20.2 Generalized estimators for linear regression models based on the extension principle
Depending on the kind of available data generalized estimators for the parameters can be obtained in the following way:
ad (a): Yx = θ0 + θ1x1 +…+ θkxk fuzzy dependent variables yi*:
For data (x1i,…,xki; yi*), i = 1(1)n the fuzzy values yi* are characterized by their characterizing functions ηi(·), i = 1(1)n.
In this situation the estimators j for θj, j = 1(1)k can be generalized in the following way:
In order to apply the extension principle first the fuzzy data y1*,…,yn* have to be combined to form a fuzzy vector y* in n.
The vector-characterizing function ζ(·,…,·) of the combined fuzzy vector y* is obtained from the characterizing functions η1(·),…,ηn(·) by application of the minimum t-norm, i.e. the values ζ(y1…, yn) are defined by
The estimators j of the regression parameters θj can now be generalized.
Using the notation xi = (xi1,…,xik) for i = 1(1)n the estimators j for θj from Theorem 19.1 ...