9
Since E[det V
n
] det Σ
µ
see e.g. Timm, 2002, p. 98, then det V
n
is a biased estimator.
Following the term of generalized variance for the covariance matrix, here we introduce definition of generalized dispersion for dispersion matrix.
Definition 2.1.4. The generalized G-dispersion of P
θ
around fixed point x is
given by det Σ
G ,x
= det Z
R
k
x − x G
−1
x − x
⊤
P
θ
dx
.
If G = I, then the generalized I-dispersion is simply the generalized dis-
persion: det Σ
x
= det Z
R
k
x − x x − x
⊤
P
θ
dx .
Moreover, if G = I and x = µ in Definition 2.1.4, then we have the generalized
dispersion the same as the generalized variance in 2.6.
For the generalized G-dispersion, the estimator is given by
[ det Σ
G ,x
= det
1 n − 1
n
X
i=1
h
X
i
− x G
−1
X
i
− x
⊤
i
.
2.2. Multivariate Dispersion Models
Jørgensen 1986 introduced dispersion models as the class of error dis- tributions for the generalized linear models GLM which has drawn a lot
of attention in the literature. The dispersion models contain many com- monly used distributions such as normal, Poisson, gamma, binomial, nega-
tive binomial, inverse Gaussian, compound Poisson, von Mises and simplex distributions Bandorff-Nielsen and Jørgensen, 1991.
The multivariate dispersion models generalize the univariate dispersion models see Appendix A which is based on the so-called deviance residuals:
r x; µ = [rx
1
; µ
1
, . . . , rx
k
; µ
k
]
⊤
, where x
j
and µ
j
denote the elements of the k-vectors x and µ respectively
and rx; µ is the univariate deviance residual.
Following Jørgensen and Lauritzen 2000 and Jørgensen 2013, we de- fine a multivariate dispersion model as follows:
Definition 2.2.1. A multivariate dispersion model DM
k
µ, Σ with position pa-
rameter µ and dispersion parameter Σ is a family distribution for x with probability density function of the form:
f x; µ, Σ = ax; Σ exp −
1 2
[rx; µ]
⊤
Σ
−1
r x; µ
, 2.7
10
where ax; Σ is a suitable function such that 2.7 is a probability density function on R
k
, µ ∈ Ω an open region in R
k
, Σ is a symmetric positive-definite k × k matrix
and rx; µ is a suitably defined vector of residuals satisfying rµ; µ = 0 for µ ∈ Ω. If ax; Σ factorizes as aΣbx, we call 2.7 a multivariate proper dispersion
model PD
k
µ, Σ as following:
f x; µ, Σ = aΣbx exp −
1 2
[rx; µ]
⊤
Σ
−1
r x; µ
. 2.8
The multivariate normal distribution is a special case of 2.8, i.e. by letting:
r x; µ = x − µ
aΣ = 2 π
−k2
[detΣ]
−12
bx = 1
we obtaine the multivariate normal distribution:
f x; µ, Σ =
1 2
π
k 2
det Σ
1 2
exp −
1 2
x − µ
⊤
Σ
−1
x − µ .
Jørgensen and Lauritzen 2000 introduced a simple notation for 2.7 by using the scaled deviance as a quadratic form of the vector of deviance
residuals,
Dx; µ, Σ = [rx; µ]
⊤
Σ
−1
r x; µ = tr
h Σ
−1
[rx; µ]
⊤
r x; µ
i .
2.9 Using notation 2.9 then the multivariate dispersion models can be written
as
f x; µ, Σ = ax; Σ exp −
1 2
Dx; µ, Σ .
They also showed that the matrix variance function corresponding to the scaled deviance D is
V
Σ
µ = 1
2 D
′′ µ,µ
x; µ, Σ = [Vµ]
1 2
Σ [Vµ]
1 2
= Σ ⊙ ˜Vµ, where Vµ = diag[Vµ
1
, . . . , Vµ
k
] denotes the diagonal variance function,
⊙ is the Hadamard element-wise product, and ˜Vµ denotes the matrix with elements [V
µ
i
]
1 2
[V µ
j
]
1 2
for i; j = 1 , . . . , k.
To generalize 2.7, Jørgensen and Lauritzen 2000 proposed a method for constructing multivariate proper dispersion models by using notation
Λ = Σ
−1
instead of Σ. Then we write ax; Λ instead of ax; Σ and Dx; µ, Λ instead of Dx; µ, Σ and so on.
The construction of multivariate proper dispersion models relies on iden-
tifying cases where the vector of deviance residuals ry; µ is such that the
11 normalizing constant aΣ depends on the parameters only through Σ. How-
ever, this technique does not work for exponential dispersion models, calling for an alternative method of construction.
2.2.1. Multivariate Exponential Dispersion Models
Exponential dispersion models are particular cases of the dispersion models. Jørgensen in his book The Theory of Dispersion Models 1997 intro-
duced two types of exponential dispersion models: additive and reproductive exponential dispersion models. The multivariate form of these models was
introduced by Jørgensen and Lauritzen 2000, and then discussed in details by Jørgensen 2013 and Jørgensen and Martínez 2013.
A. Ordinary Exponential Dispersion Models
Before we discuss these multivariate exponential dispersion models EDM parameterized by µ and Σ, we first need to understand the ordinary expo-
nential dispersion models which can be described as follow. Consider a
σ−finite measure ν on R, the ordinary cumulant generating
function CGF for a random k-vector X is defined by κθ = κθ; X = log E
e
θ
⊤
X
for θ ∈ R
k
, with effective domain Θ = {θ ∈ R
k
: κθ ∞}.
The additive form of a k-variate exponential dispersion model is denoted
by X ∼ ED
∗
µ, λ where µ is the k ×1 rate vector and λ is the positive weight pa- rameter sometimes called the index or convolution parameter. This model
is defined as a family distribution for X with probability density mass func- tion with respect to a suitable measure on R
k
as follows f
∗
x; θ, λ = a
∗
x; λ exp
h
x
⊤
θ − λκθ
i
for x ∈ R
k
, 2.10
where a
∗
x; λ is a suitable function of x and λ, and the cumulant function
κθ depends on the canonical parameter θ ∈ Θ ⊆ R
k
related to µ by the following equation
µ = EX = κ
′
θ 2.11
and λ has domain R
+
We note that the model ED
∗
µ, λ has k + 1 parameters, of which the single parameter
λ controls the covariance structure of the distribution. The mean vector of 2.10 is
λµ and the covariance matrix is λVµ with V
µ = κ
′′
◦ κ
′
µ 2.12
where κ
′′
and κ
′
denote the firts and the second derivatives of κ respectively.
12 The CGF of EDµ, λ satisfies
λκs; θ = λ [κs + θ − κθ] for s ∈ Θ − θ
2.13
which is finite for the argument s belonging to a suitable neighborhood of zero, and hence characterizes the distribution in question. It also follows
from 2.13 that the model ED
∗
µ, λ satisfies the following additive property: ED
∗
µ, λ
1
+ ED
∗
µ, λ
2
= ED
∗
µ, λ
1
+ λ
2
for λ
1
, λ
2
For each additive exponential dispersion model ED
∗
µ, λ, the corre- sponding reproductive exponential dispersion model EDµ, σ
2
is defined by the
duality transformation Y =
λ
−1
X , or
EDµ, σ
2
= λ
−1
ED
∗
µ, λ where
σ
2
= 1 λ is called the dispersion parameter.
The reproductive form of a k-variate exponential dispersion model denoted by EDµ, σ
2
with location parameter µ and dispersion parameter σ
2
is defined
as a family distribution for Y with probability density mass function on R
k
of the form
f y; θ, λ = ay; λ exp
n λ
h
y
⊤
θ − κθ
io
for y ∈ R
k
, 2.14
with respect to a suitable measure on R
k
, where ay;
λ is a suitable function
of y and λ. The random vector Y on 2.14 has mean µ and variance
VarY =
σ
2
V µ.
The model EDµ, σ
2
satisfies the following reproductive property see Jørgensen and Martínez, 2013:
I f Y
1
, . . . , Y
n
are i .i.d. EDµ, σ
2
, then 1
n
n
X
i=1
Y
i
∼ EDµ,σ
2
n.
To provide a fully flexible covariance structure corresponding to a mean vector µ and a positive-definite dispersion matrix Σ, an alternative method
for the construction of multivariate exponential dispersion model is needed. Jørgensen and Martínez 2013 discussed a detail construction for multivari-
ate exponential dispersion models which are very helpful and important for understanding this family of distribution. With this method then the covari-
ance matrix of EDM is of the form CovX = Σ ⊙ Vµ, where ⊙ denotes the Hadamard element-wise product between two matrices, and Vµ denotes
the matrix unit variance function.
13
B. Multivariate Additive Exponential Dispersion Models
The construction of multivariate additive exponential dispersion model is based on an extended convolution method, which interpolates between
the set of fully correlated pairs of variables and the set of independent mar- gins of the prescribed form so that the marginal distributions follow a given
univariate exponential dispersion model. This method explores the con- volution property of conventional additive exponential dispersion models
in order to generate the desired number of parameters, namely k means and kk + 1
2 variance and covariance parameters. Multivariate additive exponential dispersion models are particularly suitable for discrete data,
and include multivariate versions of the Poisson, binomial and negative binomial distributions.
We start by the extended convolution method in the bivariate additive case as described by Jørgensen and Martínez 2013 and Jørgensen 2013.
Suppose that we are given an ordinary bivariate additive ED
∗
µ, λ with CGF s
1
, s
2 ⊤
7→ λκs
1
, s
2
; θ
1
, θ
2
= λ[κs
1
+ θ
1
, s
2
+ θ
2
− κθ
1
, θ
2
] .
From this model we define a multivariate additive exponential dispersion bivariate case by means of the following stochastic representation for the
random vector X:
X
1
X
2
= U
11
U
12
+ U
1
+ U
2
, 2.15
where the three vectors on the right-hand side of 2.15 are assumed inde- pendent. More precisely, with θ = θ
1
, θ
2 ⊤
and the weight matrix: Λ
= λ
11
λ
12
λ
12
λ
22
, where
λ
ii
= λ
12
+ λ
i
for i = 1 , 2; the joint CGF for X is defined to be
Ks
1
, s
2
; θ, Λ = λ
12
κs
1
, s
2
; θ
1
, θ
2
+ λ
1
κs
1
, 0; θ
1
, θ
2
+ λ
2
κ0, s
2
; θ
1
, θ
2
= λ
12
κs
1
+ θ
1
, s
2
+ θ
2
+ λ
1
κs
1
+ θ
1
, θ
2
+ λ
2
κθ
1
, s
2
+ θ
2
− λ
12
+ λ
1
+ λ
2
κθ
1
, θ
2
2.16 The equation 2.16 above is interpreted as interpolating between inde-
pendence λ
12
= 0 and the maximally correlated case λ
1
= λ
2
= 0. We note that the margins have the same form as for
λκs
1
, s
2
; θ
1
, θ
2
, as seen from the two marginal CGFs as follow
Ks
1
, 0; θ, Λ = λ
12
+ λ
1
κs
1
, 0; θ
1
, θ
2
= λ
11
s
1
, 0; θ
1
, θ
2
K0 , s
2
; θ, Λ = λ
12
+ λ
2
κ0, s
2
; θ
1
, θ
2
= λ
22
, s
2
; θ
1
, θ
2
The construction hence preserves the form of the univariate margins,
14 while replacing the single parameter
λ by the three parameters of Λ, for a total of five parameters.
Using the notation 2.11, the mean vector for X is EX = diagΛµ =
λ
11
κ
′ 1
θ
1
, θ
2
λ
22
κ
′ 2
θ
1
, θ
2
, where
κ
′ i
denotes the ith partial derivative of κ for i = 1, 2, and where diagΛ
is a 2 ×2 diagonal matrix. We use the notation X ∼ ED
∗ 2
µ, Λ for the bivariate additive exponential dispersion defined by 2.16, parametrized by the rate
vector µ and weight matrix Σ.
The covariance matrix for X is VarX =
λ
11
V
11
µ λ
12
V
21
µ λ
12
V
12
µ λ
22
V
22
µ
= Λ ⊙ Vµ
where the V
ij
are elements of the unit variance function defined by 2.12. The correlation between X
1
and X
2
is CorrX
1
, X
2
= λ
12
√ λ
11
λ
22
V
12
µ p
V
11
µV
22
µ ,
which for given µ varies between zero and V
12
µ p
V
11
µV
22
µ, which may be either positive or negative, depending on the sign of V
12
µ. Now let us consider for the trivariate case, following Jørgensen 2013
we define the trivariate random vector X = X
1
, X
2
, X
3 ⊤
as the sum of six independent vectors
X
1
X
2
X
3
=
U
11
U
12
+
U
21
U
23
+
U
32
U
33
+
U
1
+
U
2
+
U
3
, 2.17
of which three terms are bivariate and three are univariate. However, rather than starting from a trivariate CGF
κs
1
, s
2
, s
3
; θ as was done by Jørgensen, 2013, Jørgensen and Martínez 2013 started from the three bivariate distri-
butions corresponding to the first three terms 2.17. For this construction to work, it is crucial that the univariate margins of the three bivariate terms are
consistent, so that for example U
11
and U
21
have distributions that belong to the same class. In order to avoid the intricacies of such a construction in
the general case, we concentrate here on the homogeneous case, where all three margins belong to the same class, for example a multivariate gamma
distribution with gamma margins.
Using µ parameterization, the trivariate exponential dispersion model
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
.
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
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
∞}.
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.
27
Stochastic model for heterogeneities in regional organ blood flow
The theory of exponential dispersion models was applied to construct a stochastic model for heterogeneities in regional organ blood flow Kendal,
2001. Regional organ blood flow exhibits a significant degree of spatial heterogeneity when measured by using labeled microspheres or by other
means. The related velocity distribution of blood flow has been character- ized by a gamma distribution. To provide a stochastic description for the
macroscopic and microscopic heterogeneities in regional organ blood flow, a scale invariant compound Poisson-gamma distribution was employed.
