AN ASYMMETRIC MULTIVARIATE LAPLACE DISTRIBUTION ´ SAMUEL KOTZ, TOMASZ J. KOZUBOWSKI, AND KRZYSZTOF PODGORSKI Abstract. We present a class of multivariate laws which is an extension of the symmetric multivariate Laplace distributions and of the univariate asymmetric Laplace distributions. The extension retains natural, asymmetric and multivariate, properties characterizing these two subclasses. The results include characterizations, mixture representations, formulas for densities and moments, and a simulation algorithm. The new family can be viewed as a subclass of hyperbolic distributions.
1. Introduction The classical univariate Laplace distribution with mean zero and variance σ 2 originally introduced by Laplace (1774) is one of the basic symmetric distribution given by the characteristic function (1)
φ(t) =
1 1+
σ 2 t2 2
or, equivalently, by the density √ (2)
f (x) =
2 −√2|x|/σ , x ∈ R, σ > 0. e 2σ
This distribution is often used for modeling phenomena with “heavier than normal tails” , see, e.g., Andrews et al. (1972), Easterling (1978), Hsu (1979), Okubo and Narita (1980), Bagchi et al. (1983), Hoaglin et al. (1983), Dadi and Marks (1987), Damsleth and El-Shaarawi (1989), and also Johnson et al. (1995) which contains a detailed list of references. In this paper we study a class of distributions which we shall call multivariate asymmetric Laplace laws. Date: January 17, 2003. Key words and phrases. Geometric compound; heavy-tail modeling; Laplace distribution; mixture; random summation; simulation. 1
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´ S. KOTZ, T. J. KOZUBOWSKI, AND K. PODGORSKI
Definition 1.1. A random vector Y in Rd has a multivariate asymmetric Laplace distribution (AL) if its ch.f. is given by (3)
Ψ(t) =
1 1 0 2 t Σt
1+
− im0 t
, t ∈ Rd ,
where m ∈ Rd , m 6= 0 and Σ is a d × d non-negative definite symmetric matrix. We shall denote this distribution by ALd (m, Σ) and write Y ∼ ALd (m, Σ). If the matrix Σ is positive-definite, the distribution is then truly d-dimensional and possesses a probability density function. If m = 0, we shall refer to the ALd (0, Σ) distribution as a symmetric multivariate Laplace law. These multivariate distributions share many properties with the univariate Laplace distribution and their marginal distributions are of the same type. One-dimensional marginal distributions are the asymmetric Laplace distributions with the ch.f. (4)
φ(t) =
1 1+
σ 2 t2 2
− iµt
,
which were introduced in Hinkley and Revankar (1977) and studied more recently in Kozubowski and Podg´orski (1999ab, 2000). This class of univariate laws arises quite naturally in various applications and has been used for modeling data exhibiting asymmetry and heavy tails [i.e. financial data, see, e.g., Madan and Seneta (1990), Madan et al. (1998)]. We note that the term multivariate Laplace law is somewhat ambiguous since it has been used for at least the following classes of multivariate distributions: • A multivariate distribution generated by a vector of i.i.d. univariate Laplace variables [Marshall and Olkin (1993), Kalashnikov (1997), Fern´andez et al. (1995)]; • A bivariate distribution with Laplace marginals introduced in Ulrich and Chen (1987); • An elliptically contoured distribution given by the ch.f. (5)
Φ(t) =
1
1+
, 1 0 2 t Σt
t ∈ Rd ,
see, e.g., McGraw and Wagner (1968), Pillai (1985), Johnson (1987), Anderson (1992), and Kotz et al. (2000). Equation (5) is simply a generalization of (1) to d-dimensions.
MULTIVARIATE LAPLACE DISTRIBUTIONS
3
• A special case (λ = 1) of the multivariate exponential power distribution with the density (6)
0
f (x) = Ce−[(x−m) Σ
−1
(x−m)]λ/2
, x ∈ Rd ,
see Fern´andez et al. (1995), Ernst (1998), Haro-Lop´ez and Smith (1999). Here, the univariate Laplace density is generalized to d-dimensions. • A multivariate distribution with the density (7)
f (x) = CK0 (||x||/β), x ∈ Rd , see Fang et al. (1990). Here, K0 is the modified Bessel function of the 3rd kind and order zero (see the Appendix). In case d = 2 (and only in this case) the ch.f. of this distribution is of the form (5). The class of asymmetric multivariate probability distributions on Rd given by the ch.f.
(3) includes the multivariate elliptically contoured Laplace distributions (5) and the univariate asymmetric Laplace distribution (4). We shall use the term asymmetric Laplace (AL) law for the members of this class. Their significance is due to the fact that they are the only distributional limits for (appropriately normalized) random sums of i.i.d. random vectors (r.v.’s) with finite second moments, (8)
X(1) + · · · + X(νp ) ,
where νp has a geometric distribution with mean 1/p (independent of the X(i) ’s): P (νp = k) = p(1 − p)k−1 , k = 1, 2, . . . , and p converges to zero. Thus, the multivariate AL laws arise quite naturally via a specific model. Since the sums such as (8) frequently appear in many applied problems in various areas, including biology, economics, insurance mathematics, reliability, and most recently in mathematical finance [see e.g. in Kalashnikov (1997) and references therein], AL distributions should have a wide variety of applications. In particular, this class seems to be appropriate for modeling heavy tailed asymmetric multivariate data while retaining finiteness of moments – in contrast with multivariate stable distributions. In this paper we shall mostly concentrate on the theory of AL laws leaving the discussion of numerous applications and statistical inference for a future work.
