- Email: [email protected]
On univariate and bivariate generalized gamma convolutions
Journal of Statistical Planning and Inference 139 (2009) 3759 -- 3765
Contents lists available at ScienceDirect
Journal of Statistical Planning and Inference journal homepage: w w w . e l s e v i e r . c o m / l o c a t e / j s p i
On univariate and bivariate generalized gamma convolutions Lennart Bondesson∗ Department of Mathematics and Mathematical Statistics, Umeå University, SE-90187 Umeå, Sweden
A R T I C L E
I N F O
A B S T R A C T
This paper has two parts. In the first part some results for generalized gamma convolutions (GGCs) are reviewed. A GGC is a limit distribution for sums of independent gamma variables. In the second part, bivariate gamma distributions and bivariate GGCs are considered. New bivariate gamma distributions are derived from shot-noise models. The remarkable property hyperbolic complete monotonicity (HCM) for a function is considered both in the univariate case and in the bivariate case. © 2009 Elsevier B.V. All rights reserved.
Available online 22 May 2009 MSC: 60E05 60E07 60G55 Keywords: Bivariate gamma distribution Generalized gamma convolution Hyperbolic complete monotonicity (HCM) Shot-noise models
1. Introduction A generalized gamma convolution (GGC) is a limit distribution for sums of independent gamma distributed random variables. The gamma distributions may have different shape and scale parameters. The GGC-class, which was introduced by Thorin (1977), is rich and it has many remarkable properties. In particular, it contains all probability densities f on (0, ∞) that are hyperbolically completely monotone (HCM); cf. Section 3. The HCM-densities turn out to be limits of densities of the form
f (x) = Cx−1
N
(1 + ci x)−i ,
x > 0 ( > 0, ci > 0, i > 0).
i=1
Apparently, the HCM-class is closed with respect to multiplication of densities. It is also closed with respect to multiplication of independent random variables with densities in the class. Many standard distributions such as the lognormal, the generalized gamma, and the generalized Beta distributions of the second kind are HCM. In Sections 2–5 a review of the main GGC-results is given. The full GGC-theory is presented in Bondesson (1992) and also in Steutel and van Harn (2004). The paper Bondesson et al. (2008), which honors Olof Thorin, gives historical information. In Section 6, bivariate gamma distributions are considered. There are many possibilities and some new ones are presented. These derive from shot-noise models. Finally in Sections 7 and 8, attempts are made to generalize the univariate GGC-theory to the bivariate case but there are some difficulties. The natural building distribution for bivariate GGCs is Cherian's (1941) gamma distribution rather than Wicksell's (1933) gamma distribution. A bivariate HCM-condition is presented. Conclusions are given in Section 9.
∗ Tel.: +46 90 7866529; fax: +46 90 7865222. E-mail address: [email protected] 0378-3758/$ - see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2009.05.015
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L. Bondesson / Journal of Statistical Planning and Inference 139 (2009) 3759 -- 3765
2. Generalized gamma convolutions In this section the GGCs are introduced. Some complex analysis is used. As is well known, a gamma distribution is a probability distribution on (0, ∞) with probability density function (pdf) f (x) =
tu xu−1 −tx e , (u)
x>0
(u > 0, t > 0).
The distribution is denoted by Gamma(u, 1/t). It has the Laplace transform (LT)
(s) =
∞
0
e−sx f (x) dx =
t t+s
u =
1 (1 + s/t)u
.
Now convolve gamma distributions with different shape parameters u and scale parameters 1/t. Equivalently, add independent gamma distributed random variables. Then a distribution with LT ⎛ ⎞ ui N N ti ti ⎝ ⎠ (s) = = exp ui log ti + s ti + s i=1
(1)
i=1
appears. Taking also limits, we obtain LTs of the form
(s) = exp −as +
log (0,∞)
t U(dt) , t+s
(2)
where a ⱖ 0 and U(dt) is a nonnegative measure on (0, ∞). The parameter a is the left-extremity of the distribution. In particular, setting in (1), ti ≡ N, ui ≡ a, and letting N → ∞, we get the LT e−as , i.e. a distribution degenerate at a. Definition 1 (Thorin, 1977). A generalized gamma convolution is a probability distribution on [0, ∞) with LT of the form (2) and 1 ∞ such that 1 t−1 U(dt) < ∞ and 0 | log t|U(dt) < ∞. The last two conditions ensure the convergence of the integral in (2). Since every gamma distribution is infinitely divisible, cf., e.g. Feller (1966, p. 177), so is every GGC. To be seen later on, many well-known distributions are in the GGC-class. For the LT of a GGC we have
(s) d log (s) = = −a − ds (s)
(0,∞)
U(dt) . t+s
(3)
The right-hand side of (3) is a function that is analytic for s ∈ C\(−∞, 0]. Moreover, the imaginary part of it is nonnegative for Im s > 0. This result leads to a characterization of LTs of GGCs.
Theorem 1. An LT (s) of a distribution on [0, ∞) corresponds to a GGC if and only if it is analytic in C\(−∞, 0] and Im[ (s)/ (s)] ⱖ 0 for Im s > 0, or, equivalently, Im[ (s)(s)] ⱖ 0 for Im s > 0. The deep `if' part of this theorem is a consequence of the Pick–Nevanlinna theory or more generally a consequence of Cauchy's integral theorem in complex analysis; cf. Bondesson (1992, pp. 20–21). The theorem implies that for instance the LTs
(s) = e−s
and
(s) = (1 + s )− (0 < ⱕ 1, > 0)
(and products of such LTs) correspond to GGCs. The first GGC is a stable distribution and the second one is a Linnik (or Mittag–Leffler) distribution on (0, ∞). 3. The HCM-condition and a consequence An important but somewhat curious condition is presented in this section. It has far-reaching consequences. Definition 2. A function f (x) ⱖ 0 on (0, ∞) is said to be hyperbolically completely monotone if, for each fixed u > 0, h(w) = f (uv)f (u/v)
with w = v + v−1
is completely monotone (CM) as a function of w, i.e. if (−1)n h(n) (w) ⱖ 0 on (0, ∞) for all integers n ⱖ 0.
