21 weight parameters
λ
ii
and λ
ij
are defined by λ
ii
= γ
ii
κ
α
θ
i
for i = 1 , . . . , k
2.26 and
λ
ij
= γ
ij
κ
1 2
α
θ
i
, θ
j
for i j,
2.27 where
κ
1 2
α
is a function defined by κ
1 2
α
θ
i
, θ
j
= α − 1
α θ
i
α − 1
α2
θ
j
α − 1
α2
. Using the parameters
λ
ii
the marginal mean are of the form λ
ii
µ
i
= λ
ii
κ
α
θ
i
α θ
i
= λ
ii
θ
i
α − 1
α−1
for i = 1 , . . . , k
2.28 and the variances are
λ
ii
κ
α
θ
i
αα − 1 θ
2 i
= λ
ii
µ
p i
for i = 1 , . . . , k.
This defines the multivariate additive Tweedie random vector X ∼ Tw
∗p k
µ, Λ, with mean vector DiagΛµ where its elements are defined by 2.28, i.e.
DiagΛµ = λ
11
θ
1
α − 1
α−1
, . . . , λ
kk
θ
k
α − 1
α−1 ⊤
and the covariance matrix for X has the form Λ ⊙ Vµ, where the elements of Λ =
λ
ij i
,j=1,...,k
are defined in 2.26 and 2.27 and Vµ has entries
V
ij
= µ
i
µ
j p
2
,
then the covariance matrix of X is
Σ = Λ ⊙ Vµ =
λ
11
µ
p 1
λ
12
µ
1
µ
2 p
2
. . . λ
1k
µ
1
µ
k p
2
λ
21
µ
2
µ
1 p
2
λ
22
µ
p 2
. . . λ
2k
µ
1
µ
k p
2
... ...
. .. ...
λ
k1
µ
k
µ
1 p
2
λ
k2
µ
k
µ
1 p
2
. . . λ
kk
µ
p k
.
The multivariate additive Tweedie model Tw
∗p k
µ, Λ satisfies the follow- ing additive property:
Tw
∗p k
µ, Λ
1
+ Tw
∗p k
µ, Λ
2
= Tw
∗p k
µ, Λ
1
+ Λ
2
22 To obtain the reproductive form Tw
k
µ, Σ, we need to use the following duality transformation:
Tw
p k
µ, Σ = λ
−1
Tw
∗p k
µ, Λ. The distribution Tw
k
µ, Σ has mean vector µ as follow µ =
θ
1
α − 1
α−1
, . . . , θ
k
α − 1
α−1 ⊤
and covariance matrix Σ
⊙ Vµ = Diagµ
p 2
Σ Diagµ
p 2
where Σ = DiagΛ
−1
Λ DiagΛ
−1
. One can see Jørgensen and Martínez 2013 and Cuenin et al. 2016 for
more details on multivariate Tweedie models. An interesting behavior of negatively correlated multivariate Tweedie distribution bivariate case was
revealed from the simulation study done by Cuenin et al. 2016. They showed that for a large negative correlation the scatter of bivariate Tweedie
distribution depicted a curve which reminds us to the inverse exponential function, also the distribution lies only on the positive side of the Euclidean
space. These behaviour appears to be new because of positive support of the multivariate Tweedie. While for large positive correlation the scatter of
bivariate Tweedie distribution depicted a straight line with positive slope as commonly seen in the same case on multivariate Gaussian distribution.
2.2.2. Multivariate Geometric Dispersion Models
The dispersion models for geometric sums are defined as two-parameter families that combine geometric compounding with an operation called
geometric tilting, in much the same way that exponential dispersion models combine convolution and exponential tilting Jørgensen, 1997, Chap.3. The
univariate geometric dispersion model was introduced by Jørgensen and Kokonendji 2011.
A. Definition and Properties A multivariate geometric sum sQ Kalashnikov, 1997, page 3, indexed
by the probability matrix Q, can be defined by
S Q
, X =
N
1
Q
X
j=1
X
j1
, . . . ,
N
k
Q
X
j=1
X
jk
⊤
2.29
where X
1
, X
2
, . . . , are i.i.d. copies of the random vector X, independent of the geometric random vector NQ = N
1
Q , . . . , N
k
Q
⊤
, and Q = q
ij i
,j=1,...,k
is a matrix of covariance parameters to be defined. The geometric random vector
23
N Q has probability mass function to be defined. Note that each of the
sums 2.29 must be ordinary geometric sums with probability parameters q
11
, . . . , q
kk
.
The geometric random vector NQ has probability mass function:
P[n
1
Q , . . . , n
k
Q] = [
P
k i=1
n
i
Q] Q
k i=1
n
i
Q
k
Y
i=1
q
n
i
Q
i
1 − q
1
− ... − q
k
where n
i
Q ∈ N = {0,1,2,...} the set of natural numbers, q
i
0, i = 1, . . . , k, and
P
k i=1
q
1
1 Sreehari and Vasudeva, 2012. For this section, we denote again the ordinary CGF for a random k-vector
X by
κs = κs; X = log E
e
s
⊤
X
for s ∈ R
k
, 2.30
with effective domain Ψ = {s ∈ R
k
: κs ∞}.
Definition 2.2.5. If X is a k-variate random vector, the geometric cumulant function
GCF for X is given by Gs = Gs; X = 1 − e
−κs
for s ∈ DG, with domain DG = {s ∈ R
k
: Gs
1} = Ψ. Recall that a CGF is a real analytic convex function, which is strictly
convex unless X is degenerate concentrated on an affine subspace of R
k
. Hence, G is also real analytic, and the domain DG, like Ψ, is convex. In
fact, the gradient G
′
s = e
−κs
κ
′
s is proportional to κ
′
s in the interior
int[DG]. Hence, by the convexity of κ, the GCF G is either sloping or cup-shaped.
When 0 ∈ int[DG], the derivatives G
n
0 = G
n
0; X are called the geo- cumulants of X
. In particular, the first geo-cumulant is the mean vector, G
′
0 = κ
′
0 = EX .
The second geo-cumulant called the geo-covariance is the k × k matrix
S X = G
′′
0 = κ
′′
0 − κ
′
κ
′⊤
0 = VarX − EXE
⊤
X
which satisfies the inequalities
−EXE
⊤
X 6 SX 6 VarX .
The geo-covariance SX satisfies a scaling relation similar to the covariance, namely for any
ℓ × k matrix A we have S
AX = ASXA
⊤
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
∞}.
