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

13.1 Test statistics and fuzzy data

For fuzzy samples x1*, … , xn* of a stochastic quantity X the value of a test statistic t(x1*, … , xn*) becomes fuzzy too. The characterizing function ψ(·) of the fuzzy value t* = t(x1*, … , xn*) is obtained by the extension principle (cf. Section 10.3).

Using the notation from the beginning of this chapter there are three possible situations:

1. The support supp(t*) is a subset of A.

2. The support supp(t*) is a subset of C.

3. supp (t*) has non-empty intersections with both A and C.

Situations (1) and (2) yield a decision as in the case of precise observations (which are usually unrealistic for continuous quantities X).

In situation (3) a decision is not possible because one cannot decide if the value t* of the test statistic is in the acceptance region or not.

This situation indicates that the information contained in the given sample is not sufficient in order to make a well-based decision.

The three possible situations are displayed in Figure 13.1. Cases (1) and (2) are supporting a test decision. Case (3) needs more data in order to come to a decision.

Figure 13.1 Testing situations with fuzzy samples.

c13f001.eps

For two-sided test problems the situations are analog.

In applications often p-values are calculated for making test decisions.

For fuzzy values t* of a test statistic t(x1, … , xn) p-values can be obtained in the following way: Based ...

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