L. Bondesson / Journal of Statistical Planning and Inference 139 (2009) 3759 -- 3765
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Equivalently, by a theorem of Bernstein, cf., e.g. Feller (1966, p. 416), f is HCM if f (uv)f (u/v) = exp(−u(v + v−1 ))Ku (d),
(4)
[0,∞)
where Ku (d) is a nonnegative measure for each u > 0. The HCM-condition first appeared in Thorin's (1977) work for the lognormal distribution though in a special case and in a different additive form. The name is motivated by the hyperbolic substitution x = uv, y = u/v. The HCM-condition is often easy to verify as will be seen in Section 4. An important result is now presented. Theorem 2 (Main HCM-theorem). If a probability density f (x) is HCM, then it is a GGC as well. Sketch of proof. We have ∞ ∞ (s)(s) = − x e−sx e−s¯y f (x)f (y) dx dy. 0
0
Now make the hyperbolic substitution x = uv, y = u/v, and use the representation (4). In that way it can be verified that Im[ (s)(s)] ⱖ 0 for Im s > 0 and hence the pdf f (x) is a GGC by Theorem 1. However, there are complications: the double integral is not well-defined for all complex s with Im s > 0. A full proof is given in Bondesson (1992, pp. 56, 57 and 71). Remark 1. If the cdf F(x) = f (x) is also a GGC.
x 0
f (y) dy is HCM, or more generally if, for any fixed > 0, the integral
x
0 (x − y)
−1
f (y) dy is HCM, then
4. The HCM-class of probability densities In this section there are first presented several examples of probability densities on (0, ∞) that are HCM and hence GGCs. Then a representation of the HCM-densities is given. Some properties of the HCM-class of densities are also expounded. The following well-known probability densities on (0, ∞) are HCM. For all the densities, C is a normalizing constant: (a) Gamma distribution: f (x) = Cx−1 e−cx
(, c > 0).
(b) Beta distribution, 2nd kind: f (x) = Cx−1
1 (1 + cx)
(, , c > 0).
(c) Generalized inverse Gaussian (GIG) distribution: 1 f (x) = Cx−1 exp −c1 x − c2 ( ∈ R, c1 , c2 > 0). x (d) Generalized gamma distribution:
f (x) = Cx−1 e−cx
(0 < ⱕ 1, , c > 0).
(e) Generalized Beta distribution, 2nd kind: f (x) = Cx−1
1 (1 + cx )
(0 < ⱕ 1, , , c > 0).
For the densities in (a)–(c) it is easy to verify that they are HCM. For instance, for the density in (b) we get f (uv)f (u/v) = C 2 u2(−1)
1 (1 + c2 u2 + cu(v + v−1 ))
and apparently the right-hand side is completely monotone with respect to w = v + v−1 . The densities in (d) and (e) can be verified to be HCM from the fact that v + v− has a CM derivative with respect to w. This property is a consequence of the non-trivial representation sin() ∞ t d (v + v− ) = dt. 2 + tw dw 1 + t 0 A very complete description of the HCM-class of densities is provided by the following theorem, the proof of which is not simple.
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Theorem 3. The HCM-class of probability densities is the class of densities that are pointwise limits of densities of the product form −i (, ci , i > 0). The HCM-densities are unimodal. Cx−1 N i=1 (1 + ci x) One may notice that, except for the factor Cx−1 , densities of this form are LTs of GGCs. Moreover, if f1 and f2 are HCM-densities then so is Cf 1 f2 . The HCM-class has nice properties, which is shown by the next theorem. Theorem 4. The HCM-class is closed with respect to multiplication and division of independent random variables and also with respect to formation of powers with exponent q, |q| ⱖ 1, of random variables, i.e. X, Y ∼ HCM (independent) ⇒ X · Y ∼ HCM,
X/Y ∼ HCM
and X q ∼ HCM.
Sketch of proof. Set f (x) = fX (x) and g(y) = yf Y (y) (also HCM). Let h be the density of X/Y. Then ∞ ∞ s h(st)h(s/t) = f (stx)f y g(x)g(y) dx dy. t 0 0 By the hyperbolic substitution x = uv, y = uv and the Bernstein representation (4), it can be verified that the double integral is CM with respect to t + t−1 for each fixed s > 0. Hence h is HCM as desired. The result on powers follows from the representation (4) and the fact that v + v− with = 1/q has a CM derivative with respect to w = v + v−1 . Since the generalized Beta distribution of the 2nd kind is the distribution of (X1 /X2 )1/ , where X1 ∼ Gamma and X2 ∼ Gamma are independent random variables, Theorem 4 immediately implies that its density is HCM. Moreover, if X1 , X2 , . . . , Xn ∼ Gamma are independent, then the density of X1 · X2 · · · Xn is HCM and hence a GGC and infinitely divisible. By the central limit theorem, it follows that the lognormal density is HCM. The lognormal distribution is also a limit of generalized gamma distributions as → 0. Remark 2. If X ∼ HCM, then log X has a logconcave, i.e. strongly unimodal, density. It is well known that when independent random variables with logconcave densities are added the sum has a logconcave density as well. This casts new light on Theorem 4 and puts it into a better known context. 5. More about GGCs The HCM-condition provides a nice real characterization of LTs of GGCs. Theorem 5. An LT (s) of a distribution on [0, ∞) corresponds to a GGC if and only if (s) is HCM. The `only if' part of Theorem 5 follows from the definition of a GGC and the fact that the HCM-property is preserved when HCM-functions are multiplied. The `if' part is mainly a consequence of Theorem 3. For example, Theorem 5 shows that the distributions with LTs e−s and (1 + s )− are GGCs since these LTs are HCM. In fact, even distributions with LTs of the form ⎛ ⎞ − M k (s) = ⎝1 + ck s ⎠ (ck > 0, 0 < k ⱕ 1) k=1
can be verified to be GGCs by Theorem 5. The `real' Theorem 5 makes it possible to avoid all complex analysis for the GGCs. It can be verified from Theorem 5 that X ∼ GGC,
Y ∼ HCM (independent) ⇒ X/Y ∼
and
XY ∼ GGC.
