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,…, xn) and ξ(·) in the vector-characterizing function of the combined fuzzy sample * and M is the observation space of X.

Remark 12.1:

The construction of the fuzzy confidence region is not an application of the extension principle. It should be noted that for classical data x1,…, xn, taking the one-point indicator functions I{xi}(·), the resulting membership function of the generalized confidence region is the indicator function of the classical confidence region κ(x1,…, xn), i.e. φ(·) = Iκ(x1,…, xn)(·).

Remark 12.2:

For the membership function φ(·) of the fuzzy confidence regions from this section under the conditions above and using the notation from before, the following holds:

To see this consider sup {ξ() : θ ∈ κ ()}.

Example 12.1

Let X ~ Exθ, θ ∈ (0; ∞) be the ...

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