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Chapter 12

a. The fuzzy confidence region can be constructed in the following way: The membership function φ(·) of the fuzzy confidence region (here a fuzzy confidence interval) Θ1 − α* for θ can be obtained using Proposition 12.1. Taking the confidence function κ(x1,…, xn) given in Example 12.1, the δ-cuts of Θ1 − α* are

The minimum is obtained for the left boundaries aδ, i of the δ-cuts of xi*, and the maximum for the right boundaries bδ, i of the δ-cuts of xi*. Therefore we obtain

The numerical result for the data from Figure 12.1 with α = 0.05 is depicted in Figure 12.2.

b. Let then

and ∃ : θκ(). Moreover by assumption ξ() = 1 , and therefore sup{ξ() : θκ()} = 1 and φ(θ) = 1 ...

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