The univariate GGC- and HCM-theory has been a success story in the sense that the theory is rather complete. Distributions on the whole real line and on the nonnegative integers are also considered in Bondesson (1992). Remark 3. There remain a few very hard problems: (i) Is the density function of each stable distribution on (0, ∞) with index ⱕ 12 HCM? This problem is treated in Bondesson (1999) with a positive answer. However, no strict proof was found. (ii) If X ∼ GGC, does it then follow that X q ∼ GGC for q ⱖ 1? √
−z (iii) Let f (s) = ( 12 )/ ( 12 + s), where (z) = 12 z(z − 1)(z/2)−z/2 (z) and (z) = ∞ n=1 n . Is f (s) HCM? If so, Riemann's famous hypothesis about the complex zeros of the -function is true.
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6. Bivariate gamma distributions In this section bivariate gamma distributions for (X1 , X2 ) are considered. There are many possibilities; cf., e.g. Hutchinson and Lai (1990). The popular copula technique, see, e.g. Nelsen (1999), which is a broad way to find multivariate distributions with given marginals, is not discussed here. Below two established bivariate gamma distributions are briefly described. Then a general probability model that is capable of producing many different bivariate gamma distributions is introduced. (a) Wicksell–Kibble distribution (Wicksell, 1933; Kibble, 1941): This distribution has bivariate LT (s, t) = E(exp(−sX 1 − tX 2 )) of the form
(s, t) = (1 + as + bt + ab(1 − 2 )st)− ,
s, t ⱖ 0.
Here a, b ⱖ 0, | | ⱕ 1, > 0. For a=b=1 the distribution appears in standardized form in which case the marginal distributions are Gamma(, 1). The coefficient ab(1 − 2 ) cannot be replaced by a coefficient larger than ab since then (s, t) fails to be an LT. The idea behind this distribution is to look at (Z12 , Z22 ), where (Z1 , Z2 ) has a bivariate normal distribution, in which case = 12 . The distribution is infinitely divisible (Vere-Jones, 1967). (b) Cherian's distribution (Cherian, 1941): Let
(s, t) =
1 1 1 . (1 + as)1 (1 + bt)2 (1 + as + bt)
For a = b = 1 this LT corresponds to, with Y, Z1 , Z2 independent, X1 = Y + Z1 ,
X2 = Y + Z2 ,
where Y ∼ Gamma(, 1), Zi ∼ Gamma(i , 1), i = 1, 2. The marginal distributions are Gamma( + i , 1), i = 1, 2. For 1 = 2 = 0, a singular bivariate gamma distribution appears. From the GGC point of view, Cherian's distribution is more attractive than the more popular Wicksell distribution. (c) Shot-noise models: There exist very few probability models leading to the univariate gamma distribution with general shape parameter in R+ . One possible model is the Cox–Ingersoll model for a diffusion process. Another is a shot-noise model dating back to Bartlett (1957). Let T1 , T2 , T3 , . . . be the random points of a Poisson point process on (0, ∞) with rate . Let V1 , V2 , V3 , . . . be associated independent exponentially distributed variables with mean a. Let X=
∞
exp(− Ti )Vi .
i=1
Then X represents the total effect shots of random magnitude Vi at Ti have at the origin 0. The distance between Ti and 0 equals Ti , so there is an exponential reduction of the effect depending on distance. By splitting the real line into small disjoint intervals of equal length and considering the Poisson process as a collection of i.i.d. Bernoulli variables, it can be shown that the LT of the distribution of X is given by ∞ X (s) = exp (V (e− t s) − 1) dt , 0
where V is the LT of the distribution of the variables Vi . Since these are exponentially distributed with mean a, V (s) = (1 + as)−1 . The substitution u = e− t then yields after some calculation
X (s) = (1 + as)−/ , i.e. X has a Gamma distribution with shape parameter / . This model offers many novel bivariate (and multivariate) generalizations. We may set exp(− 1 T)VT and X2 = exp(− 2 T)WT , X1 = T∈P1
T∈P2
where P1 and P2 are two dependent Poisson point processes on (0, ∞) and VT and WT (for the same T) are two dependent exponentially distributed variables. In an extreme case P1 = P2 . Further, the variables VT and WT are in extreme cases proportional to each other or independent. In particular, if P0 , P1 , and P2 are independent Poisson point processes with rates 0 , 1 , and 2 and (superposition)
P1 = P0 ∪ P1 ,
P2 = P0 ∪ P2 ,
WT = const × VT
and
we obtain, with a = E(VT ), b = E(WT ), the bivariate LT
X1 ,X2 (s, t) =
1
1 (1 + as)
1 /
1 2 /
(1 + bt)
(1 + as + bt)0 /
,
i.e. Cherian's bivariate gamma distribution for (X1 , X2 ) appears.