Distribution of human single nucleotide polymorphisms
A scale invariant Poisson gamma PG exponential dispersion model was used in modeling the distribution of human single nucleotide poly-
morphisms NSPs Kendal, 2003. The SNPs appear to be non-uniformly dispersed throughout the human genome, this non-uniformity can be at-
tributed to a segmented genealogical substructure within the genome, where older segments may have accumulated greater numbers of SNPs. An analy-
sis of 1.42 million human single nucleotide polymorphisms SNPs revealed an apparent power function relationship between the estimated variance
and mean number of SNPs per sample bin. By PG-EDM model the sample bins contain random Poisson distributed numbers of identical by descent
genomic segments, each with independently gamma distributed numbers of SNPs.
Geometric α-Laplace marginals in autoregressive process
Geometric Laplace distribution is one of the geometric dispersion model. Lekshmi and Jose 2004 proposed the used of geometric Laplace distribution
as stationary marginal in autoregressive process modeling. Some contexts of applications were mentioned in their paper, e.g. for modeling pooled
position errors in a large navigation system, for modeling sulphate concen- tration data with ARMA model, and for modeling financial time series data
see Hsu 1979, Damsleth and El-Shaarawi 1989, Anderson and Arnold 1993 respectively for details.
Integer valued time series modeling
A discrete dispersion model was applied in modeling autoregressive time series of count data. The model was developed in terms of a convo-
lution of Poisson and negative binomial random variables known as Pois- son–negative binomial PNB distribution Jose and Mariyamma, 2015. The
distribution was used as the marginal distribution of the time series model.
28
3. NORMAL STABLE TWEEDIE MODELS
In this chapter, we present the family of normal stable Tweedie models which is the extension of normal gamma Bernardo and Smith, 1993 and
normal inverse Gaussian Bandorff-Nielsen et al., 1982 models. Normal Stable Tweedie NST models are composed by a fixed univariate stable
Tweedie variable having a positive value domain, and the remaining random variables given the fixed one are real independent Gaussian variables with
the same variance equal to the fixed component.
A random variable X is said to be stable or to have a stable distribution if it has the property that a linear combination of two independent copies
of the variable, e.g. aX
1
+ bX
2
where X
1
and X
2
are two independent copies of X, has the same distribution with cX + d for some positive c and d ∈ R
Nolan, 2017. The stable distribution family is also sometimes referred to as the Lévy alpha-stable distribution. While Tweedie distributed variable is
a variable that belongs to the exponential dispersion family of distribution with specific power mean-variance relationships, i.e, V
µ = µ
p
see Chapter 2, Equation 2.21 for some p . Then Tweedie distribution is to the subclass
of the exponential dispersion models that admits a power variance function V
µ = µ
p
Jørgensen, 1997. Some details on univariate stable Tweedie models are given in Appendix A.
3.1. Definition and Properties
Motivated by normal gamma and normal inverse Gaussian models, Boubacar Maïnassara and Kokonendji 2014 introduced a new form of gen-
eralized variance functions which are generated by the so-called normal stable Tweedie NST models of k-variate distributions k
1. The generating σ-finite positive measure µ
α,t
on R
k
of NST models is composed by the well- known probability measure
ξ
α,t
of univariate positive σ-stable distribution
generating Lévy process X
α t
t
which was introduced by Feller 1971 as follows
ξ
α,t
dx = 1
πx
∞
X
r=0
t
r
Γ1 + αrsin−rπα
r α
r
α − 1
−r
[1 − αx]
αr
1
x
dx = ξ
α,t
xdx ,
3.1 where
α ∈ 0,1 is the index parameter, Γ. is the classical gamma function, and I
A
denotes the indicator function of any given event A that takes the value 1 if the event occurs and 0 otherwise. Parameter
α can be extended into α ∈ −∞,2] see Tweedie, 1984. For α = 2, we obtain the normal distribution
with density ξ
2 ,t
dx = 1
√ 2
πt exp
−x
2
2t dx
.
For a k-dimensional NST random vector X = X
1
, . . . , X
k ⊤
, the generating 29
30 σ-finite positive measure ν
α,t
is given by ν
α,t
dx =
ξ
α,t
dx
1 k
Y
j=2
ξ
2 ,x
1
dx
j
, 3.2
where X
1
is a univariate non-negative stable Tweedie variable and all other variables X
2
, . . . , , X
k ⊤
=: X
c 1
given X
1
are k − 1 real independent Gaussian variables with variance X
1
. By introducing power variance parameter p defined by p − 11 − α = 1
and equivalent to p = p α =
α − 2 α − 1
or α = αp =
p − 2 p − 1
, now we consider X
j
as the univariate stable component instead of X
1
for j ∈ {1,2,...,k}, t 0 the dispersion parameter of the associated Lévy process and
α = αp the index parameter which is connected to the power variance parameter, the
generating σ-finite positive measure ν
α,t
on R
k
of NST models is ν
α,t;j
dx = ξ
α,t
dx
j
Y
ℓ,j
ξ
2 ,x
j
dx
ℓ
3.3 with
α = αp ∈ [−∞,0. Since p − 11 − α = 1 then 3.1 can be expressed in term of p namely ξ
p ,t
with ξ
p ,t
= ξ
p α,t
, then equation 3.3 can be written as follows ν
p ,t:j
dx =
ξ
p ,t
dx
j
Y
ℓ,j
ξ
,x
j
dx
ℓ
. 3.4
For suitable univariate NEF F
p ,t
= F
ξ
αp,t
of stable Tweedie types, we
can interpret the multivariate NEFs G
p ,t
= G ν
αp,t
as composed by the dis-
tribution 3.4 of the random vector X = X
1
, . . . , X
k ⊤
where X
1
is a univariate stable Tweedie variable generated by xi
,x
1
with mean 0 and variance x
1
. So from Table A.1 in Appendix with S
p
⊆ [0,∞, we must retain α in [−∞,1 and the associated univariate model may be called the non-negative stable
Tweedie, which include normal Poisson models appearing as new multi- variate distribution having one discrete component.
Definition 3.1.1. For X = X
1
, . . . , X
k ⊤
a k-dimensional normal stable Tweedie random vector, it must hold that
1. X
j
is a univariate stable Tweedie random variable, and
2. X
c j
|X
j
:= X
1
, . . . , X
j−1
, X
j+1
, . . . , X
k
given X
j
follows the k − 1-variate inde- pendent normal N
k−1
, X
j
I
k−1
distribution, where I
k−1
= diag
k−1
1 , . . . , 1
denotes the k − 1 × k − 1 unit matrix.
By equation 3.4, one can obtain the cumulant function K
ν
p ,t
θ = log R
R
k
exp
31 θ
T
x
ν
p ,t
dx:
K
ν
p ,t;j
θ = log
Z
R
exp θ
j
x
j
Y
ℓ,j
Z
R
exp θ
ℓ
x
ℓ
ξ
,x
j
dx
ℓ
ξ
p ,t
dx
j
= log
Z
R
exp θ
j
x
j
Y
ℓ,æ
exp x
j
θ
2 ℓ
2
ξ
p ,t
dx
j
= log
Z
R
exp
θ
j
x
j
+ 1
2 X
ℓ,j
x
j
θ
2 ℓ
ξ
p ,t
dx
j
= log
Z
R
exp
x
j
θ
j
+ 1
2 X
ℓ,j
θ
2 ℓ
ξ
p ,t
dx
j
= tK
ξ
p ,t
θ
j
+ 1
2 X
ℓ,j
θ
2 ℓ
Here K
ξ
p ,t
is the cumulant function of univariate stable Tweedie NEF F
ξ
p ,t
generated by σ-finite positive measures ξ
p ,t
as follow: K
ξ
p ,t
= tK
ξ
p ,1
with K
ξ
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
3.5 for all
θ in their respective canonical domains
Θ ξ
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 function K
ν
p ,t;j
θ is finite for all θ = θ
1
, . . . , θ
k ⊤
in the canonical domain
Θ ν
p ,t;j
=
θ ∈
R
k
; θ
⊤
˜θ
c j
:=
θ
j
+ 1
2 X
ℓ,j
θ
2 ℓ
∈ Θ
p
3.6
where ˜θ
c j
= θ
1
, . . . , θ
j−1
, 1, θ
j+1
, . . . , θ
k
and
Θ
p
=
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 ∞.
32 For fixed p 1 and t
0, the multivariate NEF generated by ν
p ,t;j
is the set
G
p ,t;j
= Pθ; p, t; θ ∈ Θν
p ,t;j
3.7 of probability distributions
P θ; p, tdx = exp
n θ
⊤
x − K
ν
p ,t;j
θ o
ν
p ,t;j
dx .
3.8
The mean vector and the covariance matrix of G
p ,t;j
can be calculated using the first and the second derivatives of the cumulant function, i.e.
µ = K
′ ν
p ,t;j
θ and
V
G
p ,t;j
µ = K
′′ ν
p ,t;j
θµ. The followings are four examples illustrate some of the issues that may
be encountered when applying equation 3.4. Normal Poisson. For p = 1 = p−∞ in 3.4 we represent the normal Poisson
generating measure by ν
1 ,t;j
dx =
t
x
j
x
j −1
2 πx
j k−12
exp
−t−
1 2x
j
X
ℓ,j
x
2 ℓ
1
x
j
∈N\{0}
δ
x
j
dx
j
Y
ℓ,j
dx
ℓ
, where 1
A
is the indicator function of the set A. Since it is also possible to have x
j
= 0, the corresponding normal Poisson distribution is degenerated as δ
. Normal Poisson is the only NST model which has a discrete component and correlated to the continuous components.
Normal gamma. For p = 2 = p0 in 3.4 one has the generating measure of the normal gamma as follow:
ν
2 ,t;j
dx =
x
t−1 j
2 πx
j k−12
Γt exp
−x
j
− 1
2x
j
X
ℓ,j
x
2 ℓ
1
x
j
dx
1
dx
2
···x
k
Normal gamma was introduced by Casalis 1996 as a member of simple quadratic NEFs; she called it as gamma-Gaussian. This model was charac-
terized by Kokonendji and Masmoudi 2013. The bivariate case is used as prior distribution of Bayesian inference for normal distribution Bernardo
and Smith, 1993.
Normal inverse Gaussian. Setting p = 3 = p1 2 in 3.4 one has the so-called
normal inverse Gaussian family generated by
ν
3 ,t;j
dx =
tx
−k+22 j
2 π
k 2
exp
−1 2x
j
t
2
+ X
ℓ,j
x
2 ℓ
1
x
j
dx
1
dx
2
···x
k
33 It already appeared as a variance-mean mixture of a multivariate Gaus-
sian with a univariate inverse Gaussian distribution Bandorff-Nielsen et al., 1982. It can be considered as a distribution of the position of multivariate
Brownian motion at a certain stopping time. See Bandorff-Nielsen 1997, 1998 and Ølgård et al. 2005 for more details and interesting applications
in stochastic volatility modeling and heavy-tailed modeling, respectively.
Normal noncentral gamma. For p = 3 2 in 3.4 one has the generating
measure of normal noncentral gamma which can be expressed as follows: ν
3 2,t;j
dx =
x
−1 j
2 πx
j k−12
X
ℓ,j
4tx
j j
ℓΓℓ
exp
− 1
2x
j
X
ℓ,j
x
2 ℓ
1
x
j
dx
1
dx
2
···x
k
Since it is possible to have x
j
= 0 as in normal-Poisson models the corre- sponding normal distributions are degenerated as
δ .
3.2. Generalized variance function and Lévy measures
Now let p 1 and t 0, denote e
1=1 ,...,k
an orthonormal basis of R
k
, and µ = µ
1
, . . . , µ
k ⊤
the mean vector of X. Boubacar Maïnassara and Kokonendji 2014 showed the variance functions of G
p ,t;j
= G ν
p ,t;j
generated by ν
p ,t;j
is given by
V
G
p ,t;j
µ = t
1−p
µ
p−2 j
· µµ
⊤
+ Diag
k
µ
j
, . . . , µ
j
, 0
j
, µ
j
, . . . , µ
j
, 3.9
on its support
M
G
p ,t;j
= n
µ ∈ R
k
; µ
j
0 and µ
ℓ
∈ Rforℓ , j o
. 3.10
for all µ = µ
1
, . . . , µ
k ⊤
in M
G
p ,t;j
= 0 , ∞ × R
k−1
.
For p = 1 normal Poisson and j = 1, the covariance matrix of X can be
expressed as below
V
G
1 ,t;1
µ =
µ
1
µ
2
. . . µ
i
. . . µ
k
µ
2
µ
−1 1
µ
2 2
+ µ
1
. . . µ
−1 1
µ
2
µ
i
. . . µ
−1 1
µ
k
µ
2
.. .
.. .
. .. ..
. ..
. µ
i
µ
−1 1
µ
i
µ
2
. . . µ
−1 1
µ
2 i
+ µ
1
. . . µ
−1 1
µ
i
µ
k
.. .
.. .
.. .
. .. ..
. µ
k
µ
−1 1
µ
k
µ
2
. . . µ
−1 1
µ
k
µ
i
. . . µ
−1 1
µ
2 k
+ µ
1
.
The generalized variance function of any multivariate NST model presents
34 a very simpler expression than its variance function in 3.9. More precisely,
it depends solely on the first component of the mean vector with the power variance parameter p and the dimension k.
Let p = p α ≥ 1 and t 0. Then the generalized variance function of a
normal stable Tweedie model in G
p ,t;j
= G ν
p ,t;j
generated by ν
p ,t;j
is given by
det V
G
p ,t;j
µ = t
1−p
e
T j
µ
p+k−1
= t
1−p
µ
p+k−1 j
3.11
for µ ∈ M
G
p ,t;j
= 0 , ∞ × R
k−1
. The proof of equation 3.11 for j = 1 is obtained using the Schur repre-
sentation of determinant: det
γ a
⊤
a A
−1
=
γ detA − γ
−1
aa
⊤
, 3.12
where γ , 0 is a scalar, A is a k × k matrix, a and b are two vectors on R
k
. Applying the Schur representation 3.12 for j = 1 with
γ = t
1−p
e
1
⊤
µ
p
,
a = b = t
1−p
e
1
⊤
µ
p−1
µ
2
, . . . , µ
k ⊤
, A =
t
1−p
e
1
⊤
µ
p−2
µ
2
, . . . , µ
k
µ
2
, . . . , µ
k ⊤
+ e
1
⊤
µ.I
k−1
= γ
−1
aa
⊤
+ e
1
⊤
µ.I
k−1
, we obtain
det V
G
p ,t;1
µ = γ det[e
1
⊤
µ.I
k−1
] =
t
1−p
e
T 1
µ
p+k−1
. Then, it is trivial to show that for j ∈ {1,...,k} the generalized variance of
normal stable Tweedie model is given by 3.11.
