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

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 $(ZF)\mathrm{\&}(DC)$. 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.