1 = 2 = ,
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7. Bivariate GGCs The natural building distribution for bivariate GGCs seems to be Cherian's distribution rather than Wicksell's distribution. We may think of adding two-dimensional shot-noise effects induced by different independent Poisson point processes. Let Y1 , Y2 , Y3 , . . . be independent gamma variables with unit scale and different shape parameters. Consider a random vector (X1 , X2 ) with X1 =
N
c1i Yi
X2 =
and
i=1
N
c2i Yi ,
i=1
where cki ⱖ 0, k = 1, 2. The bivariate LT is
(s, t) =
N
1
i=1
(1 + c1i s + c2i t)ui
.
(5)
Remark 4. If c1i × c2i = 0 for all i, then X1 and X2 are independent. If N = 1 (and c11 , c21 > 0,) the distribution is singular. Definition 3. A bivariate GGC (BVGGC) is a limit distribution of distributions with LTs of the form (5). For a BVGGC we have that * log (s, t) * log (s, t) Im ⱖ 0, Im ⱖ0 *s *t
for Im s > 0, Im t > 0.
However, no nice converse, as in the univariate case, has been found. 8. Bivariate HCM densities Here we consider one possible definition of bivariate HCM-densities. Consider densities of the form −1 2 −1
f (x1 , x2 ) = Cx1 1
x2
1 , (1 + c1 x1 + c2 x2 )
x1 , x2 > 0.
These are formed with the help of the simple BVGGC LTs. Now make a hyperbolic substitution in each coordinate and consider the product f (u1 v1 , u2 v2 )f
u1 u2 , v1 v2
2(1 −1) 2(2 −1) u2 (1 + c1 u1 v1
= C 2 u1
2(1 −1) 2(2 −1) u2 (1 + c12 u21
= C 2 u1
u1 u2 − + c2 u2 v2 )− 1 + c1 + c2 v1 v2
+ c2 u22 + c1 u1 w1 + c2 u2 w2 + c1 c2 u1 u2 w3 )− ,
−1 where w1 = v1 + v−1 1 , w2 = v2 + v2 , w3 = v1 /v2 + v2 /v1 . The last expression is CM with respect to w1 , w2 , w3 for fixed u1 and u2 , i.e. whenever we differentiate with respect to wk , k = 1, 2, 3, the sign is changed. We take this as a definition of BVHCM.
Definition 4. A bivariate density f (x1 , x2 ) on (0, ∞)2 is called BVHCM if f (u1 v1 , u2 v2 )f (u1 /v1 , u2 /v2 ) is CM in w1 , w2 , and w3 for all fixed u1 , u2 > 0. The BVHCM-class of densities contains many well-known distributions. The class is closed with respect to multiplication of densities with suitable normalization. It also has the nice property (X, Y) ∼ BVHCM ⇒ (X q , Y q ) ∼ BVHCM
for |q| ⱖ 1.
The author has long believed that it is also closed with respect to multiplication of independent random vectors in the following sense: (X, Y), (X , Y ) ∼ BVHCM ⇒ (XX , YY ) ∼ BVHCM.
(6)
This result was derived in 1988 and put as a brief remark in the paper Bondesson (1990). However, when in 2007 inspecting the proof in hand-written notes, the author found an error. It is an open problem whether this result holds. Another strongly related question is whether the LT of a BVHCM-density always is BVHCM. The answer is positive if (6) is true. The bivariate HCM- and GGC-theory is incomplete and partly a failure story hitherto.
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9. Conclusions • The univariate GGC and HCM theories are smooth and rather complete with few remaining open problems. • The bivariate case offers many problems and difficulties and needs more research. The generalization to the multivariate case is then immediate. • In the multivariate case there is recent research but in other directions, e.g. Barndorff-Nielsen et al. (2006) and others. Acknowledgements The author thanks two referees for suggestions for improvement. References Barndorff-Nielsen, O.E., Maejima, M., Sato, K., 2006. Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations. Bernoulli 12, 1–34. Bartlett, M.S., 1957. Contribution to discussion of papers by Dr. Gani and Mr. Kendall. J. Roy. Statist. Soc. Ser. B 19, 220–221. Bondesson, L., 1990. Generalized gamma convolutions and complete monotonicity. Probab. Theory Related Fields 78, 321–333. Bondesson, L., 1992. Generalized Gamma Convolutions and Related Classes of Distributions and Densities. Lecture Notes in Statistics, vol. 76. Springer, New York. Bondesson, L., 1999. A problem concerning stable distributions. U.U.D.M Report 1999:1, Department of Mathematics, Uppsala University. Bondesson, L., Grandell, J., Peetre, J., 2008. The life and work of Olof Thorin (1912–2004). Proc. Estonian Acad. Sci. Phys. Math. 57, 18–25. Cherian, K.C., 1941. A bi-variate correlated gamma-type distribution. J. Indian Math. Soc. 5, 133–144. Feller, W., 1966. An Introduction to Probability Theory and its Applications, vol. II. Wiley, New York. Hutchinson, T.T., Lai, C.D., 1990. Continuous Bivariate Distribution, Emphasising Applications. Rumsby Scientific Publishing, Adelaide. Kibble, W.F., 1941. A two-variate gamma type distribution. Sankhya Ser. A 5, 137–150. Nelsen, R.G., 1999. An Introduction to Copulas. Lecture Notes in Statistics, vol. 139. Springer, New York. Steutel, F.W., van Harn, K., 2004. Infinite Divisibility of Probability Distributions on the Real Line. Marcel Dekker, New York. Thorin, O., 1977. On the infinite divisibility of the lognormal distribution. Scand. Actuar. J. 1977, 121–148. Vere-Jones, D., 1967. The infinite divisibility of a bivariate gamma distribution. Sankhya Ser. A 29, 421–422. Wicksell, S.D., 1933. On correlation functions of type III. Biometrika 25, 121–133.
