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
.
19
D. Multivariate Tweedie Models: an Example
In this part we discuss the multivariate Tweedie models as special cases of multivariate EDM. First we recall that the univariate Tweedie models are
particular cases of univariate EDM which admit the power unit variance function:
V µ = µ
p
2.21 with p ∈ −∞,0] ∪ [1,∞ is called the power parameter. The domain for µ is R
for p = 0 and R
+
for other values of p. The cumulant function and mean can be found for univariate Tweedie models by equating
κ
′′
θ = dµdθ = µ
p
and solving for
µ and κ. We define clearly the univariate stable Tweedie models below.
Definition 2.2.3. Let t
0 be the convolution parameter and the index parameter α ∈ [−∞,1 ∪ 1,2] intrinsically connected to the power variance parameter p ∈
−∞,0] ∪ [1,∞] by p − 11 − α = 1.
2.22
We define the univariate stable Tweedie NEFs F
p ,t
= F
ν
p ,t
generated by the σ-finite positive measures ν
p ,t
such that their cumulant functions are κ
ν
p ,t
= t
κ
ν
p ,1
with κ
ν
p ,1
θ =
exp
θ for p = 1
−log−θ for p = 2
1 2 − p
1 − pθ
p−2p−1
for 1 , 1 , 2
for all θ
in their respective canonical domains
Θ ν
p ,1
=
R for p = 0
, 1 [0
, ∞ for p
0 or 0 p 1 −∞,0 for 1 p 6 2 or p = ∞
−∞,0] for 2 p ∞.
The additive Tw
∗p
µ, λ has CGF s 7→ λ[κ
ν
α
s + θ − κ
ν
α
θ] = λκ
ν
α
θ 1 +
s θ
α
− 1 .
2.23 Some details and examples of univariate Tweedie models are given in Ap-
pendix A. In order to define the multivariate Tweedie models, we consider first
the intermediate weight parameter γ = λκ
α
θ, and write the CGF 2.23 as follows:
s 7→ γ 1 +
s θ
α
− 1 2.24
Since γ and κ
α
θ have the same sign, it follows that the domain for γ is either R
+
or R
−
, depending on the sign of α − 1α. Our starting point is the
20 bivariate singular distribution with joint CGF
s
1
, s
2 ⊤
7→ γ 1 +
s
1
θ
1
+ s
2
θ
2 α
− 1 ,
whose marginals are Tweedie distributions of the form 2.24. Now we define the multivariate Tweedie models as follow.
Definition 2.2.4. A multivariate additive Tweedie model denoted Tw
∗p k
µ, Λ is defined by the joint CGF:
Ks; θ, γ =
X
i j
γ
ij
1 + s
i
θ
i
+ s
j
θ
j α
− 1 +
k
X
i=1
γ
i
1 + s
i
θ
i α
− 1 ,
2.25 where
γ = λκ
ν
α
θ and the weight parameters γ
ij
and γ
i
all have the same sign as α − 1α.
By taking s
j
= 0 for j , i in the expression 2.25 we find that the ith marginal follows a univariate Tweedie distribution with CGF 2.24 with
θ = θ
i
and γ = γ
ii
defined by γ
ii
= X
j:j,i
γ
ij
+ γ
i
.
Table 2.2: Summary of Multivariate Tweedie Dispersion Models on R
k
with
Support S
p ,k
and Mean Domain M
p ,k
Distributions p
α = αp
S
p ,k
M
p ,k
Multivariate extreme stable p
1 α 2
R
k
, ∞
k
Multivariate Gaussian p = 0
α = 2 R
k
R
k
[Do not exist] p 1 2 α ∞
Multivariate Poisson p = 1
α = −∞ N
k
, ∞
k
Multivarite compound Poisson 1 p 2
α 0 [0
, ∞
k
, ∞
k
Multivariate non-central gamma p = 3
2 α = −1
[0 , ∞
k
, ∞
k
Multivariate gamma p = 2
α = 0 , ∞
k
, ∞
k
Multivariate positive stable p
2 α 1
, ∞
k
, ∞
k
Multivariate inverse Gaussian p = 3
α = 12 , ∞
k
, ∞
k
Multivariate extreme stable p = ∞
α = 1 R
k
R
k
The multivariate gamma distribution above is different from the one discussed by Bernard- off et al. 2008, the multivariate gamma here has the joint CGF of the form corresponds to
definition 2.2.4: Ks
, θ, Λ = − P
i j
λ
ij
log 1 −
s
i
θ
i
−
s
j
θ
j
− P
k i=1
λ
i
log 1 −
s
i
θ
i
.
For multivariate Tweedie distributions, the exponential dispersion model
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
