Abstract
In this paper, we develop fuzzy, possibilistic hypothesis tests for testing crisp hypotheses for a distribution parameter from crisp data. In these tests, fuzzy statistics are used, which are produced by the possibility distribution of the estimated parameter, constructed by the known from crisp statistics confidence intervals. The results of these tests are in much better agreement with crisp statistics than the ones produced by the respective tests of a popular book on fuzzy statistics, which uses fuzzy critical values. We also present an error that we found in the implementation of the unbiased fuzzy estimator of the variance in this book, due to a poor interpretation of its mathematical content, which leads to disagreement of some fuzzy hypotheses tests with their respective crisp ones. Implementing correctly this estimator, we produce test statistics that achieve results in hypotheses tests that are in much better agreement with the results of the respective crisp ones.
Highlights
Testing statistical hypotheses is a main branch of inferential statistics
If the value u of the test statistic of a hypothesis test is close to its critical value, the crisp test is unstable since a very small change in the sample may lead from rejection to no rejection of H0 or vice-versa, as shown in Example 1
If the value of the test statistic of a hypothesis is close to a critical value of the test, the crisp hypothesis test is unstable since a very small change in the sample may lead from rejection to no rejection of H0 or vice-versa, as shown in Examples 1 and 4
Summary
Testing statistical hypotheses is a main branch of inferential statistics (see [1]). A statistical hypothesis is an assertion about a parameter of the probability distribution of a random variable. In [30], this approach is generalized using fuzzy statistics produced by non asymptotic fuzzy estimators (see [31]) and a degree of rejection or acceptance of the null hypothesis As it is proved in [32], the possibility distribution (see [2]) induced by confidence intervals around the mode is identical to the one obtained by the maximal specificity probability–possibility transformation. Based on this principle, we develop in this paper possibilistic fuzzy statistical tests of crisp hypotheses, which lead to a possibility of rejection or acceptance of the null hypothesis for cases in which both the hypothesis and the data are crisp, whereas the other fuzzy tests (for example, those of [12]) give crisp results when applied in such cases.
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