Contents lists available at ScienceDirect
Journal of Statistical Planning and Inference journal homepage: w w w . e l s e v i e r . c o m / l o c a t e / j s p i
On univariate and bivariate generalized gamma convolutions Lennart Bondesson∗ Department of Mathematics and Mathematical Statistics, Umeå University, SE-90187 Umeå, Sweden
A R T I C L E
I N F O
A B S T R A C T
This paper has two parts. In the first part some results for generalized gamma convolutions (GGCs) are reviewed. A GGC is a limit distribution for sums of independent gamma variables. In the second part, bivariate gamma distributions and bivariate GGCs are considered. New bivariate gamma distributions are derived from shot-noise models. The remarkable property hyperbolic complete monotonicity (HCM) for a function is considered both in the univariate case and in the bivariate case. © 2009 Elsevier B.V. All rights reserved.
Available online 22 May 2009 MSC: 60E05 60E07 60G55 Keywords: Bivariate gamma distribution Generalized gamma convolution Hyperbolic complete monotonicity (HCM) Shot-noise models
1. Introduction A generalized gamma convolution (GGC) is a limit distribution for sums of independent gamma distributed random variables. The gamma distributions may have different shape and scale parameters. The GGC-class, which was introduced by Thorin (1977), is rich and it has many remarkable properties. In particular, it contains all probability densities f on (0, ∞) that are hyperbolically completely monotone (HCM); cf. Section 3. The HCM-densities turn out to be limits of densities of the form
f (x) = Cx−1
N
(1 + ci x)−i ,
x > 0 ( > 0, ci > 0, i > 0).
i=1
Apparently, the HCM-class is closed with respect to multiplication of densities. It is also closed with respect to multiplication of independent random variables with densities in the class. Many standard distributions such as the lognormal, the generalized gamma, and the generalized Beta distributions of the second kind are HCM. In Sections 2–5 a review of the main GGC-results is given. The full GGC-theory is presented in Bondesson (1992) and also in Steutel and van Harn (2004). The paper Bondesson et al. (2008), which honors Olof Thorin, gives historical information. In Section 6, bivariate gamma distributions are considered. There are many possibilities and some new ones are presented. These derive from shot-noise models. Finally in Sections 7 and 8, attempts are made to generalize the univariate GGC-theory to the bivariate case but there are some difficulties. The natural building distribution for bivariate GGCs is Cherian's (1941) gamma distribution rather than Wicksell's (1933) gamma distribution. A bivariate HCM-condition is presented. Conclusions are given in Section 9.
∗ Tel.: +46 90 7866529; fax: +46 90 7865222. E-mail address: [email protected] 0378-3758/$ - see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.jspi.2009.05.015
3760
L. Bondesson / Journal of Statistical Planning and Inference 139 (2009) 3759 -- 3765
2. Generalized gamma convolutions In this section the GGCs are introduced. Some complex analysis is used. As is well known, a gamma distribution is a probability distribution on (0, ∞) with probability density function (pdf) f (x) =
tu xu−1 −tx e , (u)
x>0
(u > 0, t > 0).
The distribution is denoted by Gamma(u, 1/t). It has the Laplace transform (LT)
(s) =
∞
0
e−sx f (x) dx =
t t+s
u =
1 (1 + s/t)u
.
Now convolve gamma distributions with different shape parameters u and scale parameters 1/t. Equivalently, add independent gamma distributed random variables. Then a distribution with LT ⎛ ⎞ ui N N ti ti ⎝ ⎠ (s) = = exp ui log ti + s ti + s i=1
(1)
i=1
appears. Taking also limits, we obtain LTs of the form
(s) = exp −as +
log (0,∞)
t U(dt) , t+s
(2)
where a ⱖ 0 and U(dt) is a nonnegative measure on (0, ∞). The parameter a is the left-extremity of the distribution. In particular, setting in (1), ti ≡ N, ui ≡ a, and letting N → ∞, we get the LT e−as , i.e. a distribution degenerate at a. Definition 1 (Thorin, 1977). A generalized gamma convolution is a probability distribution on [0, ∞) with LT of the form (2) and 1 ∞ such that 1 t−1 U(dt) < ∞ and 0 | log t|U(dt) < ∞. The last two conditions ensure the convergence of the integral in (2). Since every gamma distribution is infinitely divisible, cf., e.g. Feller (1966, p. 177), so is every GGC. To be seen later on, many well-known distributions are in the GGC-class. For the LT of a GGC we have
(s) d log (s) = = −a − ds (s)
(0,∞)
U(dt) . t+s
(3)
The right-hand side of (3) is a function that is analytic for s ∈ C\(−∞, 0]. Moreover, the imaginary part of it is nonnegative for Im s > 0. This result leads to a characterization of LTs of GGCs.
Theorem 1. An LT (s) of a distribution on [0, ∞) corresponds to a GGC if and only if it is analytic in C\(−∞, 0] and Im[ (s)/ (s)] ⱖ 0 for Im s > 0, or, equivalently, Im[ (s)(s)] ⱖ 0 for Im s > 0. The deep `if' part of this theorem is a consequence of the Pick–Nevanlinna theory or more generally a consequence of Cauchy's integral theorem in complex analysis; cf. Bondesson (1992, pp. 20–21). The theorem implies that for instance the LTs
(s) = e−s
and
(s) = (1 + s )− (0 < ⱕ 1, > 0)
(and products of such LTs) correspond to GGCs. The first GGC is a stable distribution and the second one is a Linnik (or Mittag–Leffler) distribution on (0, ∞). 3. The HCM-condition and a consequence An important but somewhat curious condition is presented in this section. It has far-reaching consequences. Definition 2. A function f (x) ⱖ 0 on (0, ∞) is said to be hyperbolically completely monotone if, for each fixed u > 0, h(w) = f (uv)f (u/v)
with w = v + v−1
is completely monotone (CM) as a function of w, i.e. if (−1)n h(n) (w) ⱖ 0 on (0, ∞) for all integers n ⱖ 0.
