7
moment exists. Then the dispersion matrix of P
θ
around fixed point x ∈ R
k
is given by
Σ
x
= Z
R
k
x − x x − x
⊤
P
θ
dx .
2.4
From 2.4, if x = µ then we have Σ
µ
= R
R
k
x − µ
x − µ
⊤
P
θ
dx which is
equal to the covariance matrix in 2.1.
Definition 2.1.2. Let G be a symmetric definite positive matrix. A generalization
of dispersion matrix of P
θ
around fixed point x
∈ R
k
is given by: Σ
G ,x
= Z
R
k
x − x G
−1
x − x
⊤
P
θ
dx .
2.5
We call 2.5 a G-dispersion matrix. Properties
Let X
1
, . . . , X
n
be a random sample of size n.
1. If G is known, the estimator of Σ
G ,x
is given by b
Σ
G ,x
= 1
n − 1
n
X
i=1
h
X
i
− x G
−1
X
i
− x
⊤
i .
In particularly, if G = I and x =
b
µ = X, the dispersion matrix estimator
is b
Σ
I ,µ
= 1
n − 1
n
X
i=1
h
X
i
− XX
i
− X
⊤
i .
Notice that this estimator is the same as the sample covariance matrix V
n
of 2.2, i.e. V
n
= b Σ
I ,µ
.
2. If G is unknown and we estimate it by ˆ G
, the G-dispersion matrix
estimator can be written as b
Σ
G ,x
= 1
n − 1
n
X
i=1
h
X
i
− x ˆ
G
−1
X
i
− x
⊤
i .
For ˆ G = V
n
, the matrix b Σ
G ,x
is also known as the matrix of Mahalanobis
squared distances Srivastava and Carter, 1983, p.232 between x and
each observation.
Proposition 2.1.1. Let Σ
G ,x
be a G-dispersion matrix of κ
θ
around fixed point
x ∈ R
k
. Then: Σ
G ,x
= Σ
G ,µ
+ µ − x G
−1
µ − x
⊤
.
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
.
