# Unbiased Fuzzy Estimators in Fuzzy Hypothesis Testing
## 4 Possibilistic Statistical Tests of Crisp Hypotheses  
We denote by 1 the rejection and by 0 the acceptance of a hypothesis \(H_{0}\) for a parameter \(\vartheta\) of the distribution of a random variable \(X\) from a sample of \(n\) observations \(X_{1},X_{2},\ldots,X_{n}\) of \(X\). Hence, a possibilistic test of a crisp hypothesis \(H_{0}\) with alternative \(H_{1}\) is a decision rule, defined as two functions: \(\varphi_{R}(X_{1},X_{2},\ldots,X_{n}):R^{n}\to[0,1]\), which gives the degree in which the observed value \(u\) of the test statistic \(U\) belongs to the rejection region (see Section 1) (the possibility of the proposition \"\(u\) is in the rejection region\"), and \(\varphi_{A}(X_{1},X_{2},\ldots,X_{n}):R^{n}\to[0,1]\), which gives the degree in which the observed value \(u\) of the test statistic \(U\) belongs to the acceptance region (see Section 1) (the possibility of the proposition \"\(u\) is in the acceptance region\"). Using the fuzzy estimator \(\widehat{\vartheta}\) of \(\vartheta\) (see Section 2) and interval arithmetics, we derive the \(\alpha\)-cuts of the possibility distribution \(\mu_{\widetilde{U}}\) of the respective fuzzy test statistic \(\widetilde{U}\). So, the possibilities of rejection or acceptance of \(H_{0}\) are as follows:  
(a) For _one sided test from the right_ (alternative hypothesis \(H_{1}:\theta>\theta_{0}\)), the possibility of rejection of \(H_{0}\) is equal to the Necessity of Strict Dominance index (see [23; 36]). \[\varphi_{R}(X_{1},X_{2},\ldots,X_{n})=\text{Ness}(\widetilde{U}>\widehat{CV}_ {1})=1-\sup_{x,y,x\leq y}\min\{\mu_{\widetilde{U}}(x),\mu_{\widehat{CV}_{1}} (y)\} \tag{19}\]  
which represents the necessity that the fuzzy set \(\widetilde{U}\) strictly dominates the fuzzy set of the right critical value \(\widehat{CV}_{1}\), the membership function of which is as follows:  
\[\mu_{\widehat{CV}_{1}}(u)=\left\{ \begin{array}{ll}1,&u=u_{c1}\\ 0,&u\
eq u_{c1}\end{array} \right. \tag{20}\]  
where \(u_{c1}\) is the critical value of the one sided from the right, defined in (1). So if the core of \(\widetilde{U}\) is at the right of \(u_{c1}\), (19) gives the following:  
\[\varphi_{R}(X_{1},X_{2},\ldots,X_{n})=1-\mu_{\widetilde{U}}^{L}(u_{c1})=1-y_{1} \tag{21}\]  
where \(\mu_{\widetilde{U}}^{L}\) is the left part of the possibility distribution \(\mu_{\widetilde{U}}\) and \(y_{1}\) is the ordinate of the point of intersection of \(\mu_{\widetilde{U}}^{L}\) with the vertical at \(u_{c1}\). Figure 3: Non-asymptotic fuzzy estimators of the variance \(\sigma^{2}\): correct unbiased (blue line), biased (green line), unbiased used in [29] (red line). The possibility of acceptance of \(H_{0}\) is as follows:  
\[\varphi_{A}(X_{1},X_{2},\ldots,X_{n})=\text{Ness}(\widetilde{U}<\widetilde{CV}_{1 })=1-\sup_{x,y,y\leq x}\min\{\mu_{\widetilde{U}}(x),\mu_{\widetilde{CV}_{1}}(y)\} \tag{22}\]  
which represents the necessity that the fuzzy set \(\widetilde{CV}_{1}\) strictly dominates the fuzzy set of \(\widetilde{U}\). So if the core of \(\widetilde{U}\) is at the left of \(u_{c1}\), (22) gives the following:  
\[\varphi_{A}(X_{1},X_{2},\ldots,X_{n})=1-\mu_{\widetilde{U}}^{R}(u_{c1})=1-y_{1} \tag{23}\]  
where \(\mu_{\widetilde{U}}^{R}\) is the right part of the possibility distribution \(\mu_{\widetilde{U}}\) and \(y_{1}\) is the ordinate of the point of intersection of \(\mu_{\widetilde{U}}^{R}\) with the vertical at \(u_{c1}\). (b) For, _the one sided test from the left_ (alternative hypothesis \(H_{1}:\theta<\theta_{0}\)), the possibility of rejection of \(H_{0}\) is equal to the Necessity of Strict Dominance index of \(\widetilde{CV}_{2}\) to \(\widetilde{U}\), where the membership function of the critical value \(\widetilde{CV}_{2}\) is as follows (\(u_{c2}\) the critical value of the one sided from the left defined in (2)):  
\[\mu_{\widetilde{CV}_{2}}(u)=\left\{ \begin{array}{ll}1,&u=u_{c2}\\ 0,&u\
eq u_{c2}\end{array} \right.