Quantitative data which cannot be characterized by precise numbers have to be characterized mathematically in a suitable way. This leads to so-called non-precise numbers which are defined by characterizing functions.

Definition 1. A characterizing function £(■) is a real function £ : R —> [0, 1] for which the so-called 5-cuts Cg [£(■)],

Remark 1. Characterizing functions are special membership functions from the theory of fuzzy sets. But non-precise numbers are more general than so-called fuzzy numbers. This is necessary to characterize fuzzy data, for example data obtained from color intensity pictures, especially X-ray data. Examples of characterizing functions are given in Figure 1.

are non-empty finite unions of bounded closed intervals, i. e.

Fig. 1. Characterizing functions

Remark 2. The one-point indicator function I{x0}(■) is the characterizing function of a precise data point xo € R. Therefore the analysis of fuzzy data contains standard statistical procedures as special case.

A main problem is the determination of the characterizing function of a fuzzy data point x*. In case of color intensities this is possible in the following way:

Let h(■) be the light intensity of a fuzzy light point on a screen. Then the characterizing function ■) is given by its values t(x) = Hl] , V x € R. max h(x)

For life time data t* the characterizing function ■) can be obtained from measurements of characteristic quantities which describe the degree of fulfilment of its objectives. Let f (■) be the function describing the degree of fulfilment of the characteristic quantity depending on the time t, then the characterizing function ■) of the fuzzy life time is given by its values max d f (t)

In case of quality of life data, which are fuzzy by nature, these data usually contain subjectivity. Therefore also the corresponding characterizing functions are subjective to a certain degree. But still they contain important information for statistical analysis.

3 Empirical reliability functions for fuzzy life times

Real life time data consist of a finite sample xf, ■ ■ ■ , x^ of non-precise numbers with corresponding characterizing functions £i(■ ), ■ ■ ■ ,£n(■ ).

The standard empirical reliability function Rn( ■) for precise data xi, ■ ■ ■ ,xn , defined by its values

is generalized in the following way:

Assuming all characterizing functions £i( ■) to be integrable, the generalized empirical reliability function Rn(■) is defined by oo

Remark 3. The generalized estimate R*( •) is a continuous function which is more realistic in case of continuous underlying life time distributions.

An example of fuzzy life times and the corresponding generalized empirical reliability function is given in Figure 2.

4 Generalized classical statistical inference for fuzzy data

Statistical estimation of derived indicators of quality of life, based on fuzzy data is possible in the following way: Let xi, • • • ,xn be classical pseudo-precise data and

be an indicator based on data xi, • • • ,xn and weights wi, • • • , wn. For standard data the indicator is a real number.

In case of fuzzy data xf, • • • , xn with corresponding characterizing functions £i( • ), • • • , £n( • ) the resulting value of the indicator becomes non-precise. In order to obtain the characterizing function n(•) of the non-precise value

I* = f (xi, • • • ,x*n ; wi ,• • • ,Wn) , the so-called extension principle from fuzzy set theory is applied. The values n(x) for all x € R of the characterizing function are defined by

Ísupj min{£i(xi), • • • Cn(^n)^ if f (xi, ■ ■ ,Xn ■ ■ ■ ,Wn) = X

0 if ^ (xi, ■ ■ ■ ,xn ): f (xi, ■ ■ ■ ,xn ;wi, ■ ■ ■ , w„) = x

The resulting non-precise value I* of the indicator is a non-precise number in the sense of section 2.

Remark 4- Although the generalized indicators are non-precise they represent valuable information concerning the considered topic.

Moreover different statistical inference procedures can be generalized to the situation of non-precise date. These methods are described in the book [Vie96].

Recent work on the generalization of the concept of p-values is published in the paper [FV04].

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