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Let
1
2
be the eigenvalues of
C
1
with the corresponding eigen-

vectors
l
1
,
l
2
. Let
j
(a)
(
x
)
be the generalized Laguerre polynomial of

degree j (Abramovitz and Stegun, 1965).

The probability density function of the Kendall shape variables for

a two dimensional object is given by:

Notice that the derivation of the distribution does not depend on

the Kronecker product form of the variance. Thus this distribution can

be used in more general situations than discussed in the methods of

moments approach. Although considering the sample sizes encoun-

tered in practice, it seems unwise to use a model with such a large

number of parameters.

3.13.3 Efficiency comparisons between method of moments and

maximum likelihood: a simulations study

All the maximal invariants are equivalent in the sense that maximum

likelihood estimators based on any of them give estimates on the same

orbit. The use of maximal invariance to eliminate nuisance parameters

usually leads to some loss of efficiency as compared to the situation

where the nuisance parameters are known. As shown earlier, the

method of moments estimators based on the maximal invariant
T
(
X
)

are easy to obtain. We next study the loss of efficiency of method of

moments as compared to maximum likelihood based estimators.

Table 3.5
gives the result of a small simulation study comparing the

method of moments based on the inter-landmark differences (EDMA),

the MLE based on the Mardia-Dryden distribution and the MLE based

on the (unobserved) data before translation or rotation. Thus the

“unobservable MLE” represents an idealized situation for comparison

provides the mean form and the covariance structures used for the

above simulation study.

The following are the percent relative root mean squared errors for

the two methods based on 100 simulations. Samples of size 30 were

generated under two different mean forms and two different covari-

corresponds to the relative root mean squared error for the maximum

likelihood estimators based on the assumption the nuisance parame-

ters are known. These values represent best achievable results.

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