Abstract
For a large class of problems for which a UMP test does not exist, there does exist a UMP unbiased test (which we sometimes abbreviate as UMPU). This is the case in particular for certain hypotheses of the form \(\theta \le \theta _0\) or \(\theta =\theta _0\), where the distribution of the random observables depends on other parameters besides \(\theta \).
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Notes
- 1.
A statement is said to hold a.e. \({\mathcal {P}}\) if it holds except on a set N with \(P(N)=0\) for all \(P\in {\mathcal {P}}\).
- 2.
See for example Section 26 of Billingsley (1995).
- 3.
Such problems are also treated in Johansen (1979), which in addition discusses large-sample tests of hypotheses specifying more than one parameter.
- 4.
A somewhat different asymptotic optimality property of these tests is established by Michel (1979).
- 5.
The comparison of two treatments as a three-decision problem or as the simultaneous testing of two one-sided hypotheses is discussed and the literature reviewed in Shaffer (2002).
- 6.
A discussion of this and alternative procedures for achieving the same aim is given by Birnbaum (1954a).
- 7.
This package can be downloaded for free from http://cran.r-project.org/.
- 8.
\(\Delta \) is equivalent to Yule’s measure of association which is \(Q=(1-\Delta )/(1+\Delta )\). For a discussion of this and related measures see Goodman and Kruskal (1954, 1959), Edwards (1963), Haberman (1982) and Agresti (2002).
- 9.
These results were conjectured by Berkson and proved by Neyman in a course on \(\chi ^2\).
- 10.
The one-sided test is of course UMP against the class of alternatives defined by the right side of (4.21), but no reasonable assumptions have been proposed that would lead to this class. For suggestions of a different kind of alternative see Gokhale and Johnson (1978).
- 11.
For a more detailed treatment of the distinction between population models [such as (i)–(iii)] and randomization models [such as (iv)], see Lehmann (1998).
- 12.
The problem of discriminating between a logistic and normal response model is discussed by Chambers and Cox (1967).
- 13.
For counterexamples when the conditions of the problem are not satisfied, see Kallenberg et al. (1984).
- 14.
For a systematic discussion of this and other concepts of dependence, see Tong (1980, Chapter 5), Kotz, Wang and Hung (1990), and Yanagimoto (1990).
- 15.
Statistical inference in these and more general Markov chains is discussed, for example, in Bhat and Miller (2002); they provide references at the end of Chapter 5.
- 16.
This distribution is tabled by Swed and Eisenhart (1943) and Gibbons and Chakraborti (1992); it can be obtained from the hypergeometric distribution [Guenther (1978)]. For further discussion of the run test, see Lou (1996).
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Lehmann, E.L., Romano, J.P. (2022). Unbiasedness: Theory and First Applications. In: Testing Statistical Hypotheses. Springer Texts in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-030-70578-7_4
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