Testing statistical hypotheses concerning the expectations of two independent normals, both with variance one. I

SummaryThe use of some objectivistic optimum properties like U.M.P.(D), M.S.(D) and M.S.S.M.P.(D) is illustrated and another property “strongest with respect to Lebesgue-measure” is introduced. The attention is restricted to very simple testing problems for two independent normals. Modifications or generalizations are mentioned sometimes, in order to stress that each of the problems to be considered is the ultimate simplification of certain situations from actual practice. The latter situations are more complicated from a technical point of view (multi-dimensionality, nuisance parameters, non-normality) but the crux is still to decide what is the most appropriate objectivistic optimum property.Special interest may be requested for (i) the optimum property “strongest w.r. to Lebesgue-measure” (Section 2), (ii) critical remarks concerning maximin and optimal-β in the sections 6 and 7, (iii) a very simple example of an inadmissible likelihood-ratio test (Section 10), (iv) the funny problem of Section 10 where a U.M.P.(D) test exists, D being the class of all size-α tests that are both similar size-α and invariant