12
Generalized confidence regions
Point estimators are based on qualitative goodness criteria, but often quantitative measures for the quality of an estimate are sought. Such estimates are confidence regions.
12.1 Confidence functions
In order to construct confidence regions (i.e. subsets Θ1−α of the parameter space Θ) which contain the true parameter θ0 with high probability 1 − α (where α is a very small positive number) so-called confidence functions are used. A confidence function
is a function which assigns to every sample x1,…, xn a subset Θ1−α of Θ such that for a mathematical sample X1,…, Xn of the observed stochastic quantity X the following is true:
For α a value of 0.05 or 0.01 is usually taken. Therefore the so-called coverage probability is 1 − α.
12.2 Fuzzy confidence regions
In the case of fuzzy samples x1*,…, xn* of a parametric stochastic model X ~ Pθ, θ ∈ Θ, for given confidence function κ : Mn → 
(Θ) a generalized confidence region for the true parameter is the fuzzy subset Θ1−α* of Θ, whose membership function φ(·) is defined in the following way:
where = (x1,…, ...
