 0) and G ∼ N (0, Σ), then X = A1/2 G+δ has characteristic function (4). The radially symmetric cases arise when Σ is a multiple of the identity matrix: if A is as above and G ∼ N (0, I), then X = A1/2 G + δ has characteristic function (3).
3.1
The amplitude distribution
Let X be a centered d-dimensional isotropic stable random vector with characteristic function exp(−γ0α |u|α ). The amplitude of X is defined by q R = |X| = X12 + · · · + Xd2 . Our primary goal is in using the distribution of R to get expressions for the density of multivariate isotropic and elliptical stable distributions. In addition, the amplitude itself arises in some 2-dimensional signal processing problems. So below we derive several expressions for the distribution of the amplitude. In dimension d = 1, isotropic is equivalent to symmetric; the d.f. of R is FR (r) = P (|X| ≤ r) = FX (r) − FX (−r) = 2FX (r) − 1 and the density is fR (r) = 2fX (r). For the rest of this section assume d ≥ 2. d When 0 < α < 2, X=A1/2 Z, where A ∼ S(α/2, 1, 2γ02 (cos πα/4)2/α , 0) is positive stable and Z ∼ N (0, I), A and Z independent. Thus d
R2 =A(Z12 + · · · + Zd2 ) = AT,
(5)
where T is chi-squared with d degrees of freedom, and independent of A. Using the standard formula for products of independent r.v., the d.f. of R can be expressed as Z ∞ 2 FA (r2 /t)fT (t)dt, FR (r) = FR (r|α, γ0 , d) = P (R ≤ r) = P (AT ≤ r ) = 0
(6)
and the density as Z fR (r) = fR (r|α, γ0 , d) =
FR0 (r)
∞
= 2r 0
fA (r2 /t)
fT (t) dt. t
(7)
A scaling argument shows FR (r|α, γ0 , d) = FR (r/γ0 |α, 1, d) and fR (r|α, γ0 , d) = fR (r/γ0 |α, 1, d)/γ0 . Figure 1 shows the graph of the density in two and three dimensions. 7
0.5
d= 2
0.0
0.1
0.2
0.3
0.4
0.8 1 1.2 1.4 1.6 1.8 2
0
2
4
6
8
10
8
10
0.5
d= 3
0.0
0.1
0.2
0.3
0.4
0.8 1 1.2 1.4 1.6 1.8 2
0
2
4
6
Figure 1: The density of the standardized (γ0 = 1) amplitude in 2 dimensions (top) and 3 dimensions (bottom) for α = 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2. Equation (5) gives a way of simulating the amplitude distribution directly, without having to generate multivariate X. It also gives an alternative way of simulating radially symmetric stable random vectors in d dimensions: let A ∼ d√ S(α/2, 1, 2γ02 (cos πα/4)2/α , 0), T ∼ χ2 (d), and s uniform on S, then X= AT s is radially symmetric α-stable with scale γ0 . In particular, in two dimensions, T is exponential and can be generated by −2 log U1 and s = (cos(2πU2 ), sin(2πU2 )), where U1 and U2 are independent U(0,1). There are other expressions for the amplitude distribution. One is a simple
8
