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Objective and Strong Objective Consistent Estimates of Unknown Parameters for Statistical Structures in a Polish Group Admitting an Invariant Metric

In: Applications of Measure Theory to Statistics

Author

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  • Gogi Pantsulaia

    (Georgian Technical University, Department of Mathematics)

Abstract

By using the notion of a Haar ambivalentHaar ambivalent set introduced by Balka, Buczolich, and Elekes (2012), essentially new classes of statistical structuresStatistical structure having objective and strong objective estimatesStrong objective estimate Objective estimate of unknown parameters are introduced in a Polish nonlocally compact group admitting an invariant metric and relations between them are studied in this chapter. An example of a weakly separated statistical structureWeakly separated statistical structure is constructed for which a question asking whether there exists a consistent estimateConsistent estimate of an unknown parameterAxiom of Dependence choices (DC) is not solvable within theAxiom of Dependence choices (DC) theory $$ (ZF)~ \& ~(DC)$$ ( Z F ) & ( D C ) . A question asking whether there exists an objective consistent estimate of an unknown parameter for any statistical structureStatistical structure in a nonlocally compact Polish groupPolish group with an invariant metric when a subjective one exists, is answered positively when there exists at least one such parameter, the preimage of which under this subjective estimateSubjective estimate is a prevalent. These results extend recent results of Pantsulaia and Kintsurashvili (2014). Some examples of objective and strong objective consistent estimatesStrong objective estimate in a compact PolishPolish group group $$\{0; 1\}^N$$ { 0 ; 1 } N are considered in this chapter. At end of this chapter we present a certain claim for theoretical statisticians in which each consistent estimation with domain in a nonlocally compact Polish groupPolish group equipped with an invariant metric must pass the certification exam on the objectivity before its practical application and we also give some recommendations.

Suggested Citation

  • Gogi Pantsulaia, 2016. "Objective and Strong Objective Consistent Estimates of Unknown Parameters for Statistical Structures in a Polish Group Admitting an Invariant Metric," Springer Books, in: Applications of Measure Theory to Statistics, chapter 0, pages 73-106, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-45578-5_5
    DOI: 10.1007/978-3-319-45578-5_5
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