´ S. KOTZ, T. J. KOZUBOWSKI, AND K. PODGORSKI
4
Formally, the AL laws form a sub-class of the geometric stable distributions. The geometric stable laws approximate geometric random sums (8) with arbitrary components, including those with infinite means [see Kozubowski and Rachev (1999ab)]. The geometric stable distributions, similarly to stable laws, have the tail behavior governed by the index of stability α ∈ (0, 2]. The AL distributions correspond to the geometric stable subclass with α = 2 [see Rachev and SenGupta (1992)]. Thus, they play an analogous role among the geometric stable laws as Gaussian distributions do among the stable laws. They also possess finite moments of all orders, and the theory is quite straightforward. However, in spite of finiteness of moments, their tails are substantially longer than those of the Gaussian laws; this, coupled with possible asymmetry, renders them to be more flexible for modeling heavy tailed asymmetric data. As mentioned in Section 2, the multivariate AL laws can be obtained as a limiting case of the generalized hyperbolic distributions, introduced by Barndorff-Nielsen (1977). Thus, some properties of AL laws can be deduced from the corresponding properties of the generalized hyperbolic laws by passing to the limit. However, direct proofs for AL laws are often simpler than for their “hyperbolic” counterparts and some of the properties are specific solely to AL laws (such as convolution properties in relation to a random summation model). Consequently, for our purposes the relation to the generalized hyperbolic laws is not of substantial significance. 2. Representations A representation of geometric stable laws discussed in Kozubowski and Panorska (1999) leads to the representation of multivariate AL laws as a location-scale mixture of multivariate normal laws. This representation allows for a simple random number generator of AL laws described below in Remark 2.3. We shall use the common notation Nd (m, Σ) to denote a ddimensional normal distribution with the mean vector m and the variance-covariance matrix Σ. Theorem 2.1. Let Y ∼ ALd (m, Σ) and let X ∼ Nd (0, Σ). Let Z be an exponentially distributed r.v. with mean 1, independent of X. Then, the following representation is valid: (9)
d
Y = mZ + Z 1/2 X.
MULTIVARIATE LAPLACE DISTRIBUTIONS
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Remark 2.1. More general normal mixtures with generalized inverse Gaussian (rather than exponential) variable Z were considered by Barndorff-Nielsen (1977). The density of a generalized inverse Gaussian distribution with parameters (λ, χ, ψ) [the GIG(λ, χ, ψ) distribution] is (10)
p(x) =
(ψ/χ)λ/2 λ−1 − 1 (χ/x+ψx) √ , x > 0, x e 2 2Kλ ( χψ)
where Kλ is the modified Bessel function of the third kind (see Appendix). The ranges of the parameters are: χ ≥ 0, ψ > 0, λ > 0; χ > 0, ψ > 0, λ = 0; χ > 0, ψ ≥ 0, λ < 0. Barndorff-Nielsen (1977) considers mixtures of the form d
Y = µ + mZ + Z 1/2 X,
(11)
where X is as above, m = Σβ for some d-dimensional vector β, and Z ∼ GIG(λ, χ, ψ). Using the notation χ = δ 2 , ψ = ξ 2 , and α2 = ξ 2 +β 0 Σβ, we have Y to be a d-dimensional generalized hyperbolic variable with index λ, whose distribution is denoted by Hd (λ, α, β, δ, µ, Σ). Taking the limiting case GIG(1, 0, 2) as the mixing distribution (which is a standard exponential) p and setting Σβ = m and µ = 0, so that δ 2 = 0, ξ 2 = 2, and α = 2 + m0 Σ−1 m, we arrive at the mixture Zm + Z 1/2 X, which is multivariate AL. Remark 2.2. By Theorem 2.1, each component Yi of an AL r.v. Y admits the representation d
Yi = mi Z + Z 1/2 σii Xi ,
(12)
where Xi is standard normal variable. We thus have EYi = mi and E(Y) = m. Furthermore, the covariance matrix of Y is Cov(Y) = Σ + mm0 . Indeed, since E(Xi Xj ) = σij and EZ 2 = 2, it follows that E(Yi Yj ) = E[(mi Z + Z 1/2 Xi )(mj + Z 1/2 Xj )] = 2mi mj + σij and Cov(Yi , Yj ) = E(Yi Yj ) − E(Yi )E(Yj ) = 2mi mj + σij − mi mj = mi mj + σij . Remark 2.3. The problem of random number generation for symmetric Laplace laws was originally posed in Devroye (1986) and reiterated in Johnson (1987): “...variate generation
´ S. KOTZ, T. J. KOZUBOWSKI, AND K. PODGORSKI
6
has not been explicitly worked out for [the bivariate Laplace and generalized Laplace distributions] in the literature.” However, simulation of generalized hyperbolic random variables, based on their normal mixture representations, was studied earlier by Atkinson (1982). Thus the problem of simulation for multivariate AL distributions is implicitly resolved. Here we present an explicit algorithm. We shall use the representation (9). The approach is quite straightforward as both exponential and multivariate normal variates are relatively easy to generate and appropriate procedures are by now implemented in many standard statistical packages. An ALd (m, Σ) generator • Generate a standard exponential variate Z. • Independently of Z, generate multivariate normal Nd (0, Σ) variate N. √ • Set Y ← m · Z + Z · N. • RETURN Y The AL laws with mean zero are elliptically contoured, since their ch.f. depends on t only through the quadratic form t0 Σt. The class of elliptically symmetric distributions consists of elliptically contoured laws with non-singular Σ and the density function (13)
f (x) = kd |Σ|−1/2 g[(x − m)0 Σ−1 (x − m)],
where g is a one-dimensional real-valued function (independent of d) and kd is a proportionality constant. We shall denote the laws with the density (13) by ECd (m, Σ, g). It is well-known [see, e.g., Fang et al. (1990)], that every r.v. Y ∼ ECd (0, Σ, g) admits the representation (14)
d
Y = RHU(d) ,
where H is a d × d matrix such that HH0 = Σ, R is a positive r.v. independent of U(d) (having the distribution of [Y 0 Σ−1 Y]1/2 ), and U(d) is a r.v. uniformly distributed on the sphere Sd = {y ∈ Rd : ||y|| = 1} (so that HU(d) is uniformly distributed on the surface of the hyperellipsoid {y ∈ Rd : y0 Σ−1 y = 1}). Our next basic result identifies the distribution of R in the class of AL distributed variables Y.