It is easy to deduce the generalized unit variance function det V
G
p;j
µ =
e
T j
µ
p+k−1
with t = 1. Table 3.1 summarizes all k-variate NST models with
support S
p
of distributions, where p = p α ≥ 1 is the power variance param-
eter, the parameter η = 1 + kp − 1 = ηp,k of the corresponding modified
Lévy measure ρν
p ,t;j
is from Proposition 3.2.1 below. Since all NST mod-
els G ν
p ,t;j
are infinitely divisible, their modified Lévy measures are of the normal gamma type for p
1 and degenerated for p = 1.
Proposition 3.2.1. Let
ν
p ,t;j
be a generating measure of an NST family for given p = p
α and t 1. Denote η = 1 + kp − 1 = ηp,k 1 the modified Lévy measure
35 parameter. Then
ρν
p ,t;j
= t
k
p − 1
−ηp,k
.ν
2 ,ηp,k
f or p 1
t
k
.δ
e
j
Q
ℓ,j
ξ
,1 ∗k
f or p = 1 3.13
Table 3.1: Summary of k-variate NST models with power variance parameter p = p
α ≥ 1, modified Lévy measure parameter η := 1 + kp − 1 and support
of distributions S
p
fixing j = 1.
Types p = p
α η = 1 + kp − 1 S
p
Normal Poisson p = 1
η = ∞ N × R
k−1
Normal compound Poisson 1 p 2 η k + 1
[0 , ∞ × R
k−1
Normal noncentral gamma p = 3
2 η = 2k + 1
[0 , ∞ × R
k−1
Normal gamma p = 2
η = k + 1 , ∞ × R
k−1
Normal positive stable p
2 1
η k + 1 , ∞ × R
k−1
Normal inverse Gaussian p = 3
η = 1 + k2 , ∞ × R
k−1
The proof of 3.13 is provided here in form of power parameter p, using Lemma 3.2.1 and following Boubacar Maïnassara and Kokonendji 2014,
Theorem 3.5. From the cumulant function 3.5 of univariate non-negative stable Tweedie we obtain the first and the second derivatives as follow
K
′ ξ
p ,1
θ =
exp θ
for p = 1 θ1 − p
−1p−1
for p ≥ 1 and also
K
′′ ξ
p ,1
θ =
exp
θ = K
′ ξ
p ,1
θ for p = 1
θ1 − p
−pp−1
= K
′ ξ
p ,1
θ
p
for p ≥ 1
Lemma 3.2.1. Let f : R → R and g,h : R
k
→ R be three functions, each is twice
differentiable and such that h = f ◦ g. Then h
′
x = f
′
gx × g
′
x and h
′′
x = f
′′
gx × g
′
xg
′
x
⊤
+ f
′
gx × g
′′
x Then fixing j = 1 and using Lemma 3.2.1 with h = K
ν
p ,t
, f = tK
ξ
p ,1
and
g θ = θ
1
+ P
k j=2
θ
2 j
2 such that g
′
θ = 1, θ
2
, . . . , θ
k ⊤
and g
′′
θ = Diag0, 1, . . . , 1,
36 we can write
K
′′ ν
p ,t
θ = ∂
2
∂θ
i
∂θ
j
tK
ξ
p ,1
gθ
i ,j=1,...,k
= t
. γ
a
⊤
a A
3.14 with
γ = K
ξ
p ,1
gθ, a = γθ
2
, . . . , θ
k ⊤
and A = γ
−1
aa
⊤
+ K
′ ξ
p ,1
gθI
k−1
. There- fore, using 3.12 it follows that
det K
′′ ν
p ,t
θ = t
k
= K
′′ ξ
p ,1
gθ K
′ ξ
p ,1
gθ
k−1
=
t
k
exp{kgθ} for p = 1
g θ1 − p
−1−kp−1
for p ≥ 1 Taking
ηp, k = 1 + kp − 1 and K
ρ
p ,t
θ = log det K
′′ ν
p ,t
θ which is
K
ρ
p ,t
θ =
k θ
1
+
1 2
P
k j=2
+ log t
k
for p = 1 −ηp,klogθ
1
−
1 2
P
k j=2
θ
2 j
+ log c
p ,k,t
for p 1
for θ ∈ Θρν
p ,t
= Θ ν
p ,t
with c
p ,k,t
= t
k
p − 1
−ηp,k
, this leads to 3.13. Recall that the Monge-Ampère equation which is generally stated as
det ψ
′′
θ = rθ where ψ is an unknown smooth function and r is a given positive function. Then from the modified Lévy measure of
ν
p ,t;j
, the Monge- Ampère equation below is considered to be the problem of the characteriza-
tion of multivariate NST models through generalized variance function
det K
′′
θ = exp n
K
ρν
p ,t;j
θ o
, p ≥ 1
3.15
where K is unknown cumulant function to be determined. See Kokonendji and Masmoudi 2013 for normal gamma model and some references of
particular cases. In this work we use 3.15 for the characterization of normal Poisson model p = 1 by generalized variance in Section 5.2.2.
4. GENERALIZED VARIANCE ESTIMATIONS OF SOME
NST MODELS
Generalized variance; i.e. the determinant of covariance matrix ex- pressed in term of mean vector; has important roles in statistical analysis of
multivariate data. The notion is introduced by Wilks Wilks 1932 as a scalar measure of multivariate dispersion and used for overall multivariate scatter.
The estimation of the generalized variance, mainly from a decision the- oretic point of view, attracted the interest of many researchers in the past
four decades; see for example Shorrock and Zidek 1976, Kubokawa and Konno 1990, Gupta and Ofori-Nyarko 1995, Iliopoulos and Kourouklis
1998 and Bobotas and Kourouklis 2013 for estimation under multivari- ate normality. In the last two decades the generalized variance has been
extended for non-normal distributions in particular for natural exponential families NEFs; see Kokonendji and Seshadri 1996, Kokonendji and Pom-
meret 2001, Kokonendji 2003 and Kokonendji and Pommeret 2007 who worked in the context of NEFs.
The uses of generalized variance also have been discussed by several authors. See for example in the theory of statistical hypothesis testing, gen-
eralized variance is used as a criterion for an unbiased critical region to have the maximum Gaussian curvature Isaacson, 1951; in the descriptive statis-
tics Goodman 1968 proposed a classification of some groups according to their generalized variances; in sampling theory it is used as a loss function
on multiparametric sampling allocation Arvanitis and Afonja, 1971.
In this chapter we discuss the ML and UMVU estimators of generalized variance of normal gamma, normal inverse Gaussian NIG and normal
Poisson models. Bayesian estimator of the generalized variance for normal Poisson is also introduced. A numerical analysis through simulation studies
is provided.
4.1. Generalized Variance Estimators
4.1.1. Maximum Likelihood Estimator
Let X
1
, . . . , X
n
be random vectors i.i.d with distribution Pθ; p, t ∈ Gν
p ,t;j
in a given NST family, i.e. for fixed for fixed j ∈ {1,2,...,k}, p ≥ 1 and t 0. Denoting X = X
1
+ ··· + X
n
n = X
1
, . . . , X
k ⊤
the sample mean.
Theorem 4.1.1. The maximum likelihood estimator MLE of the generalized vari-
ance det V
G
p ,t;j
µ is given by: T
n;k;p ,t;j
= det V
G
p ,t;j
X = t
1−p
X
j p+k−1
4.1
Proof. The ML estimator 4.1 is directly obtained from ?? by substituting µ
j
with its ML estimator X
j
. 37
38 Then for each model one has:
T
n;k;t;j
= det V
G
p ,t;j
X =
X
k j
, for normal Poisson
1 tX
k+1 j
, for normal gamma
1 t
2
X
k+2 j
, for normal inverse Gaussian For all p ≥ 1, T
n;k ,p,t;j
is a biased estimator of det V
G
p ,t;j
µ = t
1−p
µ
j p+k−1
. For
example, for p = 1 we have det V
G
p ,t;j
µ = µ
k j
, to obtain an unbiased estimator for this we need to use the intrinsic factorial moment formula
E X
j
X
j−1
X
j−2
···X
j−k+1
= µ
k j
, where X follows the univariate Poisson distribution with mean
µ
j
.
4.1.2. Uniformly Minimum Variance Unbiased Estimator
In order to avoid the lack of good properties by estimating det V
G
p ,t
µ = t
1−p
µ
k+p−1 j
with T
n;k ,p,t
, we are able to obtain directly the uniformly minimum variance and unbiased UMVU estimator U
n;k ,p,t
of det V
G
p ,t
µ.This is done through the following techniques for all integers n
k Kokonendji and Seshadri, 1996; Kokonendji and Pommeret, 2007; Kokonendji, 1994 :
U
n;k ,p,t
= C
n ,k,p,t
nX 4.2
where C
n ,k,p,t
: R
k
→ [0,∞ satisfies ν
n ,k,p,t
dx = C
n ,k,p,t
x ν
p ,nt
dx 4.3
and ν
n ,k,p,t
dx is the image measure of
1 k + 1
det 1
1 ···
1
x
1
x
2
··· x
k+1 2
ν
p ,t
dx
1
···ν
p ,t
dx
n
by the map x
1
, ···x
n
7→ x
1
+···+x
n
. The expression of C
n ,k,p,t
x for computing
the UMVU estimator U
n;k ,p,t
for p = p α ∈ [1,∞ is stated in the following
theorem.
Theorem 4.1.2. Let X
1
, ··· ,X
n
be random vectors i.i.d with distribution Pµ, G
p ,t;j
∈ G
ν
p ,t;j
in a given NST family, i.e. for fixed p ≥ 1,t 0, and having modified Lèvy measure
ρν
p ,t
satisfies 3.13 with parameter ηp, k = p + k − 1. Then
C
n ,k,p,t
x =
ν
p ,nt
∗ ρν
p ,t
dx
ν
p ,nt
dx
39 in particular, C
n ,k,p,t
x is
n
−k
x
j
x
j
− 1x
j
− 2···x
j
− k + 1, x
j
≥ k for normal Poisson
t
k
Γnt[Γnt + k + 1]
−1
x
k+1 j
for normal gamma t
k
2
−1−k2
[Γ1 + k 2]
−1
x
3 2
j
exp n
nt
2
2x
j
o ×
R
x
j
y
k 2
j
x
j
− y
j −32
exp n
−y
j
− nt
2
[2x
j
− y
j
] o
dy
j
for normal Inverse-Gaussian
Proof. From 4.3 we write: C
n ,k,p,t
x =
ν
n;k;p ,t
dx
ν
p ,nt
dx
Following Kokonendji and Pommeret, 2007, Theorem 1 and using 3.13 we have:
K
ν
n;k;p ,t
θ = nK
ν
p ,t
θ + log det K
′′ ν
p ,t
= K
ν
p ,nt
θ + K
ρν
p ,t
θ for all θ ∈ Θν
p ,1
. Then it immediately follows that ν
n;k;p ,t
= ν
p ,nt
∗ρν
p ,t
is the convolution product of
ν
p ,nt
by ρν
p ,t
. The proof for C
n ,k,p,t
x is established
by considering each group of the NST models with respect to the different values of p ≥ 1 and using 3.13.
Indeed, for p = 1 and fixing j = 1 we have ρν
1 ,t
= t
k
.δ
e
1
Q
k j=2
ξ
,1 ∗k
and
C
n ,k,1,t
x = t
k
ν
1 ,nt
∗ δ
e
1
Q
k j=2
ξ
,1 ∗k
dx
ν
1 ,nt
dx
= t
k
ξ
1 ,nt
x
1
− k ξ
1 ,nt
x
1
k
Y
j=2
Z
R
ξ
,x
1
−k
x
j
− y
j
ξ
,k
y
j
ξ
,x
1
x
j
dy
j
= t
k
x
1
nt
x
1
−k
exp−nt x
1
− knt
x
1
exp−nt × 1
= x
1
x
1
− 1...x
1
− k + 1 n
k
; because for fixed j = 2
, . . . , k the expression Wj
, x
1
, k = Z
R
ξ
,x
1
−k
x
j
− y
j
ξ
,k
y
j
ξ
,x
1
x
j
dy
j
4.4
40 is finally
Wj , x
1
, k = Z
R 1
√
2 πx
1
−k
exp
−x
j
−y
j 2
2x
1
−y
1
1 √
2 πk
exp
−y
2 j
2k 1
√ 2
πx
1
exp
−x
2 j
2x
1
dy
j
= √
x
1
√ 2
π p
kx
1
− k exp
x
2 j
2x
1
Z
R
exp
−y
2 j
+ 2
kx
j
x
1
y
j
−
kx
2 j
x
1
2
k x
1
x
1
− k
dy
j
= exp0 Z
R
ξ
,
kx1−k x1
y
j
− kx
j
x
1
dy
j
= 1 Let p = 2, then
ρν
2 ,t
= t
k
ν
2 ,η2,k
, one obtains C
n ,k,2,t
x = t
k
ν
2 ,nt
∗ ν
2 ,η2,k
dx
ν
2 ,nt
dx
= t
k
ν
2 ,nt+η2,k
dx
ν
2 ,nt
dx
= t
k
ξ
2 ,nt+η2,k
dx
1
Q
k j=2
ξ
,x
1
dx
j
ξ
2 ,nt
dx
1
Q
k j=2
ξ
,x
1
dx
j
= t
k
x
nt+ η2,k−1
1
Γ[nt + η2, k]
× Γnt
x
nt−1 1
= Γnt
Γ[nt + η2, k]
x
η2,k 1
with the modified Lévy measure parameter η2, k = k + 1.
