Statistical Shape Analysis, 2nd Edition

13 Non-parametric inference and regression

13.1 Consistency

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, …, X_n from a distribution with population mean shape [μ]. Let a sample estimator of [μ] be obtained from X₁, …, X_n and denoted by . We say that the estimator is consistent if for any > 0,

(13.1)

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 ...