O'Reilly logo

Statistical Methods for Fuzzy Data by Reinhard Viertl

Stay ahead with the world's most comprehensive technology and business learning platform.

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, tutorials, and more.

Start Free Trial

No credit card required

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

Unnumbered Display Equation

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

Unnumbered Display Equation

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

b. Let then

Unnumbered Display Equation

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

With Safari, you learn the way you learn best. Get unlimited access to videos, live online training, learning paths, books, interactive tutorials, and more.

Start Free Trial

No credit card required