An asymptotic α-test for the expectation of random fuzzy variables
Introduction
The concept of fuzzy sets was introduced by Zadeh (1965) to describe non-statistical uncertainty (inexactness, vagueness). Random fuzzy variables defined by Puri and Ralescu (1986) deal with both kinds of uncertainty: the randomness and the vagueness. These random variables generalize the concept of random closed sets (cf. Matheron, 1975).
In test theory, significance testing (α-test) is a decision about whether the distribution of a stochastic quantity ξ belongs to a class of distribution or not. This decision is based on a test variablewhereby the distribution of t(ξ1,…,ξn) is known under H0. For example for an i.i.d. normally distributed sample for the hypothesis the test statisticsis t-distributed under H0 and the hypothesis H0 is rejected ifwhere tn−1,1−α/2 is the (1−α/2)-quantile of the t-distribution with (n−1) degrees of freedom. Note that is a distance between and μ0.
One way to construct tests for fuzzy samples is given by the extension of classical α-tests to fuzzy tests by the extension principle (cf. Casals et al., 1986; Kruse and Meyer, 1987; Watanabe and Imaizumi, 1993; Viertl, 1996). But the extensions are not α-tests. Gebhardt et al. (1998) is recommended for a summary.
Since there are only trivial normally distributed random fuzzy variables, there is not a test theory with normally distributed fuzzy samples. We propose a method to test hypotheses about the expectation with respect to the limit distribution. A law of large numbers (cf. Klement et al., 1986) shows that the fuzzy expectation is again unbiased and consistently estimated by the sample mean. Now, by a central limit theorem the asymptotical distribution of the distance between the sample mean and the expectation can be calculated by an ω2-distribution (cf. Martynov, 1978). Using the asymptotical distribution, an asymptotic α-test for fuzzy data is constructed, i.e. a hypotheses will be rejected if the distance between the sample mean and the hypothetical value μ0 is greater than the (1−α)-quantile of the asymptotical ω2-distribution.
Another approach to test parametric hypotheses is given by an Hotelling-type statistics, where the usual methods of multivariate statistics are applied.
Section snippets
Fuzzy sets and the general test statistics
A fuzzy subset of is defined by its membership function . The discussion is restricted to , the class of normal compact convex fuzzy subsets of , i.e. to the class of fuzzy sets A which satisfy
(i) A is normal, i.e. is nonempty and
(ii) for α∈(0,1] the α-level sets of Aare convex and compact.
Since any α-level set of is convex and compact, the concept of support functions is used to define an -distance and to apply properties of the
Example: LR-fuzzy numbers
An important subset of is the class of LR-fuzzy numberswhere AL and AR are fuzzy numbers with the α-level sets [AL]α≔[0,L(−1)(α)] and [AR]α≔[0,R(−1)(α)], for α∈(0,1]. Here are fixed left-continuous and non-increasing functions with L(0)=R(0)=1, and . The functions L and R are called left and right shape functions, m the modal point and l,r⩾0 are, respectively, the left and right spreads of the LR-fuzzy
Distribution of the test statistics
Note that for λ1>0Therefore, only the density of ξ12+λξ22+μξ32 need be calculated for a test with LR-fuzzy numbers. Lemma 6 Let ξ1 and ξ2 be independent -distributed. Then, for λ>0, the random variable η=ξ12+λξ22 has the density functionwhere J0 is the Bessel function of order zero. Proof The density of η=ξ12+λξ22 is given by convolution bywhere δ(y) is Dirac's
Example: test with LR-fuzzy numbers
In this section, an asymptotical test for LR-fuzzy numbers is considered, using the limit distribution of . Corollary 10 Let X1,…,Xn be a sample of LR-fuzzy numbers with given covariance structure {Cmm,Cll,Crr,Clm,Crm,Clr}. Thenwhere ξ1,ξ2,ξ3 are independent -distributed random variables and λ1,λ2,λ3 are the eigenvalues of the matrix (see Theorem 5)
Hotelling-type tests
Another approach to test parametric hypothesis is given by an Hotelling-type statistics (see for example Witting and Müller-Funk, 1995) for a sample of parametric random fuzzy variablesfor some random parameters (γ1,…,γm).
If F is continuous and the hypothesis H0 is formulated parametrically byfor consistent and asymptotical Gaussian estimators the statistics
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