For p = 3 we have ρν
p ,t
= t
k
2
−η3,k
ν
2 ,η3,k
, then
C
n ,k,3,t
x = t
k
2
−η3,k
ν
3 ,nt
∗ ν
2 ,η3,k
dx
ν
3 ,nt
dx
= t
k
2
−1−k2
Z
R
k
ν
3 ,nt
x − yν
2 ,η3,k
y
ν
3 ,nt
x dy
= t
k
2
−1−k2
Z
x
1
ξ
3 ,nt
x
1
− y
1
ξ
2 ,η3,k
y
1
ξ
3 ,nt
x
1
k
Y
j=2
Z
R
ξ
,x
1
−y
1
x
j
− y
j
ξ
,y
1
y
j
ξ
,x
1
x
j
dy
1
41
= t
k
2
−1−k2
Z
x
1
ξ
3 ,nt
x
1
− y
1
ξ
2 ,η3,k
y
1
ξ
3 ,nt
x
1
× 1dy
1
= t
k
2
−1−k2
Z
x
1
nt
√
2 πx
1
−y
1 3
exp
−nt
2
2x
1
−y
1
×
y
η3,k−1 1
Γ[ η3,k]
exp−y
1 nt
q 2
πx
3 1
exp
−nt
2
2x
1
dy
1
= t
k
2
−1−k2
x
3 2
1
Γ[ η3, k]
exp nt
2
2x
1
Z
x
1
y
η3,k−1 1
x
1
− y
1 3
2
exp −y
1
− nt
2
2x
1
− y
1
dy
1
4.1.3. Bayesian Estimator
We introduce the Bayesian estimator of normal-Poisson generalized vari- ance using the conjugate prior of Poisson distribution namely gamma dis-
tribution.
Theorem 4.1.3. Let X
1
, ··· ,X
n
be random vectors i.i.d with distribution Pµ, G
1 ,t;j
∈ G
ν
1 ,t;j
a normal Poisson model. For t 0 and j ∈ {1,2,...,k} fixed, under assump-
tion of prior gamma distribution of µ
j
with parameter α 0 and β 0, the Bayesian
estimator of det V
F
t;j
µ = µ
k j
is given by
B
n ,t;j,α,β
=
α + nX
j
β + n
k
. 4.5
Proof. Let X
j1
, . . . , X
jn
given µ
j
be Poisson distribution with mean µ
j
, then the probability mass function is given by
px
ji
|µ
j
= µ
x
ji
j
x
ji
exp−µ
j
∀x
ji
∈ N. Assuming that
µ
j
follows gamma α, β, then the prior probability distribu-
tion function of µ
j
is written as f
µ
j
; α, β =
β
α
Γ α
µ
α−1 j
exp−βµ
j
, ∀µ
j
0, with Γ
α := R
∞
x
α−1
e
−x
dx. Using the classical Bayes theorem, the posterior
42 distribution of
µ
j
given an observation x
ji
can be expressed as f
µ
j
|x
ji
; α, β =
px
ji
|µ
j
f µ
j
; α, β
R
µ
j
px
ji
|µ
j
f µ
j
; α, βdµ
j
= β + 1
α+x
ji
Γ α + x
ji
µ
α+x
ji
−1 j
exp{−β + 1µ
j
} which is the gamma density with parameters
α
′
= α + x
ji
and β
′
= β + 1. Then
with random sample X
j1
, . . . , X
jn
the posterior will be gamma α + nX
j
, β + n. Since Bayesian estimator of
µ
j
is given by the expected value of the posterior distribution i.e.
α + nX
j
β + n, this concludes the proof.
4.2. Simulation Study
We implemented the generalized variance estimators on normal gamma, normal inverse Gaussian and normal Poisson models presented in previous
section on our simulation study to see further the numerical behavior of the estimators. Fixing t = 1, we carried out a Monte-Carlo simulation using
R software from R Development Core Team 2016, we considered some parameter configurations of n and k to see the effects of n and k on generalized
variance estimations. For each configuration we generated 1000 samples with dimension k = 2
, 4, 6, 8. We considered several sample sizes varied from n = 3 until n = 1000 and we fixed j = 1. To evaluate the result of the
estimations we calculated the MSE of the generalized variance estimates. We report the numerical results of the generalized variance estimations
for each model, i.e. the empirical expected value of the estimators with its standard errors Se and the empirical mean square error MSE. The
procedure of the data simulation is described in the following steps:
1. Fix k = 2, generate randomly n = k + 1 observations from univariate gamma distribution X
j
with mean µ
j
= 1 using gamma parameters: scale=shape=1.
2. For each X
j
= x
j
, generate the corresponding normal i.i.d. components from k − 1-variate normal distribution with mean zero and variance
X
j
= x
j
, we obtain the normal components of NST model X
c j
such that
X
c j
∼ N
k−1
, X
j
I
k−1
. 3. Combine X
j
and X
c j
for obtaining a k-variate normal gamma random
sample denote by X 4. Calculate the generalized variance of X using ML and UMVU also
Bayesian for p = 1, by using α = X
j
and β = k estimators. Keep the
generalized variance estimates as T
n;k ,p,1
, U
n;k ,p,1
and B
n;k ,p,1,α,β
43 5. Repeat step 1 - step 4 until 1000 times, we obtain 1000 generalized
variance values for each estimator 6. Calculate the expected values and the standard errors of the general-
ized variance estimates resulted from each estimator using the follow- ing formulas
Eb ψ =
1 1000
1000
X
i=1
b ψ
i
,
Seb ψ =
q Varb
ψ = v
u t
1 999
1000
X
i=1
b ψ
i
− Eb ψ
2
, where b
ψ is the estimate of det V
G
p ,t;j
µ using ML, UMVU and Bayesian estimators.
7. Calculate the mean square error MSE of each method over 1000 data sets using the following formula:
MSEb ψ =
h Eb
ψ − detV
G
p ,t;j
µ i
2
+ [Seb ψ]
2
8. Repeat step 1 - step 7 for n = 10 , 20, 30, 60, 100, 300, 500 and 1000.
9. Repeat step 1 - step 8 for other fixed value of k where k ∈ {2,4,6,8}. 10. Repeat step 1 - step 9 for
µ
j
= 5 .
11. Repeat step 1 - step 9 using normal inverse-Gaussian model. 12. Repeat step 1 - step 10 using normal Poisson model and with additional
small mean value µ
j
= 0 .5.
We provide the scatterplots of some generated data from normal gamma, normal inverse-Gaussian and normal Poisson for bivariate and trivariate
cases in Appendix C.
4.2.1. Normal gamma
Table 4.1 and Table 4.2 show the expected values of generalized variance estimates with their standard errors in parentheses and the means square
error values of both ML and UMVU methods for normal gamma model. By setting
µ
j
= 1 , 5 and using equation 3.11 we have generalized variance of
distribution: µ
k+1 j
= 1 and µ
k+1 j
= 5
k+1
. From the result in Table 4.1 and Table 4.2 we can observe different per-
formances of ML estimator T
n ,;k,p,t
and UMVU estimator U
n;k ,p,t
of the generalized variance. The values of T
n ,;k,p,t
converge while the values of
44 Table 4.1: The expected values with empirical standard errors and MSE of
T
n;k ,p,t
and U
n;k ,p,t
for normal-gamma with 1000 replications for given target value
µ
k+1 j
= 1 with k ∈ {2,4,6,8}.
k Target
n ET
n;k ,p,t
SeT
n;k ,p,t
EU
n;k ,p,t
SeU
n;k ,p,t
MSE T
n;k ,p,t
MSE U
n;k ,p,t
2 1
3 1.9805 3.7192
0.8912 1.6736 14.7935
2.8128 10
1.2878 1.2875 0.9756 0.9754
1.7405 0.9520
20 1.1648 0.8236
1.0085 0.7131 0.7054
0.5085 30
1.0998 0.6031 0.9978 0.5471
0.3736 0.2994
60 1.0380 0.4115
0.9881 0.3917 0.1708
0.1536 100
1.0231 0.3152 0.9931 0.3060
0.0999 0.0937
300 1.0036 0.1774
0.9936 0.1757 0.0315
0.0309 500
1.0076 0.1365 1.0016 0.1357
0.0187 0.0184
1000 0.9968 0.0951
0.9983 0.0949 0.0090
0.0090 4
1 5
4.2191 13.3899 0.8720 2.7674
189.6509 7.6750
10 2.3799 5.0869
0.9906 2.1174 27.7810
4.4837 20
1.6461 2.0572 1.0328 1.2906
4.6494 1.6668
30 1.3831 1.3505
1.0066 0.9828 1.9707
0.9660 60
1.1904 0.8014 1.0117 0.6811
0.6784 0.4640
100 1.0869 0.5706
0.9849 0.5171 0.3332
0.2676 300
1.0293 0.2938 0.9957 0.2842
0.0872 0.0808
500 1.0286 0.2296
1.0083 0.2251 0.0535
0.0507 1000
1.0137 0.1610 1.0036 0.1594
0.0261 0.0254
6 1
7 13.7175 103.5833
1.3062 9.8634 10891.2275
97.3811 10
6.6118 36.8236 1.1467 6.3866
1387.4736 40.8103
20 2.2455 4.3052
0.8670 1.6622 20.0860
2.7806 30
1.9055 3.4774 0.9905 1.8076
12.9123 3.2676
60 1.4151 1.5070
1.0092 1.0748 2.4434
1.1553 100
1.2248 0.8843 0.9972 0.7199
0.8325 0.5183
300 1.0606 0.4416
0.9894 0.4119 0.1986
0.1698 500
1.0182 0.3160 0.9765 0.3030
0.1002 0.0924
1000 1.0147 0.2200
0.9983 0.2154 0.0486
0.0464 8
1 10
10.8414 43.4986 0.6145 2.4655
1988.9796 6.2271
20 4.5993 10.9409
0.9395 2.2350 132.6591
4.9989 30
3.0371 11.7396 1.0106 3.9063
141.9670 15.2590
60 1.8257 2.9633
1.0289 1.6700 9.4631
2.7896 100
1.4218 1.3910 1.0017 0.9800
2.1128 0.9604
300 1.1233 0.6240
0.9974 0.5541 0.4046
0.3070 500
1.0872 0.4668 1.0120 0.4345
0.2255 0.1889
1000 1.0434 0.2928
1.0066 0.2824 0.0876
0.0798
U
n;k ,p,t
do not, but U
n;k ,p,t
which is the unbiased estimator always approx- imate the parameter
µ
k+1 j
= 1 and closer to the parameter than T
n ,;k,p,t
for small sample sizes, i.e. for n ≤ 30. For the two methods, the standard er-
ror of the estimates decreases when the sample size increase. Notice that the estimation interval of both estimators roughly become shorter when the
sample size increase.
45
10 20
30 60
100 300
500 1000
ML UMVU
MSE bargraph for k=2 and mu_j=1
Sample Size Mean Square Error
0.0 0.5
1.0 1.5
a k=2
10 20
30 60
100 300
500 1000
ML UMVU
MSE bargraph for k=4 and mu_j=1
Sample Size Mean Square Error
5 10
15 20
25
b k=4
10 20
30 60
100 300
500 1000
ML UMVU
MSE bargraph for k=2 and mu_j=1
Sample Size Mean Square Error
200 400
600 800
1000 1200
c k=6
10 20
30 60
100 300
500 1000
ML UMVU
MSE bargraph for k=8 and mu_j=1
Sample Size Mean Square Error
10000 20000
30000 40000
50000
d k=8
Figure 4.1: Bargraphs of the mean square errors of T
n;k ,p,t
and U
n;k ,p,t
for normal-gamma with
µ
j
= 1 and k ∈ {2,4,6,8}.
There are more important performance characterizations for an estimator than just being unbiased. The MSE is perhaps the most important of them.
It captures the bias and the variance of the estimator. For this reason, we compare the quality of the estimators using their MSE values. The
result shows that when n increases the MSE of the estimates of the two methods become similar, and they both produced almost the same result
for n = 1000. The MSE values in the tables are presented graphically in Figure 4.1 and Figure 4.2 respectively. In the figures it is obviously seen that
the performance of all estimators becomes more similar when the sample size increase. For small sample sizes, U
n;k ,p,t
always has smaller MSE, in this situation U
n;k ,p,t
is preferable than T
n;k ,p,t
. In these figures we also can observe that the difference between U
n;k ,p,t
and T
n;k ,p,t
for small sample sizes increases when the dimension increases.
