Abstract
As mentioned at the beginning of Chapter , the finite-sample theory of optimality for hypothesis testing is applied only to rather special parametric families, primarily exponential families and group families. On the other hand, asymptotic optimality will apply more generally to parametric families satisfying smoothness conditions. In particular, we shall assume a certain type of differentiability condition, called quadratic mean differentiability.
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Notes
- 1.
The definition of q.m.d. is a special case of Fréchet differentiability of the map \(\theta \rightarrow p_\theta ^{1/2}(\cdot )\) from \(\Omega \) to \(L^2(\mu )\).
- 2.
A real-valued function g defined on an interval [a, b] is absolutely continuous if \(g ( \theta ) = g (a) + \int _a^{\theta } h(x)dx\) for some integrable function h and all \(\theta \in [a,b]\); Problem 2 on p. 182 of Dudley (1989) clarifies the relationship between this notion of absolute continuity of a function and the general notion of a measure being absolute continuous with respect to another measure, as defined in Section 2.2.
- 3.
G is a limit point of a sequence \(G_n\) of distributions if \(G_{n_j}\) converges in distribution to G for some subsequence \(n_j\).
- 4.
Two real-valued sequences  \(\{ a_n \}\) and \(\{ b_n \}\) are said to be of the same order, written \(a_n \asymp b_n\) if \(|a_n / b_n |\) is bounded away from 0 and \(\infty \).
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Lehmann, E.L., Romano, J.P. (2022). Quadratic Mean Differentiable Families. In: Testing Statistical Hypotheses. Springer Texts in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-030-70578-7_14
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DOI: https://doi.org/10.1007/978-3-030-70578-7_14
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