10.3 Statistics of fuzzy data
In standard statistics basic for many inference procedures are functions of data x1, … , xn, i.e. s(x1, … , xn), where s : Mxn → N are measurable functions from the sample space to a suitable measurable space N. For a random sample x1, … , xn of a stochastic quantity (also called random variable), the stochastic quantity s(X1, … , Xn) is called statistic.
In the case of fuzzy samples x1*, … , xn* also the values s(x1*, … , xn*) become fuzzy, and the fuzziness of the value s(x1*, … , xn*) is expressed by a membership function η(·) of a fuzzy element in N.
In order to obtain the membership function η(·) first the fuzzy sample has to be combined into a fuzzy element * of the sample space. Then the extension principle can be applied to obtain η(·).
For fuzzy data x1*, … , xn* with corresponding characterizing functions ξ1(·), … , ξn(·) the values η(y) of s(x1*, … , xn*) are given by
where = (x1, … , xn) and
Frequently Mx ⊆ and N ⊆ . More generally ...