โ Scribed by Geisser, Seymour
No coin nor oath required. For personal study only.
Since UMP tests don't always exist, statisticians have proceeded to find optimal tests in more restricted classes. One such restriction is unbiasedness. Another is invariance. This chapter develops the theory of uniformly most powerful unbiased (UMPU) and invariant (UMPI) tests. When it is not possible to optimize in these ways, it is still possible to make progress, at least on the mathematical front. Locally most powerful (LMP) tests are those that have greatest power in a neighborhood of the null, and locally most powerful unbiased (LMPU) tests are most powerful in a neighborhood of the null, among unbiased tests. These concepts and main results related to them are presented here. The theory is illustrated with examples and moreover, examples are given that illustrate potential flaws. The concept of a "worse than useless" test is illustrated using a commonly accepted procedure. The sequential probability ratio test is also presented.
Whenever a UMP test exists at level a, we have shown that a 1 ร b K , that is, the power is at least as large as the size. If this were not so then there would be a test T ร ; a which did not use the data and had greater power. Further, such a test is sometimes termed as "worse than useless" since it has smaller power than the useless test T ร ; a. Now UMP tests do not always exist except in fairly restricted situations. Therefore N-P theory proposes that in the absence of a UMP test, a test should at least be unbiased, that is, 1 ร b K ! a. Further, if among all unbiased tests at level a, there exists one that is UMP then this is to be favored and termed UMPU. So for a class of problems for which UMP tests do not exist there may exist UMPU tests.
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