Abstract
The mathematical framework for statistical decision theory is provided by the theory of probability, which in turn has its foundations in the theory of measure and integration.
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Notes
- 1.
If \(\pi (z)\) is a statement concerning certain objects z, then \(\{z: \pi (z)\}\) denotes the set of all those z for which \(\pi (z)\) is true.
- 2.
The term into indicates that the range of T is in \({\mathcal {T}}\); if \(T({\mathcal {Z}})={\mathcal {T}}\), the transformation is said to be from \({\mathcal {Z}}\) onto \({\mathcal {T}}\).
- 3.
A statement that holds for all points x except possibly on a set of \(\mu \)-measure zero is said to hold almost everywhere \(\mu \), abbreviated a.e. \(\mu \), or to hold a.e. \(({\mathcal {A}}, \mu )\) if it is desirable to indicate the \(\sigma \)-field over which \(\mu \) is defined.
- 4.
We shall use this term in place of the more cumbersome “sub-\(\sigma \)-field”.
- 5.
For a proof of these relations, see for example Turnbull (1952), Section 32.
- 6.
This term is used as an alternative to the more cumbersome “set of measure zero”.
- 7.
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Lehmann, E.L., Romano, J.P. (2022). The Probability Background. In: Testing Statistical Hypotheses. Springer Texts in Statistics. Springer, Cham. https://doi.org/10.1007/978-3-030-70578-7_2
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DOI: https://doi.org/10.1007/978-3-030-70578-7_2
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