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Equivalently, by a theorem of Bernstein, cf., e.g. Feller (1966, p. 416), f is HCM if f (uv)f (u/v) = exp(−u(v + v−1 ))Ku (d),
(4)
[0,∞)
where Ku (d) is a nonnegative measure for each u > 0. The HCM-condition first appeared in Thorin's (1977) work for the lognormal distribution though in a special case and in a different additive form. The name is motivated by the hyperbolic substitution x = uv, y = u/v. The HCM-condition is often easy to verify as will be seen in Section 4. An important result is now presented. Theorem 2 (Main HCM-theorem). If a probability density f (x) is HCM, then it is a GGC as well. Sketch of proof. We have ∞ ∞ (s)(s) = − x e−sx e−s¯y f (x)f (y) dx dy. 0
0
Now make the hyperbolic substitution x = uv, y = u/v, and use the representation (4). In that way it can be verified that Im[ (s)(s)] ⱖ 0 for Im s > 0 and hence the pdf f (x) is a GGC by Theorem 1. However, there are complications: the double integral is not well-defined for all complex s with Im s > 0. A full proof is given in Bondesson (1992, pp. 56, 57 and 71). Remark 1. If the cdf F(x) = f (x) is also a GGC.
x 0
f (y) dy is HCM, or more generally if, for any fixed > 0, the integral
x
0 (x − y)
−1
f (y) dy is HCM, then
4. The HCM-class of probability densities In this section there are first presented several examples of probability densities on (0, ∞) that are HCM and hence GGCs. Then a representation of the HCM-densities is given. Some properties of the HCM-class of densities are also expounded. The following well-known probability densities on (0, ∞) are HCM. For all the densities, C is a normalizing constant: (a) Gamma distribution: f (x) = Cx−1 e−cx
(, c > 0).
(b) Beta distribution, 2nd kind: f (x) = Cx−1
1 (1 + cx)
(, , c > 0).
(c) Generalized inverse Gaussian (GIG) distribution: 1 f (x) = Cx−1 exp −c1 x − c2 ( ∈ R, c1 , c2 > 0). x (d) Generalized gamma distribution:
f (x) = Cx−1 e−cx
(0 < ⱕ 1, , c > 0).
(e) Generalized Beta distribution, 2nd kind: f (x) = Cx−1
1 (1 + cx )
(0 < ⱕ 1, , , c > 0).
For the densities in (a)–(c) it is easy to verify that they are HCM. For instance, for the density in (b) we get f (uv)f (u/v) = C 2 u2(−1)
1 (1 + c2 u2 + cu(v + v−1 ))
and apparently the right-hand side is completely monotone with respect to w = v + v−1 . The densities in (d) and (e) can be verified to be HCM from the fact that v + v− has a CM derivative with respect to w. This property is a consequence of the non-trivial representation sin() ∞ t d (v + v− ) = dt. 2 + tw dw 1 + t 0 A very complete description of the HCM-class of densities is provided by the following theorem, the proof of which is not simple.
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Theorem 3. The HCM-class of probability densities is the class of densities that are pointwise limits of densities of the product form −i (, ci , i > 0). The HCM-densities are unimodal. Cx−1 N i=1 (1 + ci x) One may notice that, except for the factor Cx−1 , densities of this form are LTs of GGCs. Moreover, if f1 and f2 are HCM-densities then so is Cf 1 f2 . The HCM-class has nice properties, which is shown by the next theorem. Theorem 4. The HCM-class is closed with respect to multiplication and division of independent random variables and also with respect to formation of powers with exponent q, |q| ⱖ 1, of random variables, i.e. X, Y ∼ HCM (independent) ⇒ X · Y ∼ HCM,
X/Y ∼ HCM
and X q ∼ HCM.
Sketch of proof. Set f (x) = fX (x) and g(y) = yf Y (y) (also HCM). Let h be the density of X/Y. Then ∞ ∞ s h(st)h(s/t) = f (stx)f y g(x)g(y) dx dy. t 0 0 By the hyperbolic substitution x = uv, y = uv and the Bernstein representation (4), it can be verified that the double integral is CM with respect to t + t−1 for each fixed s > 0. Hence h is HCM as desired. The result on powers follows from the representation (4) and the fact that v + v− with = 1/q has a CM derivative with respect to w = v + v−1 . Since the generalized Beta distribution of the 2nd kind is the distribution of (X1 /X2 )1/ , where X1 ∼ Gamma and X2 ∼ Gamma are independent random variables, Theorem 4 immediately implies that its density is HCM. Moreover, if X1 , X2 , . . . , Xn ∼ Gamma are independent, then the density of X1 · X2 · · · Xn is HCM and hence a GGC and infinitely divisible. By the central limit theorem, it follows that the lognormal density is HCM. The lognormal distribution is also a limit of generalized gamma distributions as → 0. Remark 2. If X ∼ HCM, then log X has a logconcave, i.e. strongly unimodal, density. It is well known that when independent random variables with logconcave densities are added the sum has a logconcave density as well. This casts new light on Theorem 4 and puts it into a better known context. 5. More about GGCs The HCM-condition provides a nice real characterization of LTs of GGCs. Theorem 5. An LT (s) of a distribution on [0, ∞) corresponds to a GGC if and only if (s) is HCM. The `only if' part of Theorem 5 follows from the definition of a GGC and the fact that the HCM-property is preserved when HCM-functions are multiplied. The `if' part is mainly a consequence of Theorem 3. For example, Theorem 5 shows that the distributions with LTs e−s and (1 + s )− are GGCs since these LTs are HCM. In fact, even distributions with LTs of the form ⎛ ⎞ − M k (s) = ⎝1 + ck s ⎠ (ck > 0, 0 < k ⱕ 1) k=1
can be verified to be GGCs by Theorem 5. The `real' Theorem 5 makes it possible to avoid all complex analysis for the GGCs. It can be verified from Theorem 5 that X ∼ GGC,
Y ∼ HCM (independent) ⇒ X/Y ∼
and
XY ∼ GGC.
