Abstract
The paper concerns estimates of probability measures (=p-measures) determined from a countable number of independent realizations. The main results are:
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For suitable topologies, the existence of a consistent sequence of estimates implies the existence of a sequence of estimates which is strongly consistent a. e. with respect to an arbitrary finite prior measure. In the case of a measurable parameter the existence of a consistent sequence of estimates implies strong consistency of the Bayes estimates for the quadratic loss function.
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If a family ofp-measures is separable with respect to the supremum metric, there exists a sequence of strict estimates which is uniformly consistent with respect to the supremum metric on any totally bounded subset ofp-measures.
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IfB is totally σ-bounded with respect to the supremum metric there exists a density-consistent sequence of estimates. IfB is separable with respect to the supremum metric there exists a sequence of estimates which is density consistent a. e. with respect to an arbitrary finite prior measure.
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A sequence of uniformly consistent tests exists for all compact hypotheses in the case of an abstract sample space and for all weakly closed hypotheses in the case of a separable metric sample space.
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Pfanzagl, J. On the existence of consistent estimates and tests. Z. Wahrscheinlichkeitstheorie verw Gebiete 10, 43–62 (1968). https://doi.org/10.1007/BF00572921
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DOI: https://doi.org/10.1007/BF00572921