MULTIVARIATE LAPLACE DISTRIBUTIONS
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Proposition 2.1. Let Y ∼ ALd (0, Σ), where |Σ| > 0. Then, Y admits the representation (14), where H is a d × d matrix such that HH0 = Σ, U(d) is a r.v. uniformly distributed on the sphere Sd , and R is a positive r.v. independent of U(d) with the density √ 2xd/2 Kd/2−1 ( 2x) , x > 0, (15) fR (x) = √ ( 2)d/2−1 Γ(d/2) where Kv is the modified Bessel function of the third kind defined by (40) in the Appendix. Proof. By Theorem 2.1, Y has the representation (9) with m = 0. Write Σ = HH0 , where H is a d × d non-singular lower triangular matrix. Then, the r.v. X ∼ Nd (0, Σ) from (9) has the representation X = HN, where N ∼ Nd (0, I). Further, the r.v. N, which is d
elliptically contoured, has the well known representation N = RN U(d) , where RN and U(d) are independent and U(d) is uniformly distributed on Sd while RN is positive with the density (16)
fRN (x) =
d 2d/2 Γ(d/2
+ 1)
xd−1 exp(−x2 /2), x > 0
(it is distributed as the square root of a Chi-squared r.v. with d degrees of freedom). Therefore, it is sufficient to show that Z 1/2 RN has density (15). To this end, apply a standard transformation theorem to write the density of Z 1/2 RN as Z ∞ d 1 (17) fZ 1/2 RN (y) = d/2 xd/2−2 exp(− (x2 + 2y 2 /x))dx. y 2 2 Γ(d/2 + 1) 0 Let fλ,χ,ψ be the GIG density (10) with ψ = 1, χ = 2y 2 , and λ = d/2 − 1. Then, utilizing the definition of Kλ (y) given by (40) relation (17) becomes (18)
fZ 1/2 RN (y) =
√ d ( 2y)(2y 2 )λ/2 yK λ 2d/2 Γ(d/2 + 1)
Z
∞
fλ,χ,ψ (x)dx, 0
which yields (15) since the integral equals one.
d
Remark 2.4. In the case d = 1, the r.v. U (1) takes on values ±1 with probabilities 1/2 and R = √ (1/ 2)Z, where Z is standard exponential r.v. Consequently, we arrive at the representation p σ11 /2 · ZU (1) of symmetric Laplace r.v.’s with the ch.f. (1) [see, e.g., Kozubowski and Podg´orski (2000)]. Remark 2.5. An AL r.v. Y ∼ ALd (m, Σ) can be interpreted as a value of a subordinated Gaussian process, d
Y = X(Z),
´ S. KOTZ, T. J. KOZUBOWSKI, AND K. PODGORSKI
8
where X is a d-dimensional Gaussian process with independent increments, X(0) = 0, and X(1) ∼ Nd (m, Σ). This follows immediately by evaluating the characteristic function on the right-hand side by conditioning on the exponential random variable Z. Consequently, AL distributions may be studied via the theory of (stopped) L´evy processes [see Bertoin (1996)]. 3. Densities In this section we shall study AL densities (assuming that the distribution is nonsingular). The representation given in Theorem 2.1 leads to a relation between the distribution functions and the densities of AL and multivariate normal random vectors. Let G(·) and F (·) be the d.f.’s of ALd (Σ, m) and Nd (0, Σ) r.v.’s, respectively, and let g(·) and f (·) be the corresponding densities. Corollary 3.1. Let Y ∼ ALd (m, Σ). The distribution function and the density (if it exists) of Y can be expressed as follows: G(y) =
Z
∞
F (z −1/2 y − z 1/2 m)e−z dz 0
(19)
g(y) =
Z
∞
f (z −1/2 y − z 1/2 m)z −d/2 e−z dz.
0
We can also express an AL density in terms of the modified Bessel function of the third kind defined in the Appendix. From (19), we have Z ∞ (y − zm)0 Σ−1 (y − zm) −d/2 −1/2 (20) g(y) = (2π) |Σ| exp − − z z −d/2 dz. 2z 0 In particular, −d/2
g(0) = (2π)
|Σ|
−1/2
Z
∞ 0
1 0 −1 exp −z( m Σ m + 1) z −d/2 dz, 2
which is infinite unless d = 1. For y 6= 0, we substitute w = z(1 + m0 Σ−1 m/2) into (20) obtaining 0
g(y) =
ey Σ
−1
+ 12 m0 Σ−1 m)d/2−1 (2π)d/2 |Σ|1/2
m (1
Z 0
∞
2 a exp − − z z −(d−2)/2−1 dz, 4z
q where a = (2 + m0 Σ−1 m)(y0 Σ−1 y). Taking into account the integral representation (40) of the corresponding Bessel function (see Appendix), we arrive at the following basic result.
MULTIVARIATE LAPLACE DISTRIBUTIONS
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Theorem 3.1. The density of Y ∼ ALd (m, Σ) can be expressed as follows: 0
(21)
−1
2ey Σ m g(y) = (2π)d/2 |Σ|1/2
y0 Σ−1 y 2 + m0 Σ−1 m
v/2
Kv
q
(2 +
m0 Σ−1 m)(y0 Σ−1 y)
,
where v = (2 − d)/2 and Kv (u) is the modified Bessel function of the third kind given by (40) or (41)(cf. Appendix).
