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 et al. [1], essentially new classes of statistical structures having objective and strong objective estimates of unknown parameters are introduced in a Polish nonlocally-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 "whether there exists a consistent estimate of an unknown parameter" is not solvable within the theory (ZF) & (DC). A question asking "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 R ∈ θ Following [1], a set which is neither Haar null nor prevalent is called a Haar ambivalent set. Such estimates firstly were adopted as objective infinite sample consistent estimates of a useful signal in the linear one-dimensional stochastic model (see [27], Theorem 4.1, p. 482).