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
since the nuisance parameters, R i and t i , are assumed known. Table 3.6
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.