46 Table 4.2: The expected values with empirical standard errors and MSE of
T
n;k ,p,t
and U
n;k ,p,t
for normal-gamma with 1000 replications for given target value
µ
k+1 j
= 5
k+1
with k ∈ {2,4,6,8}
k Target
n ET
n;k ,p,t
StdT
n;k ,p,t
EU
n;k ,p,t
StdU
n;k ,p,t
MSE T
n;k ,p,t
MSE U
n;k ,p,t
2 125
3 1.46E+02 1.17E+02
6.58E+01 5.27E+01 1.41E+04
6.28E+03 10
1.32E+02 5.68E+01 9.97E+01 4.30E+01
3.27E+03 2.49E+03
20 1.27E+02 3.88E+01
1.10E+02 3.36E+01 1.51E+03
1.34E+03 30
1.27E+02 3.20E+01 1.16E+02 2.91E+01
1.03E+03 9.33E+02
60 1.25E+02 2.27E+01
1.19E+02 2.16E+01 5.16E+02
5.07E+02 100
1.26E+02 1.67E+01 1.22E+02 1.62E+01
2.79E+02 2.73E+02
300 1.25E+02 9.82E+00
1.24E+02 9.72E+00 9.65E+01
9.56E+01 500
1.25E+02 7.64E+00 1.25E+02 7.59E+00
5.84E+01 5.78E+01
1000 1.25E+02 5.32E+00
1.25E+02 5.30E+00 2.83E+01
2.83E+01 4
3125 5
4.55E+03 5.13E+03 9.41E+02 1.06E+03
2.84E+07 5.89E+06
10 3.83E+03 2.90E+03
1.60E+03 1.21E+03 8.89E+06
3.79E+06 20
3.45E+03 1.77E+03 2.17E+03 1.11E+03
3.25E+06 2.16E+06
30 3.39E+03 1.37E+03
2.47E+03 9.95E+02 1.94E+06
1.42E+06 60
3.20E+03 9.56E+02 2.72E+03 8.12E+02
9.20E+05 8.23E+05
100 3.14E+03 6.98E+02
2.84E+03 6.32E+02 4.87E+05
4.78E+05 300
3.15E+03 4.17E+02 3.05E+03 4.03E+02
1.74E+05 1.69E+05
500 3.14E+03 3.18E+02
3.07E+03 3.12E+02 1.02E+05
9.99E+04 1000
3.14E+03 2.18E+02 3.11E+03 2.16E+02
4.79E+04 4.69E+04
6 78125
7 1.51E+05 2.20E+05
1.44E+04 2.10E+04 5.38E+10
4.51E+09 10
1.15E+05 1.34E+05 2.00E+04 2.33E+04
1.94E+10 3.93E+09
20 9.86E+04 7.79E+04
3.81E+04 3.01E+04 6.49E+09
2.51E+09 30
8.83E+04 5.09E+04 4.59E+04 2.64E+04
2.69E+09 1.74E+09
60 8.55E+04 3.70E+04
6.10E+04 2.64E+04 1.43E+09
9.93E+08 100
8.13E+04 2.44E+04 6.62E+04 1.98E+04
6.04E+08 5.35E+08
300 7.87E+04 1.40E+04
7.34E+04 1.31E+04 1.97E+08
1.94E+08 500
7.83E+04 1.11E+04 7.51E+04 1.06E+04
1.23E+08 1.22E+08
1000 7.88E+04 7.92E+03
7.71E+04 7.75E+03 6.31E+07
6.11E+07 8
1953125 10
3.52E+06 6.52E+06 1.99E+05 3.69E+05
4.49E+13 3.21E+12
20 2.79E+06 3.42E+06
5.70E+05 6.98E+05 1.24E+13
2.40E+12 30
2.44E+06 2.01E+06 8.13E+05 6.68E+05
4.27E+12 1.75E+12
60 2.15E+06 1.16E+06
1.21E+06 6.52E+05 1.37E+12
9.78E+11 100
2.17E+06 9.34E+05 1.53E+06 6.58E+05
9.20E+11 6.11E+11
300 2.01E+06 4.79E+05
1.79E+06 4.25E+05 2.33E+11
2.08E+11 500
1.98E+06 3.52E+05 1.84E+06 3.28E+05
1.25E+11 1.20E+11
1000 1.96E+06 2.55E+05
1.89E+06 2.46E+05 6.53E+10
6.44E+10
4.2.2. Normal inverse-Gaussian
We generated normal inverse-Gaussian model in the same way as sim- ulating the normal-gamma model. Table 4.3 shows the expected values of
generalized variance estimates with their standard errors in parentheses and the means square error values of both ML and UMVU methods in case
of normal inverse-Gaussian. By setting
µ
j
= 1 and using equation 3.11 we
47
10 20
30 60
100 300
500 1000
ML UMVU
MSE bargraph for k=2 and mu_j=5
Sample Size Mean Square Error
500 1000
1500 2000
2500 3000
a k=2
10 20
30 60
100 300
500 1000
ML UMVU
MSE bargraph for k=4 and mu_j=5
Sample Size Mean Square Error
0e+00 2e+06
4e+06 6e+06
8e+06
b k=4
10 20
30 60
100 300
500 1000
ML UMVU
MSE bargraph for k=2 and mu_j=5
Sample Size Mean Square Error
0.0e+00 5.0e+09
1.0e+10 1.5e+10
c k=6
10 20
30 60
100 300
500 1000
ML UMVU
MSE bargraph for k=8 and mu_j=5
Sample Size Mean Square Error
0e+00 1e+13
2e+13 3e+13
4e+13
d k=8
Figure 4.2: Bargraphs of the mean square errors of T
n;k ,p,t
and U
n;k ,p,t
for normal-gamma with
µ
j
= 5 and k ∈ {2,4,6,8}.
have generalized variance of distribution: µ
k+2 j
= 1. Similar with the result of generalized variance estimation for normal
gamma, the result for normal inverse-Gaussian shows that UMVU method produced better estimates than ML method for small sample sizes. When
the sample size n increases, the expected values of the estimates of the two methods become closer to the target value and they both produced almost
the same result for n = 1000. The MSE values in Table 4.3 is presented as bargraphs in Figure 4.3.
In Figure 4.3 the behavior of MSE of T
n;k ,p,t
and U
n;k ,p,t
is displayed clearly. For small sample sizes n ≤ 30, U
n;k ,p,t
is preferable than T
n;k ,p,t
. The differ- ence between the two methods when n ≤ 30 increases when k increases.
48 Table 4.3: The expected values with standard errors and MSE of T
n;k ,p,t
and U
n;k ,p,t
for normal inverse-Gaussian with 1000 replications for given target value
µ
k+2 j
= 1 and k ∈ {2,4,6,8}.
k Target
n E T
n;k ,p,t
StdT
n;k ,p,t
E U
n;k ,p,t
StdU
n;k ,p,t
MSE T
n;k ,p,t
MSE U
n;k ,p,t
2 1
3 2.0068 4.9227
0.9135 0.8235 25.2469
0.6856 10
1.4249 2.8513 1.0316 0.4388
8.3103 0.1935
20 1.5936 1.8951
1.1340 0.3718 3.9439
0.1562 30
1.3677 1.0155 1.1641 0.2668
1.1664 0.0981
60 1.0846 0.5341
1.1104 0.1856 0.2924
0.0466 100
1.0819 0.5166 1.1102 0.1675
0.2735 0.0402
300 1.0006 0.2570
1.0843 0.0919 0.0660
0.0156 500
1.0356 0.1890 1.1374 0.0727
0.0370 0.0242
1000 1.0156 0.1219
1.0116 0.0670 0.0151
0.0115 4
1 5
9.3836 30.0947 1.3196 1.1323
975.9726 1.3843
10 4.6547 13.8643
1.2837 0.8153 205.5754
0.7452 20
2.7487 5.1845 1.2963 0.6189
29.9373 0.4709
30 1.4822 2.1166
1.1854 0.4572 4.7125
0.2434 60
1.3095 1.1051 1.2560 0.3054
1.3170 0.1588
100 1.1673 0.8467
1.2264 0.2671 0.7449
0.1226 300
1.0849 0.4296 1.2542 0.1520
0.1918 0.0877
500 1.0350 0.2839
1.0762 0.0914 0.0818
0.0416 1000
1.0107 0.2080 1.0102 0.1137
0.0434 0.0337
6 1
7 20.4865 113.4633
0.9423 0.9984 13253.6508
1.0001 10
12.1032 55.7841 1.0596 0.8610
3235.1488 0.7449
20 3.4498 10.3056
1.0054 0.5933 112.2060
0.3520 30
2.1422 3.2262 1.0246 0.4970
11.7130 0.2476
60 1.8236 2.6064
1.0587 0.3744 7.4717
0.1436 100
1.2468 1.1599 1.0129 0.2643
1.4062 0.1170
300 1.0781 0.4953
1.0568 0.1596 0.2514
0.0929 500
1.0815 0.4065 1.0230 0.1110
0.1719 0.0922
1000 1.0207 0.2816
1.0204 0.0775 0.0798
0.0760 8
1 10
27.9651 106.4417 1.1645 1.2832
12056.9414 1.6737
20 10.2639 47.3683
1.2127 0.9227 2329.5787
0.8674 30
5.8903 14.2638 1.2634 0.8024
227.3707 0.7133
60 1.8667 3.2137
1.1402 0.4894 11.0792
0.2504 100
1.5251 1.8103 1.1340 0.3734
3.5530 0.1591
300 1.2059 0.8122
1.1398 0.2275 0.7021
0.1571 500
1.1817 0.6075 1.1210 0.3032
0.4021 0.1200
1000 1.0325 0.3189
1.1125 0.0564 0.1027
0.0910
4.2.3. Normal Poisson
Again by fixing j = 1 we set several sample sizes n varied from 5 until 1000 and we generated 1000 samples for each n. However, for normal Poisson
case, to see the effect of zero values proportion within X
j
, we also consider small mean values on the Poisson component because PX
j
= 0 = exp−µ
j
, then we set
µ
j
= 0 .5, 1 and 5. We also used Theorem 4.1.3 for calculating
Bayesian estimator in this simulation, we assume that the parameters of
49
10 20
30 60
100 300
500 1000
ML UMVU
MSE bargraph for k=2 and mu_j=1
Sample Size Mean Square Error
2 4
6 8
a k=2
10 20
30 60
100 300
500 1000
ML UMVU
MSE bargraph for k=4 and mu_j=1
Sample Size Mean Square Error
50 100
150 200
b k=4
10 20
30 60
100 300
500 1000
ML UMVU
MSE bargraph for k=6 and mu_j=1
Sample Size Mean Square Error
500 1000
1500 2000
2500 3000
c k=6
10 20
30 60
100 300
500 1000
ML UMVU
MSE bargraph for k=8 and mu_j=1
Sample Size Mean Square Error
2000 4000
6000 8000
10000 12000
d k=8
Figure 4.3: Bargraphs of the mean square errors of T
n;k ,p,t
and U
n;k ,p,t
for normal inverse Gaussian with
µ
j
= 1 and k ∈ {2,4,6,8}.
prior distribution depend on sample mean of the Poisson component X
j
and the space dimension k. The results are presented in Tables 4.4 - 4.9. From the expected values and
the standard errors in Tables 4.4, 4.6 and 4.8, we can observe the performance of ML T
n;k ,p,t
, UMVU U
n;k ,p,t
and Bayesian B
n;k ,p,t,α,β
estimations on the generalized variance. The values of all estimators converge; but U
n;k ,p,t
, which is the unbiased estimator, always approximates the target m
k j
more accurately than T
n;k ,p,t
and B
n;k ,p,t,α,β
for small sample sizes n 6 30. Notice that U
n;k ,p,t
can be calculated only if nX
j
k; for this reason U
n;k ,p,t
is not available for some n when m
j
= 0 .5. In this case, for other n where U
n;k ,p,t
is available, we can observe that B
n;k ,p,t,X
j
,k
is closer to U
n;k ,p,t
than T
n;k ,p,t
. Thus if
µ
j
is small and U
n;k ,p,t
is not available, we can get a good estimation
50 Table 4.4: The expected values with empirical standard errors of T
n;k ,p,t
, U
n;k ,p,t
and B
n;k ,p,t,α,β
for normal-Poisson from 1000 replications for given target value
µ
k j
= 0 .5
k
with k ∈ {2,4,6,8}, α = X
j
and β = k.
k Target
n T
n;k ,p,t
Std
T
U
n;k ,p,t
Std
U
B
n;k ,p,t,X
j
,k
Std
B
2 0.25
3 0.3930 0.5426
- 0.2515 0.3473
10 0.2868 0.2421
0.2378 0.2212 0.2410 0.2034
20 0.2652 0.1660
0.2407 0.1583 0.2416 0.1513
30 0.2642 0.1374
0.2476 0.1332 0.2480 0.1290
60 0.2598 0.0903
0.2514 0.0888 0.2515 0.0874
100 0.2534 0.0712
0.2484 0.0705 0.2485 0.0698
300 0.2495 0.0418
0.2478 0.0417 0.2478 0.0415
500 0.2491 0.0313
0.2482 0.0313 0.2482 0.0312
1000 0.2495 0.0221
0.2490 0.0221 0.2490 0.0221
4 0.0625
5 0.2999 0.8462
- 0.0592 0.1672
10 0.1696 0.3115
0.0689 0.1750 0.0646 0.1187
20 0.1089 0.1541
0.0658 0.1097 0.0638 0.0903
30 0.0886 0.0894
0.0617 0.0689 0.0613 0.0618
60 0.0774 0.0559
0.0642 0.0487 0.0639 0.0461
100 0.0704 0.0403
0.0627 0.0370 0.0627 0.0358
300 0.0643 0.0207
0.0618 0.0201 0.0618 0.0199
500 0.0635 0.0158
0.0620 0.0156 0.0620 0.0155
1000 0.0631 0.0115
0.0624 0.0114 0.0624 0.0113
6 0.015625
7 0.2792 1.2521
- 0.0152 0.0680
10 0.1212 0.3918
0.0165 0.0858 0.0128 0.0414
20 0.0427 0.0883
0.0124 0.0345 0.0119 0.0245
30 0.0356 0.0539
0.0151 0.0271 0.0145 0.0220
60 0.0236 0.0281
0.0149 0.0196 0.0147 0.0175
100 0.0211 0.0183
0.0159 0.0145 0.0158 0.0137
300 0.0173 0.0089
0.0157 0.0082 0.0157 0.0081
500 0.0166 0.0068
0.0157 0.0064 0.0157 0.0064
1000 0.0164 0.0044
0.0159 0.0043 0.0159 0.0043
8 0.00390625
10 0.0891 0.4110
- 0.0017 0.0080
20 0.0384 0.1409
0.0054 0.0288 0.0038 0.0141
30 0.0171 0.0383
0.0037 0.0107 0.0033 0.0075
60 0.0081 0.0119
0.0035 0.0058 0.0034 0.0050
100 0.0063 0.0082
0.0038 0.0053 0.0037 0.0048
300 0.0045 0.0031
0.0038 0.0027 0.0038 0.0026
500 0.0045 0.0024
0.0040 0.0022 0.0040 0.0021
1000 0.0041 0.0015
0.0039 0.0014 0.0039 0.0014
by using B
n;k ,p,t,X
j
,k
. While for µ
j
= 1 and µ
j
= 5, the Bayesian estimator with prior distribution gammaX
j
, k produces the closer estimates to the UMVU than ML method. We can improve this Bayesian estimator by using other
parameter values of prior distribution. From the MSEs in Tables 4.5,4.7 and 4.9 we can conclude that all estimators are consistent.
