Complete order statistics in parametric models

Abstract

For a given statistical model $\mathsf{P}$ it may happen that the order statistic is complete for each IID model based on $\mathsf{P}$. After reviewing known relevant results for large nonparametric models and pointing out generalizations to small nonparametric models, we essentially prove that this happens generically even in smooth parametric models.

As a consequence it may be argued that any statistic depending symmetrically on the observations can be regarded as an optimal unbiased estimator of its expectation.

In particular, the sample mean $\overline{X}_n$ is generically an optimal unbiased estimator, but, as it turns out, also generically asymptotically inefficient.