The univariate GGC- and HCM-theory has been a success story in the sense that the theory is rather complete. Distributions on the whole real line and on the nonnegative integers are also considered in Bondesson (1992). Remark 3. There remain a few very hard problems: (i) Is the density function of each stable distribution on (0, ∞) with index ⱕ 12 HCM? This problem is treated in Bondesson (1999) with a positive answer. However, no strict proof was found. (ii) If X ∼ GGC, does it then follow that X q ∼ GGC for q ⱖ 1? √
−z (iii) Let f (s) = ( 12 )/ ( 12 + s), where (z) = 12 z(z − 1)(z/2)−z/2 (z) and (z) = ∞ n=1 n . Is f (s) HCM? If so, Riemann's famous hypothesis about the complex zeros of the -function is true.
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6. Bivariate gamma distributions In this section bivariate gamma distributions for (X1 , X2 ) are considered. There are many possibilities; cf., e.g. Hutchinson and Lai (1990). The popular copula technique, see, e.g. Nelsen (1999), which is a broad way to find multivariate distributions with given marginals, is not discussed here. Below two established bivariate gamma distributions are briefly described. Then a general probability model that is capable of producing many different bivariate gamma distributions is introduced. (a) Wicksell–Kibble distribution (Wicksell, 1933; Kibble, 1941): This distribution has bivariate LT (s, t) = E(exp(−sX 1 − tX 2 )) of the form
(s, t) = (1 + as + bt + ab(1 − 2 )st)− ,
s, t ⱖ 0.
Here a, b ⱖ 0, | | ⱕ 1, > 0. For a=b=1 the distribution appears in standardized form in which case the marginal distributions are Gamma(, 1). The coefficient ab(1 − 2 ) cannot be replaced by a coefficient larger than ab since then (s, t) fails to be an LT. The idea behind this distribution is to look at (Z12 , Z22 ), where (Z1 , Z2 ) has a bivariate normal distribution, in which case = 12 . The distribution is infinitely divisible (Vere-Jones, 1967). (b) Cherian's distribution (Cherian, 1941): Let
(s, t) =
1 1 1 . (1 + as)1 (1 + bt)2 (1 + as + bt)
For a = b = 1 this LT corresponds to, with Y, Z1 , Z2 independent, X1 = Y + Z1 ,
X2 = Y + Z2 ,
where Y ∼ Gamma(, 1), Zi ∼ Gamma(i , 1), i = 1, 2. The marginal distributions are Gamma( + i , 1), i = 1, 2. For 1 = 2 = 0, a singular bivariate gamma distribution appears. From the GGC point of view, Cherian's distribution is more attractive than the more popular Wicksell distribution. (c) Shot-noise models: There exist very few probability models leading to the univariate gamma distribution with general shape parameter in R+ . One possible model is the Cox–Ingersoll model for a diffusion process. Another is a shot-noise model dating back to Bartlett (1957). Let T1 , T2 , T3 , . . . be the random points of a Poisson point process on (0, ∞) with rate . Let V1 , V2 , V3 , . . . be associated independent exponentially distributed variables with mean a. Let X=
∞
exp(− Ti )Vi .
i=1
Then X represents the total effect shots of random magnitude Vi at Ti have at the origin 0. The distance between Ti and 0 equals Ti , so there is an exponential reduction of the effect depending on distance. By splitting the real line into small disjoint intervals of equal length and considering the Poisson process as a collection of i.i.d. Bernoulli variables, it can be shown that the LT of the distribution of X is given by ∞ X (s) = exp (V (e− t s) − 1) dt , 0
where V is the LT of the distribution of the variables Vi . Since these are exponentially distributed with mean a, V (s) = (1 + as)−1 . The substitution u = e− t then yields after some calculation
X (s) = (1 + as)−/ , i.e. X has a Gamma distribution with shape parameter / . This model offers many novel bivariate (and multivariate) generalizations. We may set exp(− 1 T)VT and X2 = exp(− 2 T)WT , X1 = T∈P1
T∈P2
where P1 and P2 are two dependent Poisson point processes on (0, ∞) and VT and WT (for the same T) are two dependent exponentially distributed variables. In an extreme case P1 = P2 . Further, the variables VT and WT are in extreme cases proportional to each other or independent. In particular, if P0 , P1 , and P2 are independent Poisson point processes with rates 0 , 1 , and 2 and (superposition)
P1 = P0 ∪ P1 ,
P2 = P0 ∪ P2 ,
WT = const × VT
and
we obtain, with a = E(VT ), b = E(WT ), the bivariate LT
X1 ,X2 (s, t) =
1
1 (1 + as)
1 /
1 2 /
(1 + bt)
(1 + as + bt)0 /
,
i.e. Cherian's bivariate gamma distribution for (X1 , X2 ) appears.
