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Statistical tests for fuzzy data

Statistical test decisions are frequently based on measurable functions t(x1, … , xn) of observed samples x1, … , xn.

Denoting by MT the set of all possible values of the test statistic T = t(X1, … , Xn), for hypothesis concerning the distribution of the observed stochastic quantity X, the set MT is usually decomposed into a region of acceptance of , denoted by A, and a so-called critical region C, i.e. MT = A ∪ C with A ∩ C = . The decision rule is the following.

Let x1, … , xn be the observed sample, then:

if t(x1, … , xn) ∈ A ⇒ is accepted if t(x1, … , xn) ∈ C = MT\A ⇒ is rejected.

The subsets A and C are constructed using the probability α of an error of the first type, i.e. the probability of rejecting a true hypothesis :

For one-dimensional test ...