Abstract
Chapters 3, 4, 5, 6, and 7 were concerned with the derivation of UMP, UMP unbiased, and UMP invariant tests. Unfortunately, the existence of such tests turned out to be restricted essentially to one-parameter families with monotone likelihood ratio, exponential families, and group families, respectively. Tests maximizing the minimum or average power over suitable classes of alternatives exist fairly generally, but are difficult to determine explicitly, and their derivation in Chapter 8 was confined primarily to situations in which invariance considerations apply.
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Notes
- 1.
In general, the boundary of a set E in \(\mathrm{I}\!\mathrm{R}^k\), denoted by \(\partial E\) is defined as follows. The closure of E, denoted by \(\bar{E}\), is the set of \(x \in \mathrm{I}\!\mathrm{R}^k\) for which there exists a sequence \(x_n \in E\) with \(x_n \rightarrow x\). The set E is closed if \(E = \bar{E}\). The interior of E, denoted by \(E^{\circ }\), is the set of x such that, for some \(\epsilon > 0\), the Euclidean ball with center x and radius \(\epsilon \), defined by \(\{ y \in \mathrm{I}\!\mathrm{R}^k:~|y-x| < \epsilon \}\), is contained in E. Here \(| \cdot |\) denotes the usual Euclidean norm. The set E is open if \(E = E^{\circ }\). If \(E^c\) denotes the complement of a set E, then evidently \(E^{\circ }\) is the complement of the closure of \( { E^c}\), and so E is open if and only if \(E^c\) is closed. The boundary \(\partial E\) of a set E is then defined to be \(\bar{E} - E^{\circ } = \bar{E} \cap (E^{\circ })^c\).
- 2.
The term weak convergence (also sometimes called weak star convergence) distinguishes this type of convergence from stronger convergence concepts to be discussed later. However, the term is used because it is a special case of convergence in the weak star topology for elements in a Banach space (such as the space of signed measures on \(\mathrm{I}\!\mathrm{R}^k\)), though we will make no direct use of any such topological notions.
- 3.
When \(k =1\), we may also use the notation \(g' ( \mu )\) for the ordinary first derivative of g with respect to \(\mu \), as well as \(g'' ( \mu )\) for the second derivative.
- 4.
For discussion of this transformation, see Mudholkar (1983), Stuart and Ord, Vol. 1 (1987) and Efron and Tibshirani (1993) , p. 54. Numerical evidence supports replacing n by \(n-3\) in (11.32).
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Lehmann, E.L., Romano, J.P. (2022). Basic Large-Sample Theory. In: Testing Statistical Hypotheses. Springer Texts in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-030-70578-7_11
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DOI: https://doi.org/10.1007/978-3-030-70578-7_11
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