1 = 2 = ,
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7. Bivariate GGCs The natural building distribution for bivariate GGCs seems to be Cherian's distribution rather than Wicksell's distribution. We may think of adding two-dimensional shot-noise effects induced by different independent Poisson point processes. Let Y1 , Y2 , Y3 , . . . be independent gamma variables with unit scale and different shape parameters. Consider a random vector (X1 , X2 ) with X1 =
N
c1i Yi
X2 =
and
i=1
N
c2i Yi ,
i=1
where cki ⱖ 0, k = 1, 2. The bivariate LT is
(s, t) =
N
1
i=1
(1 + c1i s + c2i t)ui
.
(5)
Remark 4. If c1i × c2i = 0 for all i, then X1 and X2 are independent. If N = 1 (and c11 , c21 > 0,) the distribution is singular. Definition 3. A bivariate GGC (BVGGC) is a limit distribution of distributions with LTs of the form (5). For a BVGGC we have that * log (s, t) * log (s, t) Im ⱖ 0, Im ⱖ0 *s *t
for Im s > 0, Im t > 0.
However, no nice converse, as in the univariate case, has been found. 8. Bivariate HCM densities Here we consider one possible definition of bivariate HCM-densities. Consider densities of the form −1 2 −1
f (x1 , x2 ) = Cx1 1
x2
1 , (1 + c1 x1 + c2 x2 )
x1 , x2 > 0.
These are formed with the help of the simple BVGGC LTs. Now make a hyperbolic substitution in each coordinate and consider the product f (u1 v1 , u2 v2 )f
u1 u2 , v1 v2
2(1 −1) 2(2 −1) u2 (1 + c1 u1 v1
= C 2 u1
2(1 −1) 2(2 −1) u2 (1 + c12 u21
= C 2 u1
u1 u2 − + c2 u2 v2 )− 1 + c1 + c2 v1 v2
+ c2 u22 + c1 u1 w1 + c2 u2 w2 + c1 c2 u1 u2 w3 )− ,
−1 where w1 = v1 + v−1 1 , w2 = v2 + v2 , w3 = v1 /v2 + v2 /v1 . The last expression is CM with respect to w1 , w2 , w3 for fixed u1 and u2 , i.e. whenever we differentiate with respect to wk , k = 1, 2, 3, the sign is changed. We take this as a definition of BVHCM.
Definition 4. A bivariate density f (x1 , x2 ) on (0, ∞)2 is called BVHCM if f (u1 v1 , u2 v2 )f (u1 /v1 , u2 /v2 ) is CM in w1 , w2 , and w3 for all fixed u1 , u2 > 0. The BVHCM-class of densities contains many well-known distributions. The class is closed with respect to multiplication of densities with suitable normalization. It also has the nice property (X, Y) ∼ BVHCM ⇒ (X q , Y q ) ∼ BVHCM
for |q| ⱖ 1.
The author has long believed that it is also closed with respect to multiplication of independent random vectors in the following sense: (X, Y), (X , Y ) ∼ BVHCM ⇒ (XX , YY ) ∼ BVHCM.
(6)
This result was derived in 1988 and put as a brief remark in the paper Bondesson (1990). However, when in 2007 inspecting the proof in hand-written notes, the author found an error. It is an open problem whether this result holds. Another strongly related question is whether the LT of a BVHCM-density always is BVHCM. The answer is positive if (6) is true. The bivariate HCM- and GGC-theory is incomplete and partly a failure story hitherto.
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9. Conclusions • The univariate GGC and HCM theories are smooth and rather complete with few remaining open problems. • The bivariate case offers many problems and difficulties and needs more research. The generalization to the multivariate case is then immediate. • In the multivariate case there is recent research but in other directions, e.g. Barndorff-Nielsen et al. (2006) and others. Acknowledgements The author thanks two referees for suggestions for improvement. References Barndorff-Nielsen, O.E., Maejima, M., Sato, K., 2006. Some classes of multivariate infinitely divisible distributions admitting stochastic integral representations. Bernoulli 12, 1–34. Bartlett, M.S., 1957. Contribution to discussion of papers by Dr. Gani and Mr. Kendall. J. Roy. Statist. Soc. Ser. B 19, 220–221. Bondesson, L., 1990. Generalized gamma convolutions and complete monotonicity. Probab. Theory Related Fields 78, 321–333. Bondesson, L., 1992. Generalized Gamma Convolutions and Related Classes of Distributions and Densities. Lecture Notes in Statistics, vol. 76. Springer, New York. Bondesson, L., 1999. A problem concerning stable distributions. U.U.D.M Report 1999:1, Department of Mathematics, Uppsala University. Bondesson, L., Grandell, J., Peetre, J., 2008. The life and work of Olof Thorin (1912–2004). Proc. Estonian Acad. Sci. Phys. Math. 57, 18–25. Cherian, K.C., 1941. A bi-variate correlated gamma-type distribution. J. Indian Math. Soc. 5, 133–144. Feller, W., 1966. An Introduction to Probability Theory and its Applications, vol. II. Wiley, New York. Hutchinson, T.T., Lai, C.D., 1990. Continuous Bivariate Distribution, Emphasising Applications. Rumsby Scientific Publishing, Adelaide. Kibble, W.F., 1941. A two-variate gamma type distribution. Sankhya Ser. A 5, 137–150. Nelsen, R.G., 1999. An Introduction to Copulas. Lecture Notes in Statistics, vol. 139. Springer, New York. Steutel, F.W., van Harn, K., 2004. Infinite Divisibility of Probability Distributions on the Real Line. Marcel Dekker, New York. Thorin, O., 1977. On the infinite divisibility of the lognormal distribution. Scand. Actuar. J. 1977, 121–148. Vere-Jones, D., 1967. The infinite divisibility of a bivariate gamma distribution. Sankhya Ser. A 29, 421–422. Wicksell, S.D., 1933. On correlation functions of type III. Biometrika 25, 121–133.