Summary
The paper concerns sufficient conditions for the existence of least favorable distributions. In sections 2 and 3 some partially known results will be noted about the weak topology and convex hulls in the space of bounded additive set functions (contents). By a suitable dualization of the linear program for maximin tests it is shown in section 5 that for positive level the convex hull of the hypothesis (alternative) contains a content which is least favorable. This result is used in section 6 for deriving e.g. the following theorem: If there are tests of arbitrarily high power and of arbitrarily small level on a part of the hypothesis such that the complementary part of the hypothesis is uniformly dominated by a finite measure, then the least favorable hypothesis is not even a content but a measure.
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Baumann, V. Eine parameterfreie Theorie der ungünstigsten Verteilungen für das Testen von Hypothesen. Z. Wahrscheinlichkeitstheorie verw Gebiete 11, 41–60 (1968). https://doi.org/10.1007/BF00538385
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DOI: https://doi.org/10.1007/BF00538385