Abstract
A fuzzy test for testing statistical hypotheses about an imprecise parameter is proposed for the case when the available data are also imprecise. The proposed method is based on the relationship between the acceptance region of statistical tests at level ? and confidence intervals for the parameter of interest at confidence level 1 ? ?. First, a fuzzy confidence interval is constructed for the fuzzy parameter of interest. Then, using such a fuzzy confidence interval, a fuzzy test function is constructed. The obtained fuzzytest, contrary to the classical approach, leads not to a binary decision (i.e. to reject or to accept the given null hypothesis) but to a fuzzy decision showing the degrees of acceptability of the null and alternative hypotheses. Numerical examples are given to demonstrate the theoretical results, and show thepossible applications in testing hypotheses based on fuzzy observations.
Highlights
Hypothesis testing and confidence intervals play a prominent role in classical statistical texts
If the value of the parameter specified by the null hypothesis is contained in the 1 − β confidence interval the null hypothesis cannot be rejected at level β, and if it is not contained in the 1 − β confidence interval the null hypothesis can be rejected at level β
As we shall see, the fuzzy confidence set can be viewed as a statement about testing the hypothesis H : θ = θ0, which exhibits the values for which the hypothesis is accepted with degree C(θ0), i.e. {θ0 ∈ C(X ) : C(θ0) > 0} and those for which it is rejected with degree 1 − C(θ0), i.e. {θ0 ∈ C(X ) : 1 − C(θ0) > 0}, where C(X ) is a fuzzy confidence interval for the fuzzy parameter of interest (Chachi and Taheri, 2011)
Summary
Hypothesis testing and confidence intervals play a prominent role in classical statistical texts. Grzegorzewski and Hryniewicz (1997) reviewed some methods in testing statistical hypotheses in fuzzy environment, pointing out their advantages or disadvantages and practical problems. The aim of this work is to introduce a new approach to the problem of testing statistical hypotheses for fuzzy data using the relationship between confidence intervals and testing hypotheses. To do this we employ the method of constructing fuzzy confidence intervals for fuzzy parameters investigated by Chachi and Taheri (2011).
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