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

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.

40 pages, 3 figures. In Table 1 (p.4) last column is deleted; In Page 28, indications of Statistics $T_n^{(1)}$(and $T_n^{(3)}$ are corrected ((5.14) is replaced with (5.13); (5.30) is replaced with (5.19); P. 29, line 5,1 from above $T_n^{(1)}$ is replaced with $T_n^{(2)}$; The corresponding correction is entered in Table 2. AoS is deleted . Now pages are 34

References in corpus (1)

• Topological Hausdorff dimension and level sets of generic continuous functions on fractals