Abstract
We now begin the study of the statistical problem that forms the principal subject of this book, the problem of hypothesis testing. As the term suggests, one wishes to decide whether or not some hypothesis that has been formulated is correct. The choice here lies between only two decisions: accepting or rejecting the hypothesis. A decision procedure for such a problem is called a test of the hypothesis in question.
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Notes
- 1.
The standard way to remove the arbitrary choice of \(\alpha \) is to report the p-value of the test, defined as the smallest level of significance leading to rejection of the null hypothesis. This approach will be discussed toward the end of Section 3.3.
- 2.
In practice, typically neither the breaking of the r-order nor randomization is considered acceptable. The common solution, instead, is to adopt a value of \(\alpha \) that can be attained exactly and therefore does not present this problem.
- 3.
There is no loss of generality in this assumption, since one can take \(\mu =P_0+P_1\).
- 4.
- 5.
One could generalize the definition of p-value to include randomized level \(\alpha \) tests \(\phi _{\alpha }\) assuming that they are nested in the sense that \(\phi _{\alpha } (x) \le \phi _{\alpha '} (x)\) for all x and \(\alpha < \alpha '\). Simply define \(\hat{p} = \inf \{ \alpha :~ \phi _{\alpha } (X) = 1 \}\); in words, \(\hat{p}\) is the smallest level of significance where the hypothesis is rejected with probability one.
- 6.
This definition is in terms of specific versions of the densities \(p_\theta \). If instead the definition is to be given in terms of the distribution \(P_\theta \), various null-set considerations enter which are discussed in Pfanzagl (1967).
- 7.
An explicit solution for the value of \(\theta \) satisfying (3.28) may be unavailable, and one may resort to approximate numerical approaches, such as by discretization of \(\theta \) or the “automatic percentile” method of DiCiccio and Romano (1989).
- 8.
Proposed by Wolfowitz (1950).
- 9.
Suggested by Tukey (1949b).
- 10.
A discussion of the problem when this assumption is not satisfied is given by Dantzig and Wald (1951).
- 11.
See Lehmann and Stein (1948).
- 12.
Proposition 15.2 of van der Vaart (1998) provides an alternative proof in the case \(\Sigma \) is invertible.
- 13.
For more general results concerning the possibility of dispensing with randomized procedures, see Dvoretzky et al. (1951).
- 14.
For a proof of this lemma see Halmos (1974, p. 174). The lemma is a special case of a theorem of Lyapounov (1940); see Blackwell (1951a).
- 15.
Tables and approximations are discussed, for example, in Chapter 3 of Johnson and Kotz (1969).
- 16.
Cf. Neyman (1941b).
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Lehmann, E.L., Romano, J.P. (2022). Uniformly Most Powerful Tests. In: Testing Statistical Hypotheses. Springer Texts in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-030-70578-7_3
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