change of variables: setting s = r2 /t transforms (6) and (7) to Z ∞ Z ∞ rd 2 2 −2 2 s FA (s)fT (r /s)ds = d/2 FA (s)s−d/2−1 e−r /(2s) ds FR (r) = r 2 Γ(d/2) 0 0 (8) Z ∞ Z ∞ d−1 2r 2 fR (r) = 2r fA (s)s−d/2 e−r /(2s) ds s−1 fA (s)fT (r2 /s)ds = d/2 2 Γ(d/2) 0 0 (9) A third expression is from Zolotarev (1981): Z ∞ 2 α α fR (r) = d/2 (rt)d/2 Jd/2−1 (r t)e−γ0 t dt, (10) 2 Γ(d/2) 0 where Jν (·) is the Bessel function of order ν. The program STABLE evaluates (6) and (7) by numerical integration using existing routines to calculate the univariate stable d.f. FA or the univariate stable density fA . The current program works for α ≥ 0.8 and dimensions 1 ≤ d ≤ 40. The integral in (10) is more difficult to evaluate numerically, because the integrand oscillates infinitely many times, whereas the integrands in (6) and (7) do not. There are many facts about the amplitude density and d.f. We briefly list them here. Let fd (r) = fR,d (r) be the amplitude density and Fd (r) = FR,d (r) be the amplitude density in d dimensions. An argument using (8) and (9) shows r d−1 r Fd+2 (r) = Fd (r) − fd (r) and fd+2 (r) = fd (r) − fd0 (r). d d d 2 2 2 2 When α = 2, R = X1 + · · · + Xd = 2γ0 T , where T is chi-squared with d degrees of freedom. The d.f. and density are FR (r) = FT (r2 /(2γ02 )) = 1 − 2 2 2 2 2 Γ(d/2, √ 0 ))/Γ(d/2) and fR (r) = (r/γ ) fT (r /(2γ0 )). In two dimensions, √r /(4γ R = 2γ0 T is a Rayleigh distribution with density and d.f. 1 2 2 2 2 fR (r) = 2 re−r /(4γ0 ) and FR (r) = 1 − e−r /(4γ0 ) . (11) 2γ0 (Note that this is not the customary scaling for the Rayleigh, which is based on X ∼ N (0, γ02 I) and has density r/γ02 exp(−r2 /(2γ02 )) and d.f. 1−exp(−r2 /(2γ02 )).) When α = 1, the amplitude density and d.f. have explicit formula in all dimensions. The expressions in dimensions 1, 2 and 3 are: d=1 d=2 d=3
FR (r) = π2 arctan(r/γ fR (r) = π2 γ0 /(γ02 + r2 ) p 0) 2 3/2 2 2 2 FR (r) = 1 − fR (r) = γ0 r/(γ0 + r ) · γ0 / γ0 + r ¸ 2 2 4γ0 γ0 + 2r 2 γ0 r fR (r) = FR (r) = arctan(r/γ0 ) − 3π (γ02 + r2 )2 π 3(γ02 + r2 ) 9
Expressions for higher dimensions can be found using the recursion relations above. There are series expansions for fR (r) and FR (r): when 0 < α < 1 ¡ ¢ ¡ kα+d ¢ ¡ kαπ ¢ µ ¶−kα ∞ X (−1)k+1 Γ kα+2 Γ sin 2 r 2 2 2 FR (r) = 1 − (12) παΓ(d/2) k=1 k k! 2γ0 ¢ ¡ kα+d ¢ ¡ ¢µ ¡ ¶−kα−1 ∞ X Γ 2 sin kαπ (−1)k+1 Γ kα+2 1 r 2 2 fR (r) = (13) πγ0 Γ(d/2) k=1 k! 2γ0 When 1 < α < 2,
¡ ¢ µ ¶2k+d ∞ X (−1)k Γ 2k+d 4 r α ¡ ¢ FR (r) = αΓ(d/2) k=0 (2k + d) k! Γ 2k+d 2γ0 2 ¡ ¢ µ ¶2k+d−1 ∞ X (−1)k Γ 2k+d 2 r α ¡ ¢ fR (r) = αγ0 Γ(d/2) k=0 k! Γ 2k+d 2γ0 2
(14) (15)
When α < 1, (12) and (13) converges absolutely for any r > 0; when α > 1, they are asymptotic series as r → ∞. Likewise, (14) and (15) are absolutely convergent for α > 1 and an asymptotic series for α < 1 for r near 0. The fractional moments of R can be found using (5): if −d < p < α, E(Rp ) = E|X|p = E(AT )p/2 = (EAp/2 )(ET p/2 ) Γ(1 − p/α) Γ((d + p)/2) = (2γ0 )p , Γ(1 − p/2) Γ(d/2)
(16)
