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.

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 ...

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