Remark 3.1. The density (21) is a special case of a generalized hyperbolic density q 0 λ ξ exp(β (x − µ))Kd/2−λ (α δ 2 + (x − µ)0 Σ−1 (x − µ)) q (22) , (2π)d/2 |Σ|1/2 δ λ Kλ (δξ)[ δ 2 + (x − µ)0 Σ−1 (x − µ)/α]d/2−λ with λ = 1, ξ 2 = 2, δ 2 = 0, µ = 0, β = Σ−1 m, and α =
p
2 + m0 Σ−1 m (see the Remarks
following Theorem 2.1). In case δ = 0 we use the asymptotic relation (47) given in the Appendix. Remark 3.2. In the symmetric case (m = 0), we obtain from (21) q v/2 −d/2 −1/2 0 −1 −1 0 g(y) = 2(2π) y Σ y/2 |Σ| Kv 2y Σ y , which is the density of a multivariate Laplace distribution presented in Anderson (1992)1 Remark 3.3. If d = 1, we have v = 1/2 and the density becomes (23)
g(y) =
where σ 2 = Σ, µ = m, and γ =
p
1 − |y|2 (γ−µ·sign(y)) , e σ γ
µ2 + 2σ 2 . The density (23) coincides with that of a
univariate AL distribution, see Kozubowski and Podg´orski (1999ab, 2000). Remark 3.4. Suppose d = 2r+3, where r = 0, 1, 2, . . ., in which case v = (2−d)/2 = −r−1/2. Then, the Bessel function Kv (= K−v ) with v = r + 1/2 has an explicit form (45) given in the Appendix and the AL density (21) is simplified to become √ −1 0 −1 q 0 r C r ey Σ m−C y Σ y X (r + k)! p g(y) = (2C y0 Σ−1 y)−k , y 6= 0, (2π y0 Σ−1 y)r+1 |Σ|1/2 k=0 (r − k)!k! 1The density (8) in Anderson (1992) seems to contain an extra factor of √2Q.
´ S. KOTZ, T. J. KOZUBOWSKI, AND K. PODGORSKI
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p
2 + m0 Σ−1 m. If d = 3, we have r = 0 and √ −1 0 −1 0 ey Σ m−C y Σ y p g(y) = , y 6= 0. 2π y0 Σ−1 y|Σ|1/2
where v = (2 − d)/2 and C =
Remark 3.5. For d = 2, we have 0
−1
ey Σ m g(y) = K0 π|Σ|1/2 Denoting m = (m1 , m2 )0 and
q −1 −1 0 0 (2 + m Σ m)(y Σ y) .
Σ=
σ12
ρσ1 σ2
ρσ1 σ2
σ22
,
we obtain a five parameter family dependent on m1 , m2 , σ1 , σ2 , ρ with the densities of the form g(x, y) = exp ((m1 σ2 /σ1 − m2 ρ) x + (m2 σ1 /σ2 − m1 ρ) y) /(σ1 σ2 (1 − ρ2 )) p πσ1 σ2 1 − ρ2 p ·K0 C(m1 , m2 , σ1 , σ2 ) x2 σ2 /σ1 − 2ρxy + y 2 σ1 /σ2 , where C(m1 , m2 , σ1 , σ2 ) =
p
2σ1 σ2 (1 − ρ2 ) + m21 σ2 /σ1 − 2m1 m2 ρ + m22 σ1 /σ2 . σ1 σ2 (1 − ρ2 )
The parameters m1 and m2 introduce skewness. When m1 = m2 = 0, the above density could be compared to the bivariate normal density possessing the same covariance matrix Σ. √ We see that the exponential function exp(·) in the Gaussian density is replaced by K0 ( 2·) in the Laplace density. 4. Infinite divisibility properties The following result establishes infinite divisibility of AL laws and identifies their L´evy measure. Theorem 4.1. Let Y have a fully d-dimensional ALd (m, Σ) law. Then, the ch.f. of Y is of the form Ψ(t) = exp
Z Rn
e
it·x
− 1 Λ(dx)
MULTIVARIATE LAPLACE DISTRIBUTIONS
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with
−d/2 2 exp(m0 Σ−1 x) dΛ Q(x) Kd/2 (Q(x)C(Σ, m)), (x) = dx C(Σ, m) (2π)d/2 |Σ|1/2 p √ where Q(x) = x0 Σ−1 x and as above C(Σ, m) = 2 + m0 Σ−1 m. Proof. Apply Proposition 4.1 from Kozubowski and Rachev (1999b), which identifies the density of geometric stable L´evy measure, to obtain Z ∞ dΛ f (z −1/2 x − z 1/2 m)z −d/2−1 e−z dz, (x) = dx 0 where f (·) is the density of the multivariate normal distribution Nd (0, Σ) with respect to the d-dimensional Lebesgue measure, and proceed as in the computation of AL densities described in Section 3. An alternative derivation involves representing Y through a subordinated Brownian motion and applying Lemma 7, VI.2 of Bertoin (1996).
Remark 4.1. In one-dimensional case (d = 1), we obtain the L´evy measure of a univariate AL law,
! √ dΛ 2x ±1 1 , x > 0, (±x) = exp − κ dx x σ p √ where σ 2 = Σ, µ = m, and κ = 2σ/(µ + µ2 + 2σ 2 ) [see, e.g., Kozubowski and Podg´orski (2000)]. Since AL laws are infinitely divisible, we can define a L´evy process on [0, ∞) with independent increments, the Laplace motion {Y(s), s ≥ 0}, where Y(0) = 0, Y(1) is given by (3), and the ch.f. of Y(s) being (24)
Ψ(t) =
1 1 0 1 + 2 t Σt − im0 t
!s
, s > 0.
We shall refer to distributions on Rd given by (24) as generalized asymmetric Laplace (GAL) laws, and write GALd (m, Σ, s). A GAL r.v. admits mixture representation (9) where Z has a gamma distribution with the density (25)
g(x) =
xs−1 −x e . Γ(s)
´ S. KOTZ, T. J. KOZUBOWSKI, AND K. PODGORSKI
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The density corresponding to (24) can be expressed in terms of Bessel function as follows: 2 exp(m0 Σ−1 x) p(x) = (2π)d/2 Γ(s)|Σ|1/2
Q(x) C(Σ, m)
s−d/2
Ks−d/2 (Q(x)C(Σ, m)),
where Q(x) and C(Σ, m) are as above. One-dimensional GAL distributions were found useful for modeling size distributions of diamonds excavated from marine deposits in South West Africa [see Sichel (1973)] and in financial applications [see Madan and Seneta (1990) for the symmetric case, Madan et al. (1998), where it is discussed under the name a gamma variance process, and in Eberlein and Keller (1995), where the hyperbolic L´evy motion is studied]. Finally we show that an AL r.v. Y (and its probability distribution) is geometric infinitely divisible, namely if for all p ∈ (0, 1) we have d
Y=
(26)
νp X
Yp(i) ,
i=1 (i)
(i)
where νp is geometrically distributed, Yp ’s are i.i.d. for each p, and νp and (Yp ) are independent [see, e.g., Klebanov et al. (1984)]. Proposition 4.1. Let Y ∼ ALd (m, Σ). Then, Y is geometric infinitely divisible and relation (1)
(26) holds with Yp ∼ ALd (pm, pΣ). Proof. Write (26) in terms of ch.f.’s.