where the first expectation (which is finite for all for all p < α) is from Zolotarev (1986) and 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 < Re p < α, giving the Mellin transform of R. Let X be univariate strictly stable, e.g. X ∼ S(α, β, γ, 0) with α 6= 1 or X ∼ S(1, 0, γ, 0). Zolotarev (1986) shows log |X| has mean and variance µ ¶ γ E(log |X|) = γEuler (1/α − 1) + log (cos αθ0 )1/α π 2 (1 + 2/α2 ) Var(log |X|) = − θ02 12 where γEuler ≈ 0.57721 is Euler’s constant and θ0 = α−1 arctan(β tan(πα/2)). The following is a multivariate generalization of this result, it uses the digamma function ψ(z) = Γ 0 (z)/Γ(z). 10
Lemma 1 log R has moment generating function E exp(u log R) = E Ru given by (16) for −d < u < α. The mean and variance of log R are µ ¶ 1 1 1 E(log R) = log(2γ0 ) + γEuler − + ψ(d/2) α 2 2 µ ¶ 2 1 1 π 1 + ψ 0 (d/2). Var(log R) = − 2 6 α 4 4 The series expansions for the amplitude d.f. and density show: lim rα (1 − FR (r)) =
r→∞
lim rα P (|X| > r) = k1 γ0α
(17)
r→∞ k2 γ0−d
(18)
lim rα+1 fR (r) = αk1 γ0α
(19)
lim r1−d fR (r) = dk2 γ0−d
(20)
lim r−d FR (r) =
r→0 r→∞
r→0 p
sup r (1 − FR (r)) = sup rp P (|X| > r) ≤ E(Rp ), r>0
0 < p < α (21)
r>0
for positive constants k1 = k1 (α, d) = 2α
sin(πα/2) Γ((α + 2)/2)Γ((α + d)/2) , πα/2 Γ(d/2)
k2 = k2 (α, d) =
4Γ(d/α) . α2d Γ(d/2)2
We note that R is not stable, but (17) shows R is in the domain of attraction of a univariate α-stable law with β = 1. We will not pursue it here, but there are several ways of estimating γ0 and α from amplitude data: (a) maximum likelihood estimation using fR (r), (b) fractional moment methods using (16), and (c) using the first and second moments of log R and Lemma 1.
3.2
Densities of isotropic and elliptically contoured distributions
Let X be any radially symmetric (around 0) r. vector, not necessarily stable, with density fX (x) and amplitude R = |X|. The d.f. of R, FR (r) = P (|X| ≤ r), directly gives circular probabilities. The following argument gives an expression 11
for the density of X in terms of the density of R. Using polar coordinates and radially symmetry for r > 0, Z Z rZ FR (r) = P (|X| ≤ r) = fX (x)dx = fX (us)ud−1 dsdu |x|≤r 0 S Z rZ Z r = fX (u, 0, 0, . . . , 0)ud−1 dsdu = Area(S)fX (u, 0, 0, . . . , 0)ud−1 du. 0
S
0
Diffentiating shows fR (r) = Area(S)fX (r, 0, 0, . . . , 0)rd−1 . Hence for x 6= 0, fX (x) = f (|x|, 0, . . . , 0) =
fR (|x|)|x|1−d Γ(d/2) 1−d = |x| fR (|x|). Area(S) 2π d/2
The key fact here is that calculating the density of multivariate X only requires calculating the univariate function fR (r). Therefore, when X is α-stable with characteristic function (3), the above reasoning shows ¡ ¢ ½ Γ(d/2)/ ¡2π d/2 |x − δ|1−d fR (|x ¢ − δ| |α, γ0 , d) x 6= δ fX (x) = d−1 d/2 2 d Γ(d/α)/ α2 π Γ(d/2) γ0 x = δ. The value at δ uses (20). It is sometimes useful to consider the radial function h(r|α, d) = fX (r, 0, . . . , 0|α, γ0 = 1, δ = 0), which is given by ¡ ¢ ½ Γ(d/2)/ ¡2π d/2 r1−d fR (r|α,¢γ0 = 1, d) r > 0 h(r|α, d) = Γ(d/α)/ α2d−1 π d/2 Γ(d/2)2 r = 0. Then for a general isotropic α-stable X with scale γ0 and location δ, ¯ µ ¶ |x − δ| ¯¯ −d fX (x) = γ0 h α, d . γ0 ¯
(22)
