On objective and strong objective consistent estimates of unknown parameters for statistical structures in a Polish group admitting an invariant metric

TL;DR

This paper introduces new classes of statistical structures with objective and strong objective estimates of unknown parameters in Polish groups with invariant metrics, exploring their properties and existence under various conditions.

Contribution

It extends existing results by defining objective estimates using Haar ambivalent sets and analyzing their existence and properties in non-locally-compact Polish groups.

Findings

Existence of statistical structures with objective estimates in certain Polish groups.

Construction of a weakly separated structure where consistent estimation is not provable in ZF & DC.

Positive results on the existence of objective estimates when pre-images are prevalent.

Abstract

By using the notion of a Haar ambivalent set introduced by Balka, Buczolich and Elekes (2012), essentially new classes of statistical structures having objective and strong objective estimates of unknown parameters are introduced in a Polish non-locally-compact group admitting an invariant metric and relations between them are studied in this paper. An example of such a weakly separated statistical structure is constructed for which a question asking "{\it whether there exists a consistent estimate of an unknown parameter}" is not solvable within the theory $(ZF)~\&~(DC)$. A question asking "{\it whether there exists an objective consistent estimate of an unknown parameter for any statistical structure in a non-locally compact Polish group with an invariant metric when subjective one exists}" is answered positively when there exists at least one such a parameter the pre-image of which under this subjective estimate is a prevalent. These results extend recent results of authors. Some examples of objective and strong objective consistent estimates in a compact Polish group $\{0; 1\}^N$ are considered in this paper.