Abstract
This chapter develops the theory of optimal parametric tests. The Neyman–Pearson Lemma provides the most powerful test of any given size for a simple null hypothesis H 0 against a simple alternative hypothesis H 1. For one parameter exponential models such tests are uniformly most powerful (UMP) against one-sided alternatives. For two-sided alternatives here one obtains a UMP test among all unbiased tests of a given size. In multiparameter exponential models one may similarly obtain UMP unbiased tests in the presence of nuisance parameters. For statistical models which are invariant under a group of transformations all reasonable tests should be invariant under the group. The theory of UMP tests among all invariant tests is developed for linear models.
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Notes
- 1.
See Ferguson (1967, pp. 215–221).
- 2.
Note. Since \((\overline{X} -\overline{Y })/\sigma \sqrt{ \frac{1} {m} + \frac{1} {n}}\) is N(0, 1) and independent of the chi-square random variables \(U =\sum (X_{j} -\overline{X})^{2}/\sigma ^{2} +\sum (Y _{i} -\overline{Y })^{2}/\sigma ^{2}\) (having d.f. \(m - 1 + n - 2 = m + n - 2\)), by definition, \([(\overline{X} -\overline{Y })/\sigma \sqrt{ \frac{1} {m} - \frac{1} {n}}]/\sqrt{U/(m + n - 2)}\) has Student’s distribution with \(m + n - 2\) d.f.
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Bhattacharya, R., Lin, L., Patrangenaru, V. (2016). Testing Hypotheses. In: A Course in Mathematical Statistics and Large Sample Theory. Springer Texts in Statistics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-4032-5_5
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