8
PROOF . We successfully have
Σ
G ,x
= E h
X − x G
−1
X − x
⊤
i = E
h
X − µ + µ − x G
−1
X − µ + µ − x
⊤
i = E
n
X − µ + µ − x
o
G
−1
n
X − µ + µ − x
o
⊤
= E h
X − µG
−1
X − µ
⊤
i + E
h
X − µG
−1
µ − x
⊤
i +
E h
µ − x G
−1
X − µ
⊤
i + E
h
µ − x G
−1
µ − x
⊤
i = E
h
X − µG
−1
X − µ
⊤
i + E
h
X − µ
i
G
−1
µ − x
⊤
+
µ − x G
−1
E h
X − µ
⊤
i
+ µ − x G
−1
µ − x
⊤
.
Since EX = µ, then EX − µ = 0. So it follows
Σ
G ,x
= E h
X − µG
−1
X − µ
⊤
i + E
h
µ − x G
−1
µ − x
⊤
i = Σ
G
,µ
+ µ − x G
−1
µ − x
⊤
.
B. Generalized Variance and Generalized Dispersion
Generalized variance is a scalar measure of dispersion for multivariate data under normality which was introduced by Wilks 1932; it is used
for overall multivariate scatter and one can rank distinct groups and pop- ulations based on the order of their spread Wilks, 1967. The generalized
variance of a k-dimensional random vector variable X is defined as the deter- minant of its covariance matrix. It has an important role both in theoretical
and applied research. In sampling theory, Arvanitis and Afonja 1971 used generalized variance as a loss function in multiparametric sampling allo-
cation. In the theory of statistical hypothesis testing, Isaacson 1951 used generalized variance as a criterion for an unbiased critical region to have
the maximum Gaussian curvature. In the descriptive statistics, Goodman 1968 proposed a classification of some varieties of rice according to their
generalized variances. Kokonendji and Seshadri 1996 and Kokonendji and Pommeret 2007 extended the generalized variance for non-normal distri-
butions in particular for natural exponential families NEFs.
Definition 2.1.3. The generalized variance of P
θ
is given by: det Σ
µ
= det Z
R
k
x − µ
x − µ
⊤
P
θ
dx .
2.6 The estimation of generalized variance is usually based on the determi-
nant of the sample covariance matrix V
n
, i.e. [
det Σ
µ
= det V
n
= det
1 n − 1
n
X
i=1
h
X
i
− XX
i
− X
⊤
i
.
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
