The consistency of mean shape and size-and-shape estimators has been of long-standing interest. Consistency is a desirable property for an estimator, and informally a consistent estimator for a population quantity has the property that with more and more independent observations the sample estimator should become closer and closer to the true population quantity. Consider a random sample of n configurations given by X1, …, Xn from a distribution with population mean shape [μ]. Let a sample estimator of [μ] be obtained from X1, …, Xn and denoted by . We say that the estimator is consistent if for any > 0,
where d(, ) is a choice of shape distance. We write
and say that converges in probability to [μ].
Similarly a mean size-and-shape estimator is consistent if
It was shown by Lele (1993) that Procrustes mean estimators ...