Abstract
This chapter studies the problem of construction of confidence intervals for 1-parameter models. It uses the special case of the Gaussian distribution to introduce the notions of pivot and approximate pivot. Then, the determination of more general approximate pivots is discussed in the context of exponential families, by means of Wald’s method and likelihood ratios. The duality with hypothesis tests is then introduced, and used to show how optimal one-sided tests can yield optimal one-sided intervals in exponential families.
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- 1.
Note that the problem “use the data to decide if the region \(\Theta _{0}\) contains \(\theta\)” is in some sense dual to the question “use the data to find a region that is highly likely to contain \(\theta\)”.
- 2.
Since this probability obviously depends on the true value of \(\theta\)!
- 3.
The proof of this is analogous to the first part of Theorem 5.10.
- 4.
The problem is that, as we saw in Theorem 5.10, we have no guarantee in general that the region we get from inverting a test will be an interval, much less so a “one-sided” interval, unless there are further conditions.
- 5.
Recall that in that theorem we proved that the derivative of the mapping \(\vartheta \mapsto \mathbb{E}_{\vartheta }[\delta (X_{1},\ldots,X_{n})] = \mathbb{P}_{\vartheta }[\tau \geq c]\) exists and is positive for all \(\vartheta\) and all c.
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© 2016 Springer International Publishing Switzerland
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Panaretos, V.M. (2016). Confidence Intervals for Model Parameters. In: Statistics for Mathematicians. Compact Textbooks in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-28341-8_5
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DOI: https://doi.org/10.1007/978-3-319-28341-8_5
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Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-28339-5
Online ISBN: 978-3-319-28341-8
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