5. Linear transformations and conditional distributions We shall start by considering the distributions of AL vectors under linear transformations. First, we show that if Y ∼ ALd (m, Σ) then all linear combinations of components of Y are jointly AL. Proposition 5.1. Let Y = (Y1 , . . . , Yd )0 ∼ ALd (m, Σ) and let A be an l × d real matrix. Then, the random vector AY is ALl (mA , ΣA ), where mA = Am and ΣA = AΣA0 . Proof. Write 0
0
0
ΨAY (t) = Eei(AY) t = EeiY A t = ΨY (A0 t) and note that the matrix AΣA0 is non-negative definite whenever Σ is.
MULTIVARIATE LAPLACE DISTRIBUTIONS
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In particular, all univariate and multivariate marginal distributions of an AL r.v. Y are AL, as are all linear combinations of the components of Y. Corollary 5.1. Let Y = (Y1 , . . . , Yd )0 ∼ ALd (m, Σ), where Σ = (σij )di,j=1 . Then ˜ where m ˜ is a n × n ˜ = (m1 , . . . , mn )0 and Σ ˜ Σ), (i) For all n ≤ d, (Y1 , . . . , Yn ) ∼ ALn (m, matrix with σ ˜ij = σij for i, j = 1, . . . , n;
P (ii) For any b = (b1 , . . . bd )0 ∈ Rd , the r.v. Yb = dk=1 bk Yk is univariate AL(µ, σ) with √ σ = b0 Σb and µ = m0 b. Furthermore, if Y is symmetric AL, then so is Yb ; √ (iii) For all i ≤ d, Yi ∼ AL(µ, σ) with σ = σii and µ = mi . Proof. For part (i), apply Proposition 5.1 with an n × d matrix A = (aij ) such that aii = 1 and aij = 0 for i 6= j. For part (ii), apply Proposition 5.1 with l = 1 and compare the resulting ch.f. with (4). For part (iii) apply part (ii) to the standard base vectors (1, 0, . . . , 0)0 , . . . , (0, 0, . . . , 1)0 in Rd .
Remark 5.1. Part (ii) of Corollary 5.1 implies that the sum 0
Pd
k=1 Yk
has an AL distribution
0
if all Yk s are components of a multivariate AL r.v. (and thus all Yk s are univariate AL r.v.’s). On the other hand, in general a sum of i.i.d. AL r.v.’s does not have an AL distribution. Part (ii) of Corollary 5.1 shows that if Y is an AL r.v. in Rd , then all linear combinations of its components are univariate AL r.v.’s. Is the converse true? This seems to be an open problem. The following result provides a partial answer for the case where the linear combinations have either symmetric Laplace or exponential distributions. Proposition 5.2. Let Y = (Y1 , . . . , Yd )0 be a r.v. in Rd . If all the linear combinations Pd k=1 ck Yk have either symmetric Laplace or exponential distribution, then either all of them are symmetric Laplace or all are exponential and Y has an ALd (m, Σ) distribution with either m = 0 or Σ = 0, respectively. Proof. The proof follows from the corresponding result for GS laws [Kozubowski (1997), Theorem 3.3] and the fact that the ALd (m, Σ) distribution with either Σ = 0 or m = 0 are strictly geometric stable.
´ S. KOTZ, T. J. KOZUBOWSKI, AND K. PODGORSKI
14
The derivation of the conditional distributions of Y ∼ ALd (m, Σ) with a non-singular Σ is similar to that for the case of multivariate generalized hyperbolic distribution, see, e.g., Barndorff-Nielsen and Blaesild (1981). The conditional laws are not AL, but are generalized hyperbolic ones. [However, the conditional distributions can be AL if Y has a generalized AL law (24).] Theorem 5.1. Let Y ∼ GALd (m, Σ, s) have ch.f. (24) with a non-singular Σ. Let Y 0 = (Y10 , Y2 0 ) be a partition of Y into r × 1 and k × 1 dimensional sub-vectors, respectively. Let (m01 , m02 ) and
Σ=
Σ11 Σ12 Σ21 Σ22
be the corresponding partitions of m and Σ, where Σ11 is an r × r matrix. Then (i) If s = 1 (so that Y is AL), the conditional distribution of Y2 given Y1 = y1 is the generalized k-dimensional hyperbolic distribution Hk (λ, α, β, δ, µ, ∆) having the density q 0 −1 λ 2 0 ξ exp(β (y2 − µ))Kk/2−λ α δ + (y2 − µ) ∆ (y2 − µ) (27) p(y2 |y1 ) = q k/2−λ , (2π)k/2 |∆|1/2 δ λ Kλ (δξ)
δ 2 + (y2 − µ)0 ∆−1 (y2 − µ)/α
q p ξ 2 + β 0 ∆β, β = ∆−1 (m2 − Σ21 Σ−1 m ), δ = y10 Σ−1 1 11 11 y1 , q −1 µ = Σ21 Σ−1 2 + m01 Σ−1 11 y1 , ∆ = Σ22 − Σ21 Σ11 Σ12 , and ξ = 11 m1 ;
where λ = 1 − r/2, α =
(ii) If m1 = 0, then the conditional distribution of Y2 given Y1 = 0 is GALk (m2·1 , Σ2·1 , s2·1 ), where s2·1 = s − r/2, Σ2·1 = Σ22 − Σ21 Σ−1 11 Σ12 , m2·1 = m2 . Proof. By part (i) of Corollary 5.1 with n = r, the r.v. Y1 is ALr (m1 , Σ11 ). Write the densities of Y and Y1 according to (21) and simplify the ratio of the densities utilizing the familiar relations from the classical multivariate analysis: −1 −1 0 −1 Y 0 Σ−1 m = Y10 Σ−1 11 m1 + (m2 − Σ21 Σ11 m1 ) ∆ (y2 − Σ21 Σ11 y1 ), −1 −1 0 −1 Y 0 Σ−1 Y = Y10 Σ−1 11 Y1 + (Y2 − Σ21 Σ11 Y1 ) ∆ (y2 − Σ21 Σ11 y1 ), −1 −1 0 −1 m0 Σ−1 m = m01 Σ−1 11 m1 + (m2 − Σ21 Σ11 m1 ) ∆ (m2 − Σ21 Σ11 m1 ),
|Σ| = |Σ22 − Σ21 Σ−1 11 Σ12 | · |Σ11 |.
