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marks are perturbed independently of each other, that is, K as well as
D are diagonal matrices. Without loss of generality, we assume that
the first element of D is equal to 1. We show that for this model, it is
possible to obtain the estimator for K and not just its singular ver-
To begin, let us consider the case where D
I D . In the case that K
is diagonal, L
K L T has a typical form that may be exploited to obtain
an estimator of K based on the estimator of L
K L T , which we know
1 2 ,
2 2 ,…,
K 2 ) . Then the diagonal ele-
can be estimated. Let K
diag (
1 2 and all off-diagonal elements are
given by 1 2 . Thus a simple estimator of 1 2 is given by -1 times the
average of all the off-diagonal elements of the estimator of L
are given by i 2
ments of L
obtained in the previous discussion of the general case. The estimators
of 1 2 are simply obtained by subtracting the average of all the off-
diagonal elements of the estimator of 1 2
from the i -th diagonal
element of the estimator L
K L T .
Obviously, these estimators can be improved substantially by using
the structure of L
K L T in the method of moments equations directly.
The purpose of the above discussion is to show the possibility of esti-
mating the variability around each of the landmarks when certain
structures for the covariance matrix are adopted. One can also check
the reasonability of the assumed structure by looking at the estimate
of L
K L T obtained in the general case and checking if the off-diagonal
elements are very similar to each other or not. If they are very similar,
the diagonal structure may be a reasonable model.
The consistency of the method of moments estimators follows from
the well known fact that sample moments converge to the population
moments as the sample size increases (Serfling, 1980). The details can
be found in Lele (1993).
3.13.2 Maximum likelihood estimators
Dryden and Mardia (1991) derived the exact distributions of the form
coordinates and the shape coordinates. This distribution can be used to
write down the likelihood function. One can then maximize this likeli-
hood function to obtain maximum likelihood estimators of the mean
form as well as the covariance parameters. It should be noted again
that these parameters are identifiable only up to the partition induced
by v ( - ) . Such a partition was described earlier. A significant drawback
of the maximum likelihood approach is that the exact distributions of
form and shape coordinates are mathematically complex when the
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