# Unbiased Fuzzy Estimators in Fuzzy Hypothesis Testing
## 1 Introduction  
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. After the introduction of the notion of fuzzy sets by [PERSON] [2], many approaches have been proposed for fuzzy hypothesis testing, using fuzzy set theory (see [3]). The problem of testing hypotheses with fuzzy data was analyzed for the first time in [4] and in the sequel in [5, 6], where the author extended both Neyman-Pearson and Bayes theories to this framework (in [5]) and worked on the same problem in the context of fuzzy decision problems (in [6]). Fuzzy hypothesis testing with crisp data is presented in [7] in the sense of Neyman-Pearson, using the extension principle and \(\gamma\)-cuts, where fuzzy critical regions are introduced, leading to a fuzzy conclusion, as well as in [8, 9], where the author provides new definitions for the probability of type I and type II errors and presented the best test for the one-parameter exponential family. The problem of testing fuzzy hypotheses when the observations are crisp is also studied in [10], where the authors give new definitions for the probability of type I and type II errors and prove a version of the Neyman-Pearson Lemma. They also study the problem of testing hypotheses from a Bayesian point of view for which the observations are ordinary and the hypotheses are fuzzy (see [11]). In [12], testing hypotheses about the mean of a fuzzy random variable is introduced. This approach is applicable to practical situations where either the observed data or the hypotheses are fuzzy (formalized in linguistic terms). These fuzzy tests result in a fuzzy decision for the acceptance or rejection of the null hypotheses with a degree of confidence between 0 and 1. They are generalizations of the classical tests, so that they are reduced to the classical tests if both the data and the parameters are crisp. Fuzzy hypothesis tests are also developed in [13, 14, 15] for cases in which the available data are fuzzy and in [16], where the authors propose fuzzy hypothesis testing for a proportion with crisp data as the exact generalized one-tailed hypergeometric test with fuzzy hypotheses. Fuzzy hypothesistesting in the framework of the randomized and non-randomized hypergeometric test for a proportion is presented in [17]. In [18], the fuzzy \(p\)-value concept is used for testing hypotheses with fuzzy data and in [19] it is generalized on the basis of [PERSON]'s probability measure of fuzzy events [2] for testing fuzzy hypotheses with crisp data. In [20], a fuzzy \(p\)-value is obtained, using fuzzy test statistics constructed by fuzzy estimators for cases in which both the data and the hypotheses are crisp. Two different ways of making inference from set-valued data are presented in [21]. A detailed review on possibilistic interpretations of statistical tests (where hypotheses and/or data are fuzzy) and statistical decisions is presented in [22; 23] and a comprehensive review with regard to statistical properties of different approaches for calculating fuzzy \(p\)-values, in [24]. In [25], the authors demonstrate how to accomplish a fuzzy test with fuzzy data and fuzzy formulated hypotheses and discuss the defuzzification of fuzzy test decisions by means of the signed distance method. In [26], the author reviews and compares the R packages \"FPV\" and \"Fuzzy.p.value\" for hypothesis testing in fuzzy environments by using the fuzzy \(p\)-value for decision making. In [27], the authors systematically review the literature, identifying papers proposing advances in fuzzy hypothesis testing and its applications. Then, they look at each contained paper through the lens of the key research questions. In [28; 29], fuzzy hypotheses testing is developed for cases in which both the data and the hypotheses are crisp. This approach uses fuzzy critical values and fuzzy test statistics constructed by fuzzy estimators produced by a set of confidence intervals.