MULTIVARIATE LAPLACE DISTRIBUTIONS
15
Finally, verify that α2 = β 0 ∆β + ξ 2 . The proof of Part (ii) is similar.
Remark 5.2. From a definition of λ in part (i) of the theorem, this parameter can not equal one. Hence, in case of a multivariate AL distribution no conditional law can be an AL. However, in part (ii) we might have s − r/2 = 1, in which case we do obtain a conditional AL law for a multivariate generalized AL distribution. We now derive expressions for the conditional mean vector and covariance matrix via the theory of hyperbolic distributions. Proposition 5.3. Let Y have a GAL law (24) with a non-singular Σ. Let Y, m, and Σ be partitioned as in Theorem 5.1. Then −1 E(Y2 |Y1 = y1 ) = Σ21 Σ−1 11 y1 + (m2 − Σ21 Σ11 m1 )
Q(y1 ) R1−r/2 (CQ(y1 )) C
and Q(y1 ) (Σ22 − Σ21 Σ−1 11 Σ12 )R1−r/2 (CQ(y1 )) C 2 −1 0 Q(y1 ) +(m2 − Σ21 Σ−1 m )(m − Σ Σ m ) G(y1 ), 1 2 21 11 1 11 C
V ar(Y2 |Y1 = y1 ) =
where C is as above, Q(y1 ) =
q
y10 Σ−1 11 y1 ,
Rs (x) = Ks+1 (x)/Ks (x), and 2 G(y1 ) = (R1−r/2 (CQ(y1 ))R2−r/2 (CQ(y1 )) − R1−r/2 (CQ(y1 ))).
Proof. Apply Theorem 5.1 and utilize the representation (11) of the generalized hyperbolic distribution to conclude that E(Y2 |Y1 = y1 ) = µ + ∆βE(Z) and V ar(Z2 |Y1 = y1 ) = ∆β(∆β)0 V ar(Z) + ∆E(Z), where Z has the GIG(s, δ 2 , ξ 2 ) distribution (10) and µ, β, ∆, δ, and ξ are as given in Theorem 5.1. Then, apply the formulas for the moments of an GIG r.v. Z, E(Z r ) = (δ/ξ)r Ks+r (δξ)/Ks (δξ) [see, e.g., Barndorff-Nielsen and Blaesild (1981)].
´ S. KOTZ, T. J. KOZUBOWSKI, AND K. PODGORSKI
16
Remark 5.3. Note that if either m1 or m2 are non-zero then the conditional expectation of Y2 given Y1 = y1 has the non-zero second term which exhibits a certain “quadratic” dependence on y1 . For example if y1 is approaching zero, then the second term is proportional to y10 Σ−1 11 y1 [see Property A6 in the Appendix]. If m01 Σ−1 11 Σ12 = md , then by Theorem 5.1, the conditional distribution of Yd given (Y1 , . . . , Yd−1 ) is generalized hyperbolic and symmetric around µ = Σ21 Σ−1 11 y1 (β = 0 in this case), which must be the mean of the conditional distribution. Thus, conditions for the linearity of the regression of Yd on Y1 , . . . , Yd−1 , where Y = (Y1 , . . . , Yd )0 is AL are the same as those for multivariate normal laws. Proposition 5.4. Let Y = (Y1 , . . . , Yd )0 ∼ ALd (m, Σ). Write m1 = (m1 , . . . , md−1 )0 and let Σ11 Σ12 Σ= Σ21 Σ22 be a partition of Σ such that Σ11 is a d − 1 × d − 1 matrix. Then, (28)
E(Yd |Y1 , . . . , Yd−1 ) = a1 Y1 + · · · + ad−1 Yd−1 (a.s.)
if and only if (29)
Σ11 a = Σ12 and m01 a = md .
Furthermore, if |Σ| > 0, the condition (29) is equivalent to m01 Σ−1 11 Σ12 = md and a = −1 (a1 , . . . , ad−1 )0 = Σ11 Σ12 .
Proof. For a r.v. Y with a finite mean, condition (28) is equivalent to ∂Ψ(t) ∂Ψ(t) ∂Ψ(t) = a1 + · · · + ad−1 , ∂td td =0 ∂t1 td =0 ∂td−1 td =0 where Ψ is the ch.f. of Y [see, e.g., Miller (1978)]. Substitution of the AL ch.f. (3) into the above equation followed by differentiation, results in (29). In case |Σ| > 0, the solution of the first equation in (29), a = Σ−1 11 Σ12 , is also the solution of the second equation provided m01 Σ−1 11 Σ12 = md .
Remark 5.4. The regression is always linear for m = 0 (in the symmetric case).
