# The Theory of Confidence Sets

Chapter
Part of the Die Grundlehren der mathematischen Wissenschaften book series (GL, volume 202)

## Abstract

We first demonstrate the notion of a confidence interval with an example. Let ξ1,...,ξn be sample variables from an N(a,σ2 0)-distributed population, where σ0 is a given positive number and −∞<a<∞. In II, Theorem 3.2, we showed that $$(\bar \xi \, - a)\sqrt {n/{\sigma _0}}$$ is N(0,1)-distributed when a is the true parameter value1. If we define κ α for given α with 0 < α < 1 according to II.(3.3), then we get
$$p\left( { - {k_\alpha } \le \frac{{\bar \xi - a}}{{{\sigma _0}}}\sqrt n \le {k_\alpha };a} \right) = 1 - \alpha {.^2}$$
(1.1)
In place of (1.1) one can also write
$$P(\bar \xi - {\sigma _0}{k_\alpha }/\sqrt n \le a \le \bar \xi + {\sigma _0}{k_\alpha }/\sqrt n ;a) = 1 - \alpha .$$
(1.2)

## Keywords

Confidence Level Probability Measure Test Problem Critical Region True Parameter
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## References

1. 1.
For this terminology see III6.Google Scholar
2. 2.
Since the normal distribution is continuous, one can delete the equality signs within the large brackets without altering the meaning of the expression.Google Scholar
3. 3.
The notion of a confidence interval is due to J. Neyman. See J. Neyman, Ann. Math. Statist. 6, 111–116 (1935); Actualites scientifiques et industrielles 739, 25–57, Hermann & Cie, Paris 1938. See also J. Neyman, Biometrika 32, 128–150 (1941) and J. Neyman, Philos. Trans. Roy. Soc. London, Ser. A 236, 333–380 (1937).Google Scholar
4. 4.
If inf AA or sup AA the cases β = 0 or β = 1 can cause (trivial) difficulties.Google Scholar
5. 5.
However, in Theorem 2.1 strict monotonicity is required. If this is not the case, the uniqueness of the construction is lost.Google Scholar
6. 6.
Confidence intervals for the binomial parameter were first given by J. Clopper and E. S. Pearson, Biometrika 26, 404–413 (1934). Also see O. Bunke, Wiss. Z. Humboldt-Univ. Berlin, Math.-Natur. Reihe 9, 335–363 (1959/60).
7. 7.
Also see E. Ricker, J. Amer. Statist. Assoc. 32, 349–356 (1937).
8. 8.
9. 9.
For simplicity we have assumed that for each 1-β, 0 ⫅ β ⫅ 1, such regions exist.Google Scholar
10. 10.
See the arguments following III, Theorem 4.2 as well as the appendix p. 481 ff.Google Scholar
11. 11.
12. 12.
See R. Borges, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 1, 47–69 (1962).