15 ED
∗ 3
µ, Λ is defined via the joint CGF for the vector X as follows:
Ks
1
, s
2
, s
3
; µ, Λ = λ
12
κs
1
, s
2
; µ
1
, µ
2
+ λ
13
κs
1
, s
3
; µ
1
, µ
3
+ λ
23
κs
2
, s
3
; µ
2
, µ
3
+ λ
1
κs
1
, µ
1
+ λ
2
κs
2
, µ
2
+ λ
3
κs
3
, µ
3
This definition satisfies the requirement that each marginal distribution belongs to the univariate model ED
∗
µ, λ. For example, the CGF of the first margin is
Ks
1
, 0, 0; µ, Λ = λ
12
κs
1
, 0; µ
1
, µ
2
+ λ
13
κs
1
, 0; µ
1
, µ
3
+ λ
23
κ0, 0; µ
2
, µ
3
+ λ
1
κs
1
, µ
1
+ λ
2
κ0, µ
2
+ λ
3
κ0, µ
3
= λ
12
κs
1
, µ
1
+ λ
13
κs
1
, µ
1
+ λ
1
κs
1
, µ
1
While all three bivariate marginal distributions are of the form 2.16. For example, the CGF of the joint distribution of X
1
and X
2
is Ks
1
, s
2
, 0; µ, Λ = λ
12
κs
1
, s
2
; µ
1
, µ
2
+ λ
13
κs
1
, 0; µ
1
, µ
3
+ λ
23
κs
2
, 0; µ
2
, µ
3
+ λ
1
κs
1
, µ
1
+ λ
2
κs
2
, µ
2
+ λ
3
κ0, µ
3
= λ
12
κs
1
, s
2
; µ
1
, µ
2
+ λ
1
+ λ
13
κs
1
, µ
1
+ λ
2
+ λ
23
κs
2
, µ
2
, which has rate vector
µ
1
, µ
2 ⊤
and weight matrix given by the upper left 2 × 2 block of Λ.
Based on these considerations the construction for bivariate and trivari- ate cases, the multivariate exponential dispersion model ED
∗ k
µ, Λ for gen- eral k is defined as the following.
Definition 2.2.2. An additive k-variate exponential dispersion model ED
∗ k
µ, Λ with k × 1 rate vector µ = µ
1
, . . . , µ
k ⊤
and k × k weignt matrix Λ = {λ
ij
}
k i
,j=1
, is a family distribution with a joint cumulant generating function of the form
Ks; µ, Λ =
X
i j
λ
ij
κs
i
, s
j
; µ
i
, µ
j
+
k
X
i=1
λ
i
κs
i
, µ
i
, 2.18
where all weights λ
ij
and λ
i
are positive, with λ
ii
= P
j;j,i
λ
ij
+ λ
i
, The mean vector of 2.18 is
EX = diagΛµ,
where diag{Λ} is a k × k diagonal matrix. By arguments similar to those given in the trivariate case above it can be shown that each univariate margin
belongs to the univariate model ED
∗
µ, λ, and that all marginal distributions for a subset of the k variables are again of the form 2.18.
ED
∗ k
µ, Λ of 2.18 also satisfies the following generalized additive prop-
16 erty
ED
∗ k
µ, Λ
1
+ ED
∗ k
µ, Λ
2
= ED
∗ k
µ, Λ
1
+ Λ
2
. 2.19
From 2.18 we find that for i , j the ijth covariance is CovX
1
, X
j
= λ
ij
V µ
i
, µ
j
, where V
µ
i
, µ
j
= κ
′′
, 0; µ
i
, µ
j
denotes the second mixed derivative of κ·,·;µ
1
, µ
2
at zero. Generalizing 2.2, the covariance matrix for X may hence be ex- pressed as a Hadamard product,
VarX = Λ ⊙ Vµ, where the matrix unit variance function V now has diagonal elements V
µ
i
and off-diagonal elements V µ
i
, µ
j
. This construction done by Jørgensen 2013 gives us exactly the desired
number of parameters, namely k rates and kk + 1 2 covariance parameters.
C. Multivariate Reproductive Exponential Dispersion Models
The reproductive form of multivariate exponential dispersion model is constructed by applying the so-called duality transformation to a given mul-
tivariate additive exponential dispersion model. The reproductive form is particularly suited for continuous data, and includes the multivariate nor-
mal distribution as a special case, along with new multivariate forms of gamma, inverse Gaussian and other Tweedie distributions.
A multivariate exponential dispersion model in its reproductive form is parameterized by a k-vector of means µ and a symmetric positive-definite
k × k dispersion matrix Σ. We shall now derive the reproductive form of the bivariate exponential dispersion model by means of an analogy of the
duality transformation used in connection with 2.2. Hence we define random vector Y = Y
1
, Y
2 ⊤
as follow: Y
1
Y
2
= X
1
λ
11
X
2
λ
22
, with mean vector
EY =
µ
1
µ
2
and covariance matrix
CovY =
1 λ
11
V
11
µ
λ
12
λ
11
λ
22
V
12
µ
λ
12
λ
11
λ
22
V
21
µ
1 λ
22
V
22
µ
=
Σ ⊙ Vµ,
17 where Σ is the symmetric positive-definite matrix defined by
Σ =
1 λ
11
λ
12
λ
11
λ
22
λ
12
λ
11
λ
22
1 λ
22
.
