24 As noted by Klebanov et al. 1985, the exponential distribution plays the
role of degenerate distribution for multivariate geometric sums. Here we let Expµ denote the distribution with GCF
Gs = s
⊤
µ for s
⊤
µ 1
which for µ , 0 corresponds to a unit exponential variable multiplied onto the vector µ, while µ = 0 corresponds to the degenerate distribution at 0.
We refer to GCFs of the form 2.2 as the degenerate case. Since SX = 0 for X
∼ Expµ, we may interpret the operator SX = VarX − EXE
⊤
X as a general measure of the deviation of the random vector X or its distribution
from exponentiality. The multivariate Tweedie characteristic as special cases of exponential
dispersion models can also be applied to the family of geometric dispersion models, which can be so-called multivariate geometric Tweedie models. The
theoretical aspects of this new class of distribution need to be explored and would become an interesting work.
2.2.3. Multivariate Discrete Dispersion Models
Discrete dispersion models are two-parameter families obtained by com- bining convolution with a factorial tilting operation. Using the factorial
cumulant generating function, Jørgensen and Kokonendji 2016 introduced a dilation operator, generalizing binomial thinning, which may be viewed
as a discrete analogue of scaling. The discrete Tweedie factorial dispersion models are closed under dilation, which in turn leads to a discrete Tweedie
asymptotic framework where discrete Tweedie models appear as dilation limits.
A. Definition and Properties
Before looking to the definition, let us recall that the CGF 2.30 satisfies the linear transformation law
κt, AX + b = κAt; X + bt where A is a diagonal matrix.
Definition 2.2.6. If X is a k-variate random vector, and s a k-vector with non-
negative elements, we use the notation f
X
= f
X
1
1
. . . f
X
k
k
. The multivariate factorial cumulant generating function FCGF is defined by
Ct = Ct; X = log E
h
1 + t
X
i =
κ log1 + t; X for t −1
see Johnson et al., 2005, p. 4, where 1 is a vector of ones, and the inequality t −1 is understood element-wise. The effective domain of
ζ is define by DC = {t −1 : Ct
∞}.
25 We should note that C, like
κ, characterizes convolution additively, i.e.
for independent random variables X and Y we have Ct; X + Y = Ct
, X + Ct, Y.
The CGF
κ is a real analytic convex function and strictly convex unless X is
degenerate. Hence C is also real analytic, and the domain DC, like Ψ, is an interval. The derivative C
′
t = κ
′
log1 + t 1 + t has the same sign as
κ
′
log1 + t on intDC. Hence, by the convexity of κ, the FCGF C is either monotone or u-shaped.
When 0 ∈ int[DC], the derivatives C
n
0 = C
n
0; X are called the factorial-cumulants of X
. The first factorial-cumulant is the mean vector, C
′
0 =
κ
′
0 = EX ,
and the second factorial-cumulant is the k × k dispersion matrix
S X = C
′′
0 = CovX − diagEX
with entries S
ij
X = SX
i
for i = j CovX
i
, X
j
for i , j .
Jørgensen and Kokonendji 2011 introduced a new definition of multi- variate overunder dispersion based on the dispersion matrix, namely that
the random vector X is equidispersed if SX = 0, X is called overunderdispersed if the dispersion matrix SX is positivenegative semidefinite, i.e. SX has
at least one positivenegative eigenvalue, respectively. Also, we say that the dispersion of X is indefinite if SX has both positive and negative eigenvalues.
The dispersion matrix SX satisfies a scaling properties with respect to dilation. For a random vector X, the dilation linear combination c X with 1 × k
vector c is defined as follows: Ct; c · X = Cc
⊤
t; X
provided that the right-hand side is a univariate FCGF. The mean and dispersion matrix of a dilation linear combination are given by
Ec · X = cEX and
S c · X = cSXc
⊤
respectively. It follows that if c · X is equidispersed for some c , 0, then the dispersion matrix SX is singular. The reverse implication holds if the
vector c 0 is such that cSXc
⊤
= 0 . Similarly, for an ℓ × k matrix A ≥ 0 we
define A · X by Ct; A X = CA
⊤
t ; X
, again provided that the right-hand side is an FCGF.
26
B. Multivariate Poisson Tweedie models
Multivariate Poisson Tweedie models Jørgensen and Kokonendji, 2011 are special cases of multivariate discrete dispersion models. The models
are considered as a new class of multivariate Poisson-Tweedie mixtures, which is based on the multivariate Tweedie distributions of Jørgensen and
Martínez 2013. Consider the k-variate Tweedie distribution Y ∼ Tw
p
µ, Σ with mean vector µ and covariance matrix
CovY = Diagµ
p 2
Σ Diagµ
p 2
. 2.31
Table 2.3: Summary of Multivariate Poisson Tweedie Models
Model p
Type
Multivariate Neyman Type A p = 1
+ Multivariate Poisson-Compound Poisson 1
p 32 +
Multivariate Pólya-Aeppli p = 3
2 +
Multivariate Poisson-compound Poisson 3
2 p 2 +
Multivariate negative binomial p = 2
+ Multivariate factorial discrete stable
p 2
+ Multivariate Poisson-inverse Gaussian
p = 3 +
The multivariate Poisson-Tweedie model PT
p
µ, Σ is defined as a Poisson mixture
X |Y ∼ independent PoY
i
for i = 1 , . . . , k,
where X
1
, . . . , X
k
are assumed conditionally independent given Y . The mul-
tivariate Poisson-Tweedie model has univariate Poisson-Tweedie margins, X
i
∼ PT
p
µ
i
, σ
ii
where σ
ij
denote the entries of Σ . The mean vector is µ and
the dispersion matrix is 2.31 positive-definite making the distribution overdispersed. The covariance matrix for X has the form
CovX = Diagµ + Diagµ Σ Diagµ,
making it straightforward to fit multivariate Poisson-Tweedie regression models using quasi-likelihood. The multivariate Poisson-Tweedie model
satisfies the following dilation property:
DiagcPT
p
µ, Σ = PT
p
h
Diagcµ, Diagc
1−p2
Σ
Diagc
1−p2
i ,
where c is a k-vector with positive elements. 2.2.4. Applications
Here we give some examples from literatures which have emphasized the application of a certain dispersion model. The followings are illustra-
tive descriptions of the applications of exponential, geometric and discrete dispersion models.