In this simulation, the proportion of zero values in the samples increases
51 Table 4.5: The empirical mean square errors MSE of T
n;k ,p,t
, U
n;k ,p,t
and B
n;k ,p,t,α,β
for normal-Poisson from 1000 replications with µ
j
= 0 .5,
k ∈ {2,4,6,8}, α = X
j
and β = k.
k n
MSET
n;k ,p,t
MSEU
n;k ,p,t
MSEB
n;k ,p,t,X
j
,k
2 3
0.3148 -
0.1206 10
0.0600 0.0491
0.0415 20
0.0278 0.0251
0.0229 30
0.0191 0.0177
0.0166 60
0.0083 0.0079
0.0076 100
0.0051 0.0050
0.0049 300
0.0017 0.0017
0.0017 500
0.0010 0.0010
0.0010 1000
0.0005 0.0005
0.0005 4
5 0.7724
- 0.0280
10 0.1085
0.0306 0.0141
20 0.0259
0.0120 0.0082
30 0.0087
0.0048 0.0038
60 0.0033
0.0024 0.0021
100 0.0017
0.0014 0.0013
300 0.0004
0.0004 0.0004
500 0.0003
0.0002 0.0002
1000 0.0001
0.0001 0.0001
6 7
1.6371 -
0.0046 10
0.1646 0.0074
0.0017 20
0.0085 0.0012
0.0006 30
0.0033 0.0007
0.0005 60
0.0009 0.0004
0.0003 100
0.0004 0.0002
0.0002 300
0.0001 0.0001
0.0001 500
0.0000 0.0000
0.0000 1000
0.0000 0.0000
0.0000 8
10 0.1762
- 0.0001
20 0.0210
0.0008 0.0002
30 0.0016
0.0001 0.0001
60 0.0002
0.0000 0.0000
100 0.0001
0.0000 0.0000
300 0.0000
0.0000 0.0000
500 0.0000
0.0000 0.0000
1000 0.0000
0.0000 0.0000
when the mean of the Poisson component becomes smaller. For normal- Poisson distribution with
µ
j
= 0 .5, we have many zero values in the samples.
However, this situation does not affect the generalized variance estimation as we can see that T
n;k ,p,t
, U
n;k ,p,t
and B
n;k ,p,t,α,β
have the same behavior for all values of
µ
j
. The MSE in Table 4.5, 4.7 and 4.9 are displayed as graphs presented in
Figure 4.4, Figure 4.5 and Figure 4.6. From those figures we conclude that
52
10 20
30 60
100 300
500 1000
ML UMVU
BAYES
MSE bargraph for k=2 and mu_j=0.5
Sample Size Mean Square Error
0.00 0.01
0.02 0.03
0.04 0.05
a k=2
10 20
30 60
100 300
500 1000
ML UMVU
BAYES
MSE bargraph for k=4 and mu_j=0.5
Sample Size Mean Square Error
0.00 0.02
0.04 0.06
0.08 0.10
b k=4
10 20
30 60
100 300
500 1000
ML UMVU
BAYES
MSE bargraph for k=6 and mu_j=0.5
Sample Size Mean Square Error
0.00 0.05
0.10 0.15
c k=6
10 20
30 60
100 300
500 1000
ML UMVU
BAYES
MSE bargraph for k=8 and mu_j=0.5
Sample Size Mean Square Error
0.00 0.05
0.10 0.15
d k=8
Figure 4.4: Bargraphs of the mean square errors of T
n;k ,p,t
, U
n;k ,p,t
and B
n;k ,p,t,α,β
for normal-Poisson with µ
j
= 0 .5 and k ∈ {2,4,6,8}.
the U
n;k ,p,t
is preferable than T
n;k ,p,t
because it always has smaller MSE val- ues when sample sizes are small; i.e. n 6 30. In this situation, the difference
between U
n;k ,p,t
and T
n;k ,p,t
increases when the dimension k increases. The lack of U
n;k ,p,t
when n kX
j
can be solved by using Bayesian estimator as an alternative for obtaining a better estimation than ML estimator. How-
ever, UMVU is the only unbiased estimator while ML and Bayesian are asymptotically unbiased.
Notice that for µ
j
= 5 the MSE values of Bayesian estimator for n ≥ 30 are greater than the ML estimator. This kind of behavior also happens for
µ
j
= 10 we do not present the result here. Hence we conclude that ML estimator is better than the Bayesian estimator for large values of
µ
j
.
53
Table 4.6: The expected values with empirical standard errors of T
n;k ,p,t
, U
n;k ,p,t
and B
n;k ,p,t,α,β
for normal-Poisson from 1000 replications for given target value
µ
k j
= 1 with k ∈ {2,4,6,8}, α = X
j
and β = k.
k Target
n T
n;k ,p,t
Std
T
U
n;k ,p,t
Std
U
B
n;k ,p,t,X
j
,k
Std
B
2 1
3 1.3711 1.4982
1.0349 1.3130 0.8775 0.9589
10 1.0810 0.6589
0.9817 0.6286 0.9083 0.5536
20 1.0424 0.4471
0.9925 0.4363 0.9498 0.4074
30 1.0329 0.3817
0.9996 0.3756 0.9694 0.3583
60 1.0184 0.2661
1.0017 0.2639 0.9858 0.2576
100 1.0066 0.2016
0.9966 0.2006 0.9870 0.1977
300 1.0112 0.1153
1.0079 0.1151 1.0045 0.1146
500 0.9986 0.0942
0.9966 0.0941 0.9946 0.0938
1000 0.9998 0.0641
0.9988 0.0641 0.9978 0.0640
4 1
5 2.6283 5.0058
1.0721 2.7753 0.5192 0.9888
10 1.7362 2.2949
1.0422 1.6267 0.6617 0.8746
20 1.3276 1.1713
1.0073 0.9588 0.7782 0.6866
30 1.2274 0.8892
1.0167 0.7750 0.8482 0.6145
60 1.1111 0.5643
1.0085 0.5250 0.9170 0.4657
100 1.0647 0.4448
1.0038 0.4260 0.9471 0.3957
300 1.0245 0.2389
1.0043 0.2354 0.9846 0.2296
500 1.0092 0.1889
0.9972 0.1872 0.9854 0.1845
1000 1.0013 0.1272
0.9953 0.1267 0.9894 0.1257
6 1
7 4.5153 12.8404
0.9378 4.0255 0.2452 0.6974
10 3.6865 8.1473
1.1642 3.3992 0.3893 0.8603
20 1.9674 2.9034
1.0227 1.7467 0.5462 0.8061
30 1.5605 1.8825
0.9901 1.3133 0.6362 0.7675
60 1.2954 1.0360
1.0220 0.8541 0.8075 0.6458
100 1.2084 0.7824
1.0462 0.6957 0.9043 0.5855
300 1.0621 0.3793
1.0109 0.3641 0.9621 0.3436
500 1.0294 0.2778
0.9992 0.2710 0.9699 0.2617
1000 1.0185 0.1939
1.0034 0.1915 0.9885 0.1882
8 1
10 7.4252 31.4171
0.9507 7.0716 0.1444 0.6111
20 2.9347 6.6901
0.8934 2.6109 0.2938 0.6698
30 2.1114 4.0742
0.9266 2.1005 0.4142 0.7992
60 1.5445 1.8064
0.9988 1.2576 0.6477 0.7575
100 1.2712 1.0323
0.9704 0.8197 0.7437 0.6039
300 1.0814 0.5222
0.9863 0.4820 0.8998 0.4345
500 1.0569 0.3933
0.9998 0.3748 0.9458 0.3520
1000 1.0290 0.2634
1.0007 0.2571 0.9732 0.2491
54
Table 4.7: The empirical mean square errors MSE of T
n;k ,p,t
, U
n;k ,p,t
and B
n;k ,p,t,α,β
for normal-Poisson from 1000 replications with µ
j
= 1, k ∈ {2,4,6,8}, α = X
j
and β = k.
k n
MSET
n;k ,p,t
MSEU
n;k ,p,t
MSEB
n;k ,p,t,X
j
,k
2 3
2.3824 1.7252
0.9344 10
0.4407 0.3955
0.3149 20
0.2017 0.1904
0.1685 30
0.1468 0.1411
0.1293 60
0.0711 0.0697
0.0665 100
0.0407 0.0403
0.0392 300
0.0134 0.0133
0.0131 500
0.0089 0.0089
0.0088 1000
0.0041 0.0041
0.0041 4
5 27.7093
7.7075 1.2089
10 5.8085
2.6480 0.8794
20 1.4793
0.9193 0.5206
30 0.8424
0.6008 0.4007
60 0.3308
0.2757 0.2238
100 0.2021
0.1815 0.1594
300 0.0577
0.0554 0.0529
500 0.0358
0.0351 0.0342
1000 0.0162
0.0161 0.0159
6 7
177.2319 16.2084
1.0560 10
73.5952 11.5816
1.1131 20
9.3656 3.0514
0.8557 30
3.8580 1.7250
0.7214 60
1.1606 0.7300
0.4541 100
0.6556 0.4861
0.3520 300
0.1477 0.1327
0.1195 500
0.0780 0.0734
0.0694 1000
0.0379 0.0367
0.0356 8
10 1028.3183
50.0103 1.1055
20 48.5011
6.8281 0.9473
30 17.8342
4.4177 0.9819
60 3.5595
1.5815 0.6979
100 1.1392
0.6728 0.4304
300 0.2793
0.2325 0.1988
500 0.1579
0.1405 0.1268
1000 0.0702
0.0661 0.0628
55
10 20
30 60
100 300
500 1000
ML UMVU
BAYES
MSE bargraph for k=2 and mu_j=1
Sample Size Mean Square Error
0.0 0.1
0.2 0.3
0.4
a k=2
10 20
30 60
100 300
500 1000
ML UMVU
BAYES
MSE bargraph for k=4 and mu_j=1
Sample Size Mean Square Error
1 2
3 4
5
b k=4
10 20
30 60
100 300
500 1000
ML UMVU
BAYES
MSE bargraph for k=6 and mu_j=1
Sample Size Mean Square Error
10 20
30 40
50 60
70
c k=6
10 20
30 60
100 300
500 1000
ML UMVU
BAYES
MSE bargraph for k=8 and mu_j=1
Sample Size Mean Square Error
200 400
600 800
1000
d k=8
Figure 4.5: Bargraphs of the mean square errors of T
n;k ,p,t
, U
n;k ,p,t
and B
n;k ,p,t,α,β
for normal-Poisson with µ
j
= 1 and k ∈ {2,4,6,8}.
56
Table 4.8: The expected values with empirical standard errors of T
n;k ,p,t
, U
n;k ,p,t
and B
n;k ,p,t,α,β
normal-Poisson from 1000 replications for given target value
µ
k j
= 5
k
with k ∈ {2,4,6,8}, α = X
j
and β = k.
k Target
n T
n;k ,p,t
Std
T
U
n;k ,p,t
Std
U
B
n;k ,p,t,X
j
,k
Std
B
2 25
3 26.59 13.49
24.93 13.06 17.02 8.63
10 25.50 7.04
25.00 6.97 21.42 5.92
20 25.30 5.13
25.05 5.10 23.06 4.67
30 25.14 4.07
24.97 4.06 23.59 3.82
60 25.10 2.84
25.01 2.83 24.29 2.75
100 25.14 2.25
25.09 2.25 24.65 2.21
300 25.02 1.28
25.01 1.28 24.86 1.27
500 25.01 1.01
25.00 1.01 24.91 1.01
1000 25.00 0.68
25.00 0.68 24.95 0.68
4 625
10 722.35 379.32
643.63 348.05 275.30 144.56
20 656.31 267.59
618.66 256.15 384.72 156.86
30 643.63 213.01
618.63 206.83 444.80 147.21
60 640.46 148.59
627.87 146.40 528.56 122.63
100 635.85 113.86
628.30 112.85 565.60 101.28
300 627.40 64.85
624.90 64.65 603.00 62.33
500 624.47 49.86
622.97 49.77 609.73 48.68
1000 623.99 36.45
623.24 36.42 616.56 36.02
6 15625
10 21464.28 21253.41
16234.02 17018.96 2266.48 2244.22
20 18437.60 11153.66
15946.65 9905.34 5118.94 3096.66
30 17479.01 8777.20
15851.79 8099.08 7126.43 3578.58
60 16272.58 5688.96
15485.73 5460.96 10143.13 3546.08
100 16036.24 4356.63
15564.94 4250.16 12000.40 3260.19
300 15724.19 2492.89
15567.97 2472.28 14244.22 2258.26
500 15670.97 1874.69
15577.30 1865.36 14764.51 1766.25
1000 15708.09 1336.94
15661.09 1333.60 15245.44 1297.56
8 390625
10 695388.58 1228177.84
420480.34 830043.27 13526.61 23890.36
20 536931.48 478032.35
411437.57 381388.82 53753.78 47857.21
30 451337.25 314617.29
376214.48 268982.30 88537.24 61717.36
60 423826.51 195474.33
386517.24 180478.34 177727.86 81970.41
100 419253.53 153381.74
396662.57 146155.19 245277.48 89733.50
300 394950.63 83679.72
387661.84 82331.65 328601.12 69621.99
500 392989.13 63923.39
388617.32 63302.46 351700.23 57207.36
1000 390946.84 43047.19
388764.23 42836.90 369748.20 40713.01
57
Table 4.9: The empirical mean square errors MSE of T
n;k ,p,t
, U
n;k ,p,t
and B
n;k ,p,t,α,β
for normal-Poisson from 1000 replications with µ
j
= 5, k ∈ {2,4,6,8}, α = X
j
and β = k.
k n
MSET
n;k ,p,t
MSEU
n;k ,p,t
MSEB
n;k ,p,t,X
j
,k
2 3
1.8452E+02 1.7062E+02
1.3821E+02 10
4.9851E+01 4.8634E+01
4.7817E+01 20
2.6385E+01 2.6036E+01
2.5606E+01 30
1.6613E+01 1.6486E+01
1.6607E+01 60
8.0631E+00 8.0272E+00
8.0455E+00 100
5.0817E+00 5.0602E+00
4.9898E+00 300
1.6345E+00 1.6329E+00
1.6328E+00 500
1.0213E+00 1.0208E+00
1.0211E+00 1000
4.6392E-01 4.6384E-01
4.6441E-01 4
10 1.5336E+05
1.2149E+05 1.4319E+05
20 7.2587E+04
6.5653E+04 8.2341E+04
30 4.5719E+04
4.2820E+04 5.4142E+04
60 2.2319E+04
2.1442E+04 2.4339E+04
100 1.3081E+04
1.2745E+04 1.3786E+04
300 4.2109E+03
4.1801E+03 4.3684E+03
500 2.4859E+03
2.4808E+03 2.6029E+03
1000 1.3299E+03
1.3296E+03 1.3687E+03
6 10
4.8580E+08 2.9002E+08
1.8349E+08 20
1.3231E+08 9.8219E+07
1.1997E+08 30
8.0477E+07 6.5646E+07
8.5032E+07 60
3.2784E+07 2.9841E+07
4.2626E+07 100
1.9149E+07 1.8068E+07
2.3767E+07 300
6.2243E+06 6.1154E+06
7.0063E+06 500
3.5166E+06 3.4819E+06
3.8601E+06 1000
1.7943E+06 1.7798E+06
1.8277E+06 8
10 1.6013E+12
6.8986E+11 1.4277E+11
20 2.4992E+11
1.4589E+11 1.1577E+11
30 1.0267E+11
7.2559E+10 9.5066E+10
60 3.9313E+10
3.2589E+10 5.2044E+10
100 2.4346E+10
2.1398E+10 2.9178E+10
300 7.0210E+09
6.7873E+09 8.6942E+09
500 4.0918E+09
4.0112E+09 4.7878E+09
1000 1.8532E+09
1.8385E+09 2.0934E+09
58
10 20
30 60
100 300
500 1000
ML UMVU
BAYES
MSE bargraph for k=2 and mu_j=5
Sample Size Mean Square Error
10 20
30 40
a k=2
10 20
30 60
100 300
500 1000
ML UMVU
BAYES
MSE bargraph for k=4 and mu_j=5
Sample Size Mean Square Error
20000 40000
60000 80000
100000 140000
b k=4
10 20
30 60
100 300
500 1000
ML UMVU
BAYES
MSE bargraph for k=6 and mu_j=5
Sample Size Mean Square Error
0e+00 1e+08
2e+08 3e+08
4e+08
c k=6
10 20
30 60
100 300
500 1000
ML UMVU
BAYES
MSE bargraph for k=8 and mu_j=5
Sample Size Mean Square Error
0.0e+00 5.0e+11
1.0e+12 1.5e+12
d k=8
Figure 4.6: Bargraphs of the mean square errors of T
n;k ,p,t
, U
n;k ,p,t
and B
n;k ,p,t,α,β
for normal-Poisson with µ
j
= 5 and k ∈ {2,4,6,8}.
