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Complete order statistics in parametric models. (English) Zbl 0880.62009

Summary: For a given statistical model \({\mathcal P}\) it may happen that the order statistic is complete for each i.i.d. model based on \({\mathcal P}\). 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.

MSC:

62B05 Sufficient statistics and fields
62G30 Order statistics; empirical distribution functions
62F10 Point estimation
62A01 Foundations and philosophical topics in statistics
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