Abstract
The criteria discussed so far, unbiasedness and invariance, suffer from the disadvantage of being applicable, or leading to optimum solutions, only in rather restricted classes of problems. We shall therefore turn now to an alternative approach, which potentially is of much wider applicability.
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Notes
- 1.
A different definition of local minimaxity is given by Giri and Kiefer (1964).
- 2.
See Sierpinski (1920).
- 3.
These assumptions are essentially equivalent to the condition that the group G is amenable . Amenability and its relationship to the Hunt–Stein Theorem are discussed by Bondar and Milnes (1982) and (with a different terminology) by Stone and von Randow (1968).
- 4.
When the null hypothesis parameter space is described as a number of inequalities about means being satisfied, the problem is known in econometrics as testing moment inequalities; for a review, see Canay and Shaikh (2017).
- 5.
The existence of maximin tests is established in considerable generality in Cvitanic and Karatzas Karatzas (2001).
- 6.
Locally optimal tests for multiparameter hypotheses are given in Gupta and Vermeire (1986).
- 7.
Due to John Pratt.
- 8.
An interesting example of a type-D test is provided by Cohen and Sackrowitz (1975), who show that the \(\chi ^2\)-test of Chapter 16.3 has this property. Type D and E tests were introduced by Isaacson (1951).
- 9.
Due to Fritz Scholz.
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Lehmann, E.L., Romano, J.P. (2022). The Minimax Principle. In: Testing Statistical Hypotheses. Springer Texts in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-030-70578-7_8
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