Let Y be d-dimensional α-stable elliptically contoured with scale γ(u) = d (uT Σu)1/2 and shift vector δ. Then Y=A1/2 G + δ, where positive A ∼ S(α/2, 1, 2(cos πα/4)2/α , 0) and G ∼N(0, Σ) as above. It is well known that d G=Σ1/2 G1 , where Σ1/2 is from the Cholesky decomposition of Σ and G1 ∼N(0,I) d has independent standard normal components. Hence Y=A1/2 Σ1/2 G1 + δ = Σ1/2 A1/2 G1 + δ := Σ1/2 X + δ, where X is radially symmetric α-stable. Hence Y is an affine transformation of X, so (22) shows ¯ ´ ³ ¯ −1/2 −1/2 (23) fY (y) = | det Σ| h |Σ (y − δ)| ¯ α, d . 12
3.3
Statistical analysis as elliptical stable
We first describe ways of assessing a d-dimensional data set to see if it is approximately sub-Gaussian and then estimating the parameters of a sub-Gaussian vector. These methods are illustrated using the 30 stocks that make up the Dow Jones index. First perform a one dimensional stable fit to each coordinate of the data using ˆ i = (ˆ one of the univariate estimation methods to get estimates θ αi , βˆi , γˆi , δˆi ). If the αi ’s are significantly different, then the data is not jointly α-stable, so it cannot be sub-Gaussian. Likewise, if the βi ’s are not all close to 0, then the distribution is not symmetric and it cannot be sub-Gaussian. If the αi ’s are all close, form a P pooled estimate of α = ( di=1 αi )/d = average of the indices of each component. Then shift the data by δˆ = (δˆ1 , δˆ2 , . . . , δˆd ) so the distribution is centered at the origin. Next, test for sub-Gaussian behavior. This can be approached by examining two dimensional projections because of the following result. If X is a ddimensional sub-Gaussian α-stable random vector, then every two dimensional projection Y = (Y1 , Y2 ) = (a1 · X, a2 · X) (24) (a1 , a2 ∈ Rd ) is a 2-dimensional sub-Gaussian α-stable random vector. Conversely, suppose X is a d-dimensional α-stable random vector with the property that every two dimensional projection of form (24) is non-singular sub-Gaussian, then d-dimensional X is non-singular sub-Gaussian α-stable. Thus is suffices to assess multivariate data by looking at two dimensional distributions. While one can’t do this for all projections, one can check pairs visually by looking at scatter plots or by plotting the bivariate estimated directional scale function of the projected data and compare it to the scale function of the estimated elliptical fit. Estimating the d(d + 1)/2 parameters (upper triangular part) of R can be done in two ways. For the first method, set rii = γi2 , i.e. the square of the scale parameter of the i-th coordinate. Then estimate rij by analyzing the pair (Xi , Xj ) and take rij = (γ 2 (1, 1) − rii − rjj )/2, where γ(1, 1) is the scale parameter of h(1, 1), (Xi , Xj )i = Xi +Xj . This involves estimating d+d(d−1)/2 = d(d+1)/2 one dimensional scale parameters. For the second method, note that E exp(ihu, Xi) = exp(−(uRuT )α/2 ), so X X [− ln E exp(ihu, Xi)]2/α = uRuT = u2i rii + 2 ui uj rij . i
13
i