5. CHARACTERIZATIONS OF NORMAL POISSON
MODELS
A normal-Poisson model is a special case of normal stable Tweedie NST models. Similar to all NST models, this model was introduced by Boubacar
Maïnassara and Kokonendji for particular case of j, i.e. j = 1. They de- fined that a k-variate normal-Poisson model is composed by distribution of
random vector X = X
1
, . . . , X
k ⊤
where X
1
is a univariate non-negative Pois- son variable and X
2
, . . . , . . . , X
k ⊤
=: X
c 1
given X
1
are k − 1 real independent Gaussian variables with variance X
1
. 5.1.
Definition and Properties
Following Boubacar Maïnassara and Kokonendji 2014 with j = 1 we here describe the family of multivariate normal-Poisson
j
models.
Definition 5.1.1. Let j ∈ {1,2,...,k} and k 1. For a k-dimensional normal-Poisson
random vector X, it must hold two conditions:
1. X
j
is a univariate Poisson random variable 2. X
1
, . . . , X
j−
, X
j+1
. . . , X
k ⊤
given X
j
=: X
c j
|X
j
= x
j
, follows the k−1-variate normal N
k−1
, X
j
I
k−1
distribution, where I
ℓ
= diag
ℓ
1 , . . . , 1 denotes the
ℓ × ℓ unit matrix. In order to satisfy the second condition we need X
j
0. But in practice it is possible to have X
j
= 0 in the Poisson component. In this case, the corresponding normal components are degenerated as the Dirac mass
δ which makes their values become 0s. In the previous chapter it has been
shown that zero values in X
j
do not affect the estimation of the generalized variance of normal-Poisson. Theoretically, the problem may happens when
we have many zero values in the sample; that is the distribution of normal components will change into Dirac distribution.
Then by Definition 5.1.1 and Equation 3.4, for a fixed power of convo- lution t
0 and given j ∈ {1,2,...,k}, denote F
t;j
= F
ν
1 ,t;j
with ν
1 ,t;j
:= ν
∗t 1
,j
. The NEF Kotz et al., 2000, Chapter 54 of a k-dimensional normal-Poisson
j
random vector X is generated by
ν
1 ,t;j
dx =
t
x
j
x
j −1
2 πx
j k−12
exp
−t−
1 2x
j
X
ℓ,j
x
2 ℓ
1
x
j
∈N\{0}
δ
x
j
dx
j
Y
ℓ,j
dx
ℓ
, 5.1
where I
A
is the indicator function of the set A. Since t 0 then ν
1 ,t;j
is known to be an infinitely divisible measure; see, e.g., Sato 1999. For simplicity,
henceforth we replace ν
1 ,t;j
by ν
t;j
. 59
60 The cumulant function which is the logarithm of the Laplace transform
of ν
t;j
is:
K
ν
t;j
θ = log Z
R
k
expθ
⊤
x ν
t;j
dx
= log Z
R
exp θ
j
x
j
Y
ℓ,j
Z
R
exp θ
ℓ
x
ℓ
ν
N0,x
j
dx
ℓ
ν
Pt
dx
j
= log Z
R
exp θ
j
x
j
Y
ℓ,j
exp x
j
θ
2 ℓ
2
ν
Pt
dx
j
= t exp
θ
j
+ 1
2 X
ℓ,j
θ
2 ℓ
5.2
where ν
N0,x
j
= 1
2 πx
j 1
2
exp −
1 2x
j
x
2 ℓ
and ν
Pt
= t
x
j
x
j −1
exp−t1
x
j
∈N\{0}
δ
x
j
.
The function K
ν
t;j
θ is finite for all θ in the canonical domain
Θ ν
t;j
=
θ ∈
R
k
; θ
⊤
˜θ
c j
:= θ
j
+ 1
2 X
ℓ,j
θ
2 ℓ
5.3
with θ = θ
1
, . . . , θ
k ⊤
and ˜θ
c j
:= θ
1
, . . . , θ
j−1
, θ
j
= 1 , θ
j+1
, . . . , θ
k ⊤
. 5.4
The probability distribution of normal-Poisson
j
which is a member of NEF is given by
Pθ; ν
t;j
dx = exp{θ
⊤
x − K
ν
t;j
θ}ν
t;j
dx .
From 5.2 we can calculate the first derivative of the cumulant function that produces a k-vector as the mean vector of F
t;j
, and also its second derivative which is a k × k matrix that represents the covariance matrix. The
first derivative of the cumulant function with respect to θ is obtained as follow:
61
K
′ ν
t;j
θ = ∂
∂θ
s
K
ν
t;j
θ
s=1 ,...,k
=
t
θ
1
exp
θ
j
+
1 2
P
ℓ,j
θ
2 ℓ
...
t θ
j−1
exp
θ
j
+
1 2
P
ℓ,j
θ
2 ℓ
t exp
θ
j
+
1 2
P
ℓ,j
θ
2 ℓ
t
θ
j+1
exp
θ
j
+
1 2
P
ℓ,j
θ
2 ℓ
...
t θ
k
exp
θ
j
+
1 2
P
ℓ,j
θ
2 ℓ
= tθ exp
θ
j
+ 1
2 X
ℓ,j
θ
2 ℓ
θ
1
... θ
j−1
1 θ
j+1
... θ
k
.
= t exp
θ
j
+ 1
2 X
ℓ,j
θ
2 ℓ
θθ
c j
= K
ν
t;j
θ × ˜θ
c j
Then the second derivative of the cumulant function with respect to θ is obtained as follow:
62
K
′′ ν
t
θ = ∂
∂θ
s
K
′ ν
t
θ
s=1 ,...,k
= t exp
θ
j
+ 1
2 X
ℓ,j
θ
2 ℓ
1 +
θ
2 1
θ
1
θ
2
θ
1
θ
3
. . . θ
1
θ
j−1
θ
1
θ
1
θ
j+1
. . . θ
1
θ
k
θ
2
θ
1
1 + θ
2 2
θ
2
θ
3
. . . θ
2
θ
j−1
θ
2
θ
2
θ
j+1
. . . θ
2
θ
k
θ
3
θ
1
θ
2
θ
3
1 + θ
2 3
. . . θ
3
θ
j−1
θ
3
θ
3
θ
j+1
. . . θ
3
θ
k
... . ..
... θ
j−1
θ
1
θ
j−1
θ
2
θ
j−1
θ
3
. . . 1 +
θ
2 j−1
θ
j−1
θ
j−1
θ
j+1
. . . θ
j−1
θ
k
θ
1
θ
2
θ
3
. . . θ
j−1
1 θ
j+1
. . . θ
k
θ
j+1
θ
1
θ
j+1
θ
2
θ
j+1
θ
3
. . . 1 + θ
j+1
θ
j−1
θ
j+1
1 + θ
2 j+1
. . . θ
j+1
θ
k
... . ..
... θ
k
θ
1
θ
k
θ
2
θ
k
θ
3
. . . θ
k
θ
j−1
θ
k
θ
k
θ
j+1
. . . 1 + θ
2 k
= t exp
θ
j
+ 1
2 X
ℓ,j
θ
2 ℓ
θ
1
... θ
j−1
1 θ
j+1
... θ
k
h θ
1
··· θ
j−1
1 θ
j+1
··· θ
k
i +
1 0 ··· ··· 0 0
0 1 ··· 0 0
0 0 1
0 ··· 0 0 ... ...
j
... ... ... ...
1 ... ...
0 0 ··· 1 0
0 0 ··· ··· 0 1
63 Using notations in 5.4 we obtain:
K
′′ ν
t;j
θ = K
ν
t;j
θ × ˜θ
c j
˜θ
c⊤ j
+ I
j
k
,
with I
j
k
= diag
k
1 , . . . , 1, 0
j
, 1, . . . , 1. The cumulant function presented in 5.2 and its derivatives are functions of the canonical parameter θ. For practical
calculation we need to use the following mean parameterization:
Pµ; F
t;j
:= Pθµ; ν
t;j
,
where θµ is the solution in θ of the equation µ = K
′ ν
t;j
θ. For fixed j ∈ {1,2,...,k}, the variance function of a normal-Poisson
j
model which is the variance-covariance matrix in term of mean parameterization
is obtained through the second derivative of the cumulant function, i.e. V
F
t;j
µ = K
′′ ν
t;j
θµ. Then we have
V
F
t;j
µ = 1
µ
j
µµ
⊤
+ diag
k
µ
j
, . . . , µ
j
, 0
j
, µ
j
, . . . , µ
j
5.5 on its support
M
F
t;j
= n
µ ∈ R
k
; µ
j
0 and µ
ℓ
∈ Rforℓ , j o
. 5.6
Consequently, its generalized variance function is
det V
F
t;j
µ = µ
k j
with µ ∈ M
F
t;j
. 5.7
Equation 5.7 expresses that the generalized variance of normal-Poisson
j
model depends mainly on the mean of the Poisson component. The infinite divisibility property of normal-Poisson is very useful for
its characterization by generalized variance. Regarding to this property, Kokonendji and Seshadri 1994 and Hassairi 1999 found an interesting
representation that we state it in the following proposition.
Proposition 5.1.1. If
ν is an infinitely divisible measure generating an NEF F = F
ν on R
k
, then there exists a positive measure ρν on R
k
such that
det K
′′ ν
θ = exp K
ρν
θ, for all θ ∈ Θν. The positive measure ρν is called the modified Lévy measure of ν.
For F
t;j
of normal-Poisson
j
model, the modified Lévy measure that satis- fies Proposition 5.1.1 is given by
ρν
t;j
= t
k
δ
e
j
Y
ℓ,j
N0,1
∗k
, 5.8
64
where e
i i=1
,...,k
an orthonormal basis of R
k
and N0,1 is the standard uni- variate normal probability measure. It comes from the cumulant function of
ρν
t;j
which is
K
ρν
t;j
θ = k
t
θ
j
+ 1
2 X
ℓ,j
θ
2 ℓ
t
=: k × θ
⊤
˜θ
c j
t
. By implementing Proposition 5.1.1 into normal-Poisson
j
model we obtain
det K
′′ ν
t;j
θ = t exp k × θ
⊤
˜θ
c j
. 5.9
We use 5.9 for characterizing normal-Poisson by generalized variance. The problem in this characterization is that for given information in the right-
hand side of 5.9, we need to find the cumulant function K in the left-hand side of 5.9 such that the determinant of its second derivative equals to the
Laplace transform exp K
ρν
t;j
θ.
5.2. Characterizations
In order to characterize normal-Poisson models through its generalized variance function from 5.9 back to 5.2 and then to 5.1, up to some
elementary operations of NEFs via Proposition 5.2.1 below, it is also inter- esting to have their characterizations by variance functions from 5.5 back
to 5.2.
5.2.1. Characterization by Variance Function
We here state the first result as follows.
Theorem 5.2.1. Let k ∈ {2,3,...} and t 0. If an NEF F
t;j
satisfies 5.5 for a given j ∈ {1,2,...,k} then, up to affinity, F
t;j
is a normal-Poisson
j
model. To proof the Theorem 5.2.1, we need the following well-known properties
of NEFs.
Proposition 5.2.1. Let
ν and e
ν be two σ-finite positive measures on R
k
such that F = F
ν, e F = F
e
ν and µ ∈ M
F
.
i If there exists d , c ∈ R
k
× R such that e νdx = exp{hd,xi + c}νdx, then
F = e F: Θ
e ν
= Θ
ν
− d and K
e ν
θ = K
ν
θ + d + c; for
e µ = µ
∈ M
F
, V
e F
e
µ = V
F
µ and det V
e F
e
µ = det V
F
µ. ii If
e ν = ϕ
∗
ν with ϕx = Ax + b, then: Θ
e
ν = A
⊤
Θ ν and K
e ν
θ = K
ν
A
⊤
θ+
b
⊤
θ; for e
µ = Aµ + b ∈ ϕM
F
, V
e F
e
µ = AV
F
ϕ
−1
e
µA
⊤
and det V
e F
e
µ = det A
2
det V
F
µ.
65 iii If
e ν = ν
∗t
is the t-th convolution power of ν for t 0, then, for
e µ = tµ
∈ tM
F
, V
e F
e
µ = tV
F
t
−1
e µ and
det V
e F
e µ = t
k
det V
F
µ.
