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v
L
(
M
,
~
K
,
~
parameter space
,
if and only if
v
L
(
M
,
K
,
D
)
D
)
v
L
*
(
M
,
~
K
,
~
D
)
.
Consider first the statement
L
1
K
L
T
1
v
L
*
(
M
,
K
,
D
)
L
1
~
K
L
T
1
if and only if
L
2
~
L
2
K
L
T
2
K
L
T
2
.
L
1
~
L
2
~
1
~
L
1
K
L
T
1
K
L
T
1
⇒Α
K
L
T
1
A
T
K
L
T
1
A
T
K
L
T
2
K
L
T
2
.
L
Α
L
L
1
2
The converse implication is proved in the same fashion. The rest of
the proof follows along the same lines.
Another maximal invariant suggested in the literature is form coor-
dinates (also known as size and shape coordinates; Bookstein, 1986;
Kendall, 1989). Its properties are studied in detail by Dryden and
Mardia (1991). This maximal invariant does not include reflection as
part of the group transformation. Dryden and Mardia (1991) also study
the maximal invariant under an additional component of scaling in
order to consider only “shape coordinates”. The exact distributions of
“form” as well as “shape” coordinates are obtained in Dryden and
Mardia (1991). Identifiability issues are not dealt with explicitly in any
of these papers. More details on the exact distribution of the shape
coordinates are discussed later in this chapter. Rao and Suryawanshi
(1996, 1998) and Rao (2000) describe a maximal invariant consisting of
all possible angles. This is a maximal invariant under the group of
transformations which includes scaling.
3.13 Estimation of parameters
Having established the identifiability of certain parameters, the next
natural question is whether these parameters can be estimated in a
practically suitable and statistically desirable fashion. There have
been various methods suggested in the literature. The most commonly
used method is based on Generalized Procrustes Analysis. The review
article by Goodall (1991) provides a description of this method.
Alternatively, maximum likelihood estimation can be conducted using
the exact shape distributions derived by Mardia and Dayden (1998) or
estimators can be constructed via the method of moments (Stoyan,
1990; Lele, 1993).
Method of moments estimators are based on the moments of the
distribution of the maximal invariant described in the previous sec-
tion. Lele (1993) showed that under the assumption that
D
=
I
, the
estimating functions based on the method of moments have a unique,
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