MULTIVARIATE LAPLACE DISTRIBUTIONS
17
6. Unimodality and related representation All univariate AL distributions are unimodal with the mode at zero [see Kozubowski and Podg´orski (2000)]. A natural extension of univariate unimodality is star unimodality in Rd . This property requires that for a distribution with continuous density f the density be non-increasing along the rays emanating from zero. A useful criterion for star unimodality due to Dharmadhikari and Joag-Dev (1988) states that a distribution with continuous density f on Rd is star unimodal about zero if and only if f (ux) ≤ f (tx) whenever 0 < t < u < ∞ and x 6= 0 (it follows directly from the proof [see Dharmadhikari and Joag-Dev (1988)] that the criterion remains valid for densities discontinuous at zero as well). We shall show below that all fully d-dimensional AL laws are star unimodal about zero. Proposition 6.1. The distribution of Y ∼ ALd (m, Σ), where |Σ| > 0, is star unimodal about 0. Proof. Assume that d > 1 and let x 6= 0. For t > 0 let h(t) = log g(tx), where g is the density of Y given by (21). Write h(t) = log C1 + C2 t + v log t + log Kv (C3 t), where v = 1 − d/2 and C1 =
2(x0 Σ−1 x)v/2 > 0, (2π)d/2 |Σ|1/2 (2 + m0 Σ−1 m)v/2 C2 = m0 Σ−1 x ∈ R,
C3 =
p
p
2 + m0 Σ−1 m
x0 Σ−1 x > 0.
To show that h is a non-increasing function of t, we calculate its derivative with respect to t: (30)
d v K 0 (C3 t) h(t) = C2 + + v C3 . dt t Kv (C3 t)
Using the properties (43) - (44) of the Bessel function Kv (listed in the Appendix) we can write (30) as (31)
d Kv−1 (C3 t) h(t) = C2 − C3 . dt Kv (C3 t)
18
´ S. KOTZ, T. J. KOZUBOWSKI, AND K. PODGORSKI
If C2 < 0, (31) implies that h0 (t) ≤ 0, since the Bessel function Kv is always positive and C3 > 0. Otherwise, write Σ−1 = Q0 Q and use the Cauchy-Schwarz inequality to obtain p p |C2 | = |(Qm)0 (Qx)| ≤ ||Qm|| · ||Qx|| = m0 Σ−1 m x0 Σ−1 x < C3 . This together with (31) implies that h0 (t) ≤ 0 whenever Kv−1 (C3 t) ≥ 1. Kv (C3 t) Since Kv (x) = K−v (x) for all v, the later inequality is equivalent to K−v (C3 t) ≤ K−v+1 (C3 t). By Property A3 of Bessel functions cited in the Appendix, the above inequality holds for all t > 0 (since −v ≥ 0 as d > 1).
Remark 6.1. It also follows that every AL r.v. Y is linear unimodal about 0, that is every linear combination c0 Y is univariate unimodal about zero [see Definition 2.3 of Dharmadhikari and Joag-Dev (1988)]. This follows from part (iii) of Corollary 5.1 since all univariate AL laws are unimodal about zero. 7. Stability properties In this section we shall discuss briefly characterizations of the AL laws related to stability properties under appropriate summation schemes. The results of this section do not follow from the theory of generalized hyperbolic distributions since the latter laws do not in general possess convolution properties (except for some special cases such as the normal inverse Gaussian or the normal variance gamma models). 7.1. Limits of random sums. Our first result, which can serve as an alternative definition of the AL laws, shows that they are the only possible limits of random sums (8) of i.i.d. r.v.’s with finite second moments. It follows from the Transfer Theorem for random summation [see, e.g., Rosi´ nski (1976)] and its converse [see Szasz (1972)] combined with the classical Central Limit Theorem for i.i.d. r.v.’s with a finite covariance matrix. Proposition 7.1. Let νp be a geometrically distributed r.v. with mean 1/p, where p ∈ (0, 1). A random vector Y has an AL distribution in Rd if and only if there exists an independent of
MULTIVARIATE LAPLACE DISTRIBUTIONS
19
νp sequence {X(i) } of i.i.d. random vectors in Rd with finite covariance matrix, and ap > 0, bp ∈ Rd , such that (32)
ap
νp X
d
(X(j) + bp ) −→ Y,
as p → 0.
j=1
Next, we derive a normalization leading to convergence in (32). Theorem 7.1. Let X(j) be i.i.d. random vectors in Rd with mean vector m and covariance matrix Σ. For p ∈ (0, 1), let νp be a geometric r.v. with mean 1/p, and independent of the sequence (X(j) ). If ap = p1/2 and bp = m(p1/2 − 1), then as p → 0, (33)
ap
νp X
d
(X(j) + bp ) → Y ∼ ALd (m, Σ).
j=1
Proof. By the Cram´er-Wald device [see, e.g., Billingsley (1968)] (33) is equivalent to c0 ap
νn X
d
(X(j) + bp ) → c0 Y, as p → 0,
j=1
for all c in Rd . Denoting Wj = c0 (X(j) − m), µ = c0 m, bp = p1/2 µ, and Y = c0 Y, we obtain (34)
ap
νp X
d
(Wj + bp ) → Y ∼ AL(µ, σ), as p → 0.
j=1
The variables Wj are i.i.d. with mean zero and variance σ 2 = c0 Σc, and Y is an AL r.v. with the ch.f. ψ(t) =
1 . 1 + σ 2 t2 /2 − iµt
Writing (34) in terms of ch.f.’s results in peipµt φ(p1/2 t) → ψ(t), 1 − (1 − p)eipµt φ(p1/2 t)
(35)
where φ and ψ are the ch.f.’s of Wj and Y , respectively. Taking the reciprocals of both sides in (35) we obtain 1 − (1 − p)eipµt φ(p1/2 t) → 1 + σ 2 t2 /2 − iµt. peipµt φ(p1/2 t) Since the factor eipµt φ(p1/2 t) converges to zero as p → 0, one can write (splitting the numerator): e−ipµt − 1 1 − (1 − p)φ(p1/2 t) + = I + II → 1 + σ 2 t2 /2 − iµt. p p
´ S. KOTZ, T. J. KOZUBOWSKI, AND K. PODGORSKI
20
Indeed, as p → 0 we have I=
e−ipµt − 1 sin(pµt) cos(pµt) − 1 = −iµt + pµt → −iµt, p pµt pµt
and the convergence (36)
II =
1 − (1 − p)φ(p1/2 t) → 1 + σ 2 t2 /2 p
follows from Breiman (1993), Theorem 8.44. (To see the latter, note that since Wj has the first two moments, its ch.f. can be written as φ(u) = 1 + iuEWj +
(iu)2 (EXj2 + δ(u)), 2
where δ is a bounded function of u such that limu→0 δ(u) = 0. Applying the above with u = p1/2 t to the left-hand side of (36) we obtain t2 2 pt2 2 (σ + δ(p1/2 t)) + 1 − (σ + δ(p1/2 t)), 2 2 which converges to 1 + t2 σ 2 /2 as p → 0). 7.2. Stability under random summation. Stable r.v.’s with index α possess the wellknown stability property: X is α-stable if and only if for any n ≥ 2 (37)
d
X(1) + · · · + X(n) = n1/α X + dn ,
where X(i) ’s are i.i.d. copies of X and dn is a vector in Rd [see, e.g., Samorodnitsky and Taqqu (1994)]. Asymmetric Laplace random vectors admit a similar characterization with respect to geometric summation. Theorem 7.2. Let Y, Y (1) , Y (2) , . . . be i.i.d. r.v.’s in Rd with finite second moments, and let νp be a geometrically distributed random variable independent of the sequence {Y (i) , i ≥ 1}. Then (38)
a(p)
νp X
d
(Y (i) + b(p)) = Y, p ∈ (0, 1),
i=1
where a(p) > 0 and b(p) ∈ Rd if and only if Y ∼ ALd (m, Σ) with either Σ = 0 or m = 0. The normalizing constants are of the form a(p) = p1/α ,
b(p) = 0.
