11
Generalized point estimators
In this chapter classical point estimators based on samples x1,…, xn of a stochastic quantity X are considered. Such estimators are functions 
: Mn → N, where M is the observation space of X (i.e. the set of all possible values of X) and Mn is the sample space for a sample of size n(N, Q) is a measurable space, where N is the set of all possible values for the quantity to be estimated.
Frequently the quantity to be estimated is a parameter θ of a parametric stochastic model X ~ Pθ, θ ∈ Θ or a quantile, or another characteristic value of the unknown distribution P0 of X.
In the following suitable classical estimators are considered, fulfilling standard quality requirements such as unbiasedness, efficiency, maximum likelihood, consistency, and others.
11.1 Estimators based on fuzzy samples
In reality frequently the observed samples are n fuzzy numbers x1*,…, xn* with corresponding characterizing functions ξ1(·),…,ξn(·). Therefore a classical estimator
becomes a function (x1*,…, xn*) of n fuzzy variables. Therefore by the extension principle (cf. Chapter ) the value (x1*,…, xn*) becomes fuzzy too. The characterizing function η(·) of the fuzzy value (x