Proposition 5.2.1 shows that the generalized variance function det V
F
µ of F is invariant for any element of its generating measure Part i and for
affine transformation
ϕx = Ax + b such that det A = ±1, in particular for a translation x 7→ x+b Part ii. Sometimes we use terminology type to call an
NEF F as a particular model up to affinity Part ii and convolution power Part iii.
Proof. Without loss of generality, first we assume that t = 1 with flashback to the identifiability of Poisson component. Also, we here fix j = 1 while the
proof for other j ∈ {2,...,k} with k 1 can be deduced by permutation.
First let us remember this following well-known block-wise inversion : γ
a
⊤
a A
−1
= γ
−1
+ γ
−2
a
⊤
S
−1
a −γ
−1
a
⊤
S
−1
−γ
−1
S
−1
a S
−1
5.10
with S = A − γ
−1
aa
⊤
. Now let F = F
ν be an NEF satisfies 5.5 and 5.6 for t = 1 and j = 1.
Using the block-wise inversion of 5.10 into V
F
µ in 5.5 for j = 1, one has : h
V
F
µ i
−1
=
1 µ
1
+
1 µ
3 1
P
k ℓ=2
µ
2 ℓ
−
1 µ
2 1
µ
c 1
⊤
−
1 µ
2 1
µ
c 1
1 µ
1
I
k−1
5.11
with µ
1
0 and µ
c 1
:= µ
2
, . . . , µ
k ⊤
∈ R
k
. Since µ = K
′ ν
θ and V
F
µ = K
′′ ν
θ, then by writting θ in terms of µ one gets
V
F
µ = h
θ
′
µ i
−1
which implies θµ =
Z h
V
F
µ i
−1
dµ.
For θ ∈ Θ := θM
F
such that M
F
:= n
µ ∈ R
k
; µ
1
0 and µ
ℓ
∈ Rwithℓ , 1 o
, there exists a function
ϕ : R
k
→ R such that θ
′
µ = ∂
2
ϕµ ∂µ
i
∂µ
ℓ i
,ℓ=1,2,...,k
. 5.12
Using 5.12 into 5.11 for getting the first informations on Poisson compo- nent, we have
∂
2
ϕµ ∂µ
2 1
= 1
µ
1
+ 1
µ
3 1
k
X
ℓ=2
µ
2 ℓ
66 and then
∂ϕµ ∂µ
1
= log µ
1
− 1
2 µ
2 1
k
X
ℓ=2
µ
2 ℓ
+ f
µ
2
, . . . , µ
k
, 5.13
where f : R
k−1
→ R is an analytical function to be determined. Note that since
µ
1
0 then log µ
1
and 1 2µ
2 1
in 5.13 are well-defined. Derivative of 5.13 with respect to
µ
ℓ
gives ∂
2
ϕ ∂µ
1
µ
ℓ
= − µ
ℓ
µ
2 1
+ ∂ f µ
2
, . . . , µ
k
∂µ
ℓ
. 5.14
Expression 5.14 is equal to the 1, ℓth element of
h
V
F
µ i
−1
in 5.11, that is −
µ
ℓ
µ
2 1
+ ∂ f µ
2
, . . . , µ
k
∂µ
ℓ
= − µ
ℓ
µ
2 1
; therefore,
∂ f µ
2
, . . . , µ
k
∂µ
ℓ
= 0 for all ℓ ∈ {2,...,k} implies f µ
2
, . . . , µ
k
= c
1
a real constant. Thus, 5.13 becomes
∂ϕ ∂µ
1
= log µ
1
− 1
2 µ
2 1
k
X
ℓ=2
µ
2 ℓ
+ c
1
5.15 and its primitive can be written as
ϕµ = µ
1
log µ
1
− µ
1
+ 1
2 µ
1 k
X
ℓ=2
µ
2 ℓ
+ c
1
µ
1
+ h
µ
2
, . . . , µ
k
, 5.16
where h : R
k−1
→ R is an analytical function to be determined. From now on, complete informations of the model i.e. normal components begin to show
itself. The two first derivatives of 5.16 with respect to µ
ℓ
give, respectively, ∂ϕµ
∂µ
ℓ
= µ
ℓ
µ
1
+ ∂hµ
2
, . . . , µ
k
∂µ
ℓ
, ∀ℓ ∈ {2,...,k}
5.17 and
∂
2
ϕµ ∂µ
2 ℓ
= 1
µ
1
+ ∂h
2
µ
2
, . . . , µ
k
∂µ
2 ℓ
, ∀ℓ ∈ {2,...,k}.
5.18 Expression 5.18 is equal to the diagonal
ℓ, ℓth element of h
V
F
µ i
−1
in 5.11 for all
ℓ ∈ {2,...,k}, hence we have 1
µ
1
+ ∂
2
h µ
2
, . . . , µ
k
∂µ
2 ℓ
= 1
µ
1
. Consequently,
∂
2
h µ
2
, . . . , µ
k
∂µ
2 ℓ
= 0 and ∂hµ
2
, . . . , µ
k
∂µ
ℓ
= c
ℓ
a real con-
67 stant for all
ℓ ∈ {2,...,k}. Then, equation 5.17 becomes ∂ϕµ
∂µ
ℓ
= µ
ℓ
m
1
+ c
ℓ
∀ℓ ∈ {2,...,k}. 5.19
Using equation 5.15 and 5.19 one obtains
θµ =
logµ
1
− 1
2 µ
2 1
k
X
ℓ=2
µ
2 ℓ
, µ
2
m
1
, . . .
, µ
k
µ
1
⊤
+ c
1
, . . . , c
k ⊤
or
θµ =
θ
1
= log µ
1
− 1
2 µ
2 1
P
k ℓ=2
µ
2 ℓ
+ c
1
θ
ℓ
= µ
ℓ
µ
1
+ c
ℓ
, ℓ = 2, . . . , k.
5.20
From 5.20, each θ
ℓ
belongs to R for ℓ ∈ {1,2,...,k} because µ
1
0 and µ
ℓ
∈ R for
ℓ ∈ {2,...,k}. Thus, one has ΘM
F
=: Θ ⊆ R
k
and also µ
1
= exp
θ
1
− c
1
+ 1
2
k
X
ℓ=2
θ
ℓ
− c
ℓ 2
, 5.21
µ
ℓ
= θ
ℓ
− c
ℓ
exp
θ
1
− c
1
+ 1
2
k
X
ℓ=2
θ
ℓ
− c
ℓ 2
. 5.22
Since µ =
∂K
ν
θ ∂θ
, then using 5.21 one can obtain K
ν
θ as follow:
K
ν
θ = Z
∂K
′ ν
θ ∂θ
1
d θ
1
= exp
θ
1
− c
1
+ 1
2
k
X
ℓ=2
θ
ℓ
− c
ℓ 2
+ gθ
2
, . . . , θ
k
, 5.23
where g : R
k−1
→ R is an analytical function to be determined. Again, derivative of 5.23 with respect to
θ
ℓ
produces
∂K
ν
θ ∂θ
ℓ
= θ
ℓ
− c
ℓ
exp
θ
1
− c
1
+ 1
2
k
X
ℓ=2
θ
ℓ
− c
ℓ 2
+ ∂gθ
2
, . . . , θ
k
∂θ
ℓ
which is equal to 5.22; then, one gets ∂gθ
2
, . . . , θ
k
∂θ
ℓ
= 0 for all ℓ ∈ {2,...,k}
implying g θ
2
, . . . , θ
k
= C a real constant. Finally, it ensues from it that we have
K
ν
θ = exp
θ
1
− c
1
+ 1
2
k
X
ℓ=2
θ
ℓ
− c
ℓ 2
+C.
68
By Proposition 5.2.1 one can see that, up to affinity, this K
ν
is a normal- Poisson
1
cumulant function as given in 5.2 with t = 1 on its corresponding support 5.3. Theorem 5.2.1 is therefore proven by using the analytical
property of K
ν
. 5.2.2.
Characterization by Generalized Variance Function
Before stating our next result, let us briefly recall that, for an unknown
smooth function K : Θ ⊆ R
k
→ R, k 2, the Monge–Ampère equation is de- fined by
det K
′′
θ = gθ 5.24
where K
′′
= D
2 ij
K
i ,j=1,...,k
denotes the Hessian matrix of K and g is a given
positive function see e.g. Gutierrez 2001. The class of equation 5.24 given g has been a source of intense investigations which are related to
many areas of mathematics. Note also that explicit solutions of 5.24, even if in particular situations of g, remain generally challenging problems. We
can refer to Kokonendji and Seshadri 1996; Kokonendji and Masmoudi 2013, 2006 for some details and handled particular cases.
We now state the next result in the following sense.
Theorem 5.2.2. Let F
t;j
= F
ν
t;j
be an infinitely divisible NEF on R
k
k 1 for
given j ∈ {1,2,...,k} such that 1. Θ
ν
t;j
= R
k
, and
2. det K
′′ ν
t;j
θ = t exp k × θ
⊤
˜θ
c j
for θ and ˜θ
c j
given as in 5.4. Then F
t;j
is of normal-Poisson
j
type. The proof of Theorem 5.2.2 is given below. A reformulation of this
theorem, by changing the canonical parameterization into the mean param- eterization, is stated in the following theorem without proof.
Theorem 5.2.3. Let j ∈ {1,2,...,k} fixed and F
t;j
= F
ν
t;j
be an infinitely divisible NEF on R
k
such that
1. M
F
t;j
= n
µ ∈ R
k
; µ
j
0 and µ
ℓ
∈ Rwithℓ , j o
, and
2. det V
F
t;j
µ = µ
k j
. Then F
t;j
is of normal-Poisson
j
type. Theorem 5.2.2 can be viewed as the solution to a particular Monge-
Ampère equation. Whereas Theorem 5.2.3 is interesting for generalized variance estimations.
69 Proof. To proof Theorem 5.2.2 is to solve the Monge-Ampère equation prob-
lem of normal-Poisson models item 2 of the theorem. For that purpose, we need three propositions which are already used in Kokonendji and Mas-
moudi 2006 and Kokonendji and Masmoudi 2013 and we provide the propositions below for making the paper as self-contained as possible.
Proposition 5.2.2. If
ν is an infinitely divisible measure on R
k
, then there exist a symmetric non-negative definite d × d matrix Σ with rank r 6 k and a positive
measure ψ on R
k
such that
K
′′ ν
θ = Σ + Z
R
k
xx
⊤
expθ
⊤
x ψdx.
See, e.g., Gikhman and Skorokhod 2004, page 342.
The above expression of K
′′ ν
θ is an equivalent of the Lévy-Khinchine for- mula see e.g. Bertoin 1996; Kokonendji and Khoudar 2006; Sato 1999; thus,
Σ
comes from a Brownian part and the rest L
′′ ψ
θ := R
R
k
xx
⊤
expθ
⊤
x ψdx
corresponds to jumps part of the associated Lévy process through the Lévy measure
ψ.
Proposition 5.2.3. Let A and B be two k × k matrices. Then
detA + B =
X
S⊂{1,2,...,k}
det A
S
′
det B
S
, with S
′
= {1,2,...,k} \ S and A
S
= a
ij i
,j∈S
2
for A = a
ij i
,j∈{1,2,...,k}
2
. See Muir 1960.
Proposition 5.2.4. Let f : R
k
→ R be a C
2
map. Then, f is an affine polynomial if and only if
∂
2
f θ∂θ
2 i
= 0 ,
for i = 1 , . . . , k.
See Bar-Lev, et al. 1994, Lemma 4.1. Without loss of generality, we assume t = 1 in Theorem 5.2.2. Letting
F
j
= F
ν
j
in Theorem 5.2.2 for fixed j ∈ {1,2,...,k}, we have to solve the following equation with respect to
ν
j
or its characteristic function:
det K
′′ ν
j
θ = exp
k ·
θ
j
+ 1
2 X
ℓ,j
θ
2 ℓ
, ∀θ ∈ R
k
. 5.25
From Proposition 5.2.2 relative to the representation of infinitely divisible distribution, the unknown left member of Equation 5.25 can be written as
det K
′′ ν
j
θ = det Σ
+ Z
R
k
xx
⊤
expθ
⊤
x ψdx
= det h
Σ + L
′′ ψ
θ i
. 5.26
70 For S = {i
1
, i
2
, . . . , i
ℓ
}, with 1 6 i
1
i
2
··· i
ℓ
6 k, a non-empty subset of
{1,2,...,k}, and τ
S
: R
k
→ R
ℓ
the map defined by τ
S
x = x
i
1
, x
i
2
, . . . , x
i
ℓ
⊤
, we define
ψ
S
the image measure of H
ℓ
dx
1
, . . . , dx
ℓ
= 1
ℓ det [
τ
S
x
1
. . . τ
S
x
ℓ
]
2
ψdx
1
. . . ψdx
ℓ
by ψ
ℓ
: R
k ℓ
→ R
k
, x
1
, . . . , x
ℓ
7→ x
1
+ x
2
+ ··· + x
ℓ
. By Proposition 5.2.3 and Expression 5.26 the modified Lévy measure
ρν in 5.1.1 can be expressed as
ρν
j
= det Λ δ
+ X
∅,S⊂{1,2,...,k}
det Λ
S
′
ψ
S
, 5.27
where Λ is a diagonal representation of Σ in an orthonormal basis e = e
i i=1
,...,k
see, e.g., Hassairi 1999, page 384. Since Σ is the Brownian part, then it corresponds to the k − 1 normal components from the right member
of 5.25; that implies r = rankΣ = k − 1 and detΣ = 0. Therefore detΛ = 0 with Λ = diag
λ
1
, λ
2
, . . . , λ
k
such that λ
j
= 0 and λ
ℓ
0 for all ℓ , j. For all non-empty subsets S of {1,2,...,k} there exist real numbers α
S
0 such that det Λ
S
′
ψ
S
=
Y
iS
λ
i
ψ
S
= α
S
h δ
e
j
∗ N0,1e
c j
i
∗k
, 5.28
where e
c j
= e
1
, . . . , e
j−1
, e
j+1
, . . . , e
k
denotes the induced orthonormal basis of
e without component e
j
; i.e. k − 1 is the dimension of e
c j
. With respect to Kokonendji and Masmoudi 2006, Lemma 7 for making