We denote the model corresponding to Y by ED
2
µ, Σ which has 2 means and 3 variance-covariance parameters. We hence call this five-parameter
family a bivariate reproductive exponential dispersion model. To obtain a general multivariate reproductive form of exponential dis-
persion models, we use a generalization of the ordinary duality transfor- mation 2.2. By using a k × k diagonal matrix diagΛ from above, define
the reproductive multivariate exponential dispersion model ED
k
µ, Σ by the extended duality transformation
ED
k
µ, Σ = DiagΛ
−1
ED
∗ k
µ, Λ 2.20
The reproductive model ED
k
µ, Σ has mean vector µ and dispersion matrix Σ = diagΛ
−1
Λ diagΛ
−1
. It satisfies the following reproductive property, generalizing 2.19, namely that for Y
1
, . . . , Y
n
i.i.d. ED
k
µ, Σ, 1
n
k
X
i=1
Y
i
= ED
k
µ, Σn
The covariance matrix for ED
k
µ, Σ has the form Σ ⊙ Vµ, e.g. for k = 3:
Σ ⊙ Vµ =
σ
11
V
11
µ σ
12
V
12
µ σ
13
V
13
µ σ
21
V
21
µ σ
22
V
22
µ σ
23
V
23
µ σ
31
V
31
µ σ
32
V
32
µ σ
33
V
33
µ
=
1 λ
11
V
11
µ
λ
12
λ
11
λ
22
V
12
µ
λ
13
λ
11
λ
33
V
13
µ
λ
12
λ
11
λ
22
V
21
µ
1 λ
22
V
22
µ
λ
23
λ
22
λ
33
V
23
µ
λ
13
λ
11
λ
33
V
31
µ
λ
23
λ
22
λ
33
V
32
µ
1 λ
33
V
33
µ
According to the duality transformation 2.20, each additive exponential dispersion model ED
∗ k
µ, Λ has a corresponding reproductive counterpart. The inverse duality transformation is given by
ED
∗ k
µ, Λ = DiagΛ ED
k
µ, DiagΛ
−1
Λ DiagΛ
−1
by which the additive form ED
∗ k
µ, Λ may be recovered from the reproduc- tive form.
A summary of the additive and reproductive forms of exponential dis- persion model is shown in Table 2.1.
18 Table 2.1: Ordinary and multivariate exponential dispersion models
Form Type
Symbol Mean Vector Covariance Matrix
Additive Ordinary
ED
∗
µ, λ λµ
λVµ
Multivariate ED
∗ k
µ, Λ diagΛµ
Λ ⊙ Vµ
Reproductive Ordinary EDµ, σ
2
µ σ
2
V µ
Multivariate ED
k
µ, Σ µ
Σ ⊙ Vµ
To give a final remark to this part, we introduce a proposition below.
Proposition 2.2.1. An ordinary additive exponential dispersion model ED
∗
µ, λ is a particular case of multivariate additive exponential dispersion models ED
∗ k
µ, Λ with specific Λ, and also, an ordinary reproductive exponential dispersion model
EDµ, σ
2
is a particular case of multivariate reproductive exponential dispersion models ED
k
µ, Σ with specific Σ.
Proof. Let X be a multivariate additive exponential dispersion random vec- tor, i.e. X ∼ ED
∗ k
µ, Λ with Λ = {λ
ij
= λ}
i ,j=1,...,k
or
Λ =
λ λ . . . λ
λ λ . . . λ ... ... ... ...
λ λ . . . λ
= λ
1 1
. . . 1 1 1
. . . 1 ... ... ... ...
1 1 . . . 1
=
λ J
k
.
Then the mean vector of X is: EX = DiagΛµ = λµ
1
, . . . , λµ
k ⊤
= λµ
and its covariance matrix
Λ ⊙ Vµ =
λV
11
µ λV
12
µ . . . λV
1k
µ
λV
21
µ λV
22
µ . . . λV
2k
µ ...
... . ..
...
λV
k1
µ λV
k2
µ . . . λV
kk
µ
=
λVµ.
Those are the mean vector and the covariance matrix of an ordinary additive exponential dispersion model see Table 2.1, then we can write X ∼ ED
∗
µ, λ.
Let X be a multivariate reproductive exponential dispersion model X ∼
ED
k
µ, Σ with Σ = {σ
2 ij
= σ
2
}
i ,j=1,...,k
= σ
2
J
k
, then its covariance matrix is
Σ
⊙ Vµ =
σ
2
V
11
µ σ
2
V
12
µ . . . σ
2
V
1k
µ σ
2
V
21
µ σ
2
V
22
µ . . . σ
2
V
2k
µ ...
... . ..
... σ
2
V
k1
µ σ
2
V
k2
µ . . . σ
2
V
kk
µ
= σ
2
V
µ
which is the covariance matrix of ordinary reproductive exponential disper- sion models, then X ∼ EDµ,σ
2
.
