# zbMATH — the first resource for mathematics

Testing statistical hypotheses with vague data. (English) Zbl 0948.62010
A definition of fuzzy test for testing statistical hypotheses with vague data is proposed. Then the general method for the construction of fuzzy tests for hypotheses concerning an unknown parameter against one-sided or two-sided alternative hypotheses is shown. This fuzzy test, contrary to the classical approach, leads not to the binary decision: to reject or to accept given null hypotheses, but to a fuzzy decision showing a grade of acceptability of the null and the alternative hypotheses, respectively. However, it is a natural generalization of the traditional test, i.e. if the data are precise, not vague, we get a classical statistical test with the binary decision. A measure of fuzziness of the considered fuzzy test is suggested and the robustness of that test is also discussed.

##### MSC:
 62F03 Parametric hypothesis testing 62C99 Statistical decision theory 62F99 Parametric inference
##### Keywords:
fuzzy data; fuzzy test; fuzzy decision
Full Text:
##### References:
  Arnold, B.F., Statistical tests optimally meeting certain fuzzy requirements on the power function and on the sample size, Fuzzy sets and systems, 75, 365-372, (1995) · Zbl 0851.62025  Casals, R.; Gil, M.A., A note on the operativeness of neyman – pearson tests with fuzzy information, Fuzzy sets and systems, 30, 215-220, (1989) · Zbl 0665.62008  Casals, R.; Gil, M.A.; Gil, P., On the use of Zadeh’s probabilistic definition for testing statistical hypotheses from fuzzy information, Fuzzy sets and systems, 20, 175-190, (1986) · Zbl 0611.62018  Casals, R.; Gil, M.A.; Gil, P., The fuzzy decision problem: an approach to the problem of testing statistical hypotheses with fuzzy information, Eur. J. oper. res., 27, 371-382, (1986) · Zbl 0605.62018  Delgado, M.; Verdegay, J.L.; Vila, M.A., Testing fuzzy hypotheses. A Bayesian approach, (), 307-316  Grzegorzewski, P.; Hryniewicz, O., Testing hypotheses in fuzzy environment, Mathware and soft computing, 4, 203-217, (1997) · Zbl 0893.68139  Kruse, R., The strong law of large numbers for fuzzy random variables, Inform. sci., 28, 233-241, (1982) · Zbl 0571.60039  Kruse, R.; Meyer, K.D., Statistics with vague data, (1987), Riedel Dordrecht · Zbl 0663.62010  Kruse, R.; Meyer, K.D., Confidence intervals for the parameters of a linguistic random variable, (), 113-123  H. Kwakernaak, Fuzzy random variables, part I: definitions and theorems, Inform. Sci. 15 (1978) 1-15; Part II: algorithms and examples for the discrete case, Inform. Sci. 17 (1979) 253-278. · Zbl 0438.60004  Lehmann, E.L., Testing statistical hypotheses, (1986), Wiley New York · Zbl 0608.62020  Puri, M.L.; Ralescu, D.A., Fuzzy random variables, J. math. anal. appl., 114, 409-422, (1986) · Zbl 0592.60004  Saade, J., Extension of fuzzy hypothesis testing with hybrid data, Fuzzy sets and systems, 63, 57-71, (1994) · Zbl 0843.62004  Saade, J.; Schwarzlander, H., Fuzzy hypothesis testing with hybrid data, Fuzzy sets and systems, 35, 197-212, (1990) · Zbl 0713.62010  Son, J.Ch.; Song, I.; Kim, H.Y., A fuzzy decision problem based on the generalized neyman – pearson criterion, Fuzzy sets and systems, 47, 65-75, (1992) · Zbl 0757.62012  Stein, W.E.; Talati, K., Convex fuzzy random variables, Fuzzy sets and systems, 6, 271-283, (1981) · Zbl 0467.60005  Watanabe, N.; Imaizumi, T., A fuzzy statistical test of fuzzy hypotheses, Fuzzy sets and systems, 53, 167-178, (1993) · Zbl 0795.62025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.