MULTIVARIATE LAPLACE DISTRIBUTIONS
21
Proof. The result follows from a similar characterization of strictly geometric stable laws [see Kozubowski (1997), Theorem 3.1] and the fact that the only geometric stable laws with finite second moments are the ALd (m, Σ) laws with either Σ = 0 or m = 0.
7.3. Stability of deterministic sums. A deterministic sum of i.i.d. AL r.v.’s scaled by an appropriate random variable has the same distribution as each component of the sum. Theorem 7.3. Let Bm , where m > 0, have a Beta(1, m) distribution and let {X(i) } be a sequence of i.i.d. random vectors with finite second moments. Then, the following statements are equivalent: d
1/2
(i) For all n ≥ 2, X(1) = Bn−1 (X(1) + · · · + X(n) ). (ii) X(1) is ALd (m, Σ) with either Σ = 0 or m = 0. Proof. The result follows from the corresponding result for strictly geometric stable (GS) laws presented in Kozubowski and Rachev (1999b) and the fact that ALd (m, Σ) distributions with either Σ = 0 or m = 0 are strictly GS [The result for strictly GS laws can be deduced from Pakes (1992)]. Our final stability property of AL laws was derived in one-dimensional case by Pillai (1985). Proposition 7.2. Let Y, Y (1) , Y (2) , and Y (3) be ALd (m, Σ) r.v.’s with either Σ = 0 or m = 0. Let p ∈ (0, 1), and let I be an indicator random variable with P (I = 1) = p, and P (I = 0) = 1 − p. Then, (39)
d
Y = p1/α IY (1) + (1 − I)(Y (2) + p1/α Y (3) ), p ∈ (0, 1),
where the variables are independent. Proof. For any c ∈ Rd the variables c0 Y, c0 Y (1) , c0 Y (2) , and c0 Y (3) are univariate AL(µ, σ) with either µ = 0 or σ = 0 (see Corollary 5.1). Consequently, by the result in the onedimensional case [see, Pillai (1985)], we obtain d
c0 Y = p1/α Ic0 Y (1) + (1 − I)(c0 Y (2) + p1/α c0 Y (3) ),
´ S. KOTZ, T. J. KOZUBOWSKI, AND K. PODGORSKI
22
or, equivalently, d
c0 Y = c0 (p1/α IY (1) + (1 − I)(Y (2) + p1/α Y (3) )), which implies (39).
8. Appendix Here we collect several relevant results for the modified Bessel function of the third kind with index λ ∈ R, denoted Kλ (·). We refer the reader to Abramowitz and Stegun (1965), Olver (1974), and Watson (1962) for further details. The following two representations of the function Kλ (u) have been repeatedly utilized in the text [see, e.g., Watson (1962, pp.183), Abramowitz and Stegun (1965, pp. 376)]. 1 u λ Kλ (u) = 2 2
(40)
(41)
(u/2)λ Γ(1/2) Kλ (u) = Γ(λ + 1/2)
Z
∞
t
0
Z
−λ−1
u2 exp −t − 4t
dt, u > 0,
∞
e−ut (t2 − 1)λ−1/2 dt, u > 0, λ ≥ −1/2. 1
Property A1. The Bessel function Kλ (u) is continuous and positive function of λ ≥ 0 and u > 0. Property A2. For any fixed λ ≥ 0, the function Kλ (u) is positive and decreasing in u on the interval (0, ∞). Property A3. For any fixed u > 0, the function Kλ (u) is positive and increasing in u on the interval (0, ∞). Property A4. For any λ ≥ 0 and u > 0, the function Kλ (u) satisfies the relations (42)
Kλ (u) = Kλ (−u), 2λ Kλ (u) + Kλ−1 (u), u
(43)
Kλ+1 (u) =
(44)
Kλ−1 (u) + Kλ+1 (u) = −2Kλ0 (u).
MULTIVARIATE LAPLACE DISTRIBUTIONS
23
Property A5. For λ = r + 1/2, where r is a non-negative integer, r r π −u X (r + k)! (45) Kr+1/2 (u) = e (2u)−k . 2u (r − k)!k! k=0
In particular, (46)
K1/2 (u) =
r
π −u e . 2u
Property A6. If λ is fixed, then (47)
as x → 0+ , Kλ (x) ∼ Γ(λ)2λ−1 x−λ (λ > 0),
K0 (x) ∼ log(1/x).
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´ S. KOTZ, T. J. KOZUBOWSKI, AND K. PODGORSKI
Department of Engineering, Management & System Engineering, The George Washington University, Washington, DC 20052 E-mail address:
[email protected] Department of Mathematics, University of Nevada, Reno, NV 89557 E-mail address:
[email protected] Department of Mathematical Sciences, IUPUI, Indianapolis, IN 46202-3216 E-mail address:
[email protected]
