# Unbiased Fuzzy Estimators in Fuzzy Hypothesis Testing
## 4 Possibilistic Statistical Tests of Crisp Hypotheses
\tag{24}\]  
\[\varphi_{R}(X_{1},X_{2},\ldots,X_{n})=\text{Ness}(\widetilde{U}<\widetilde{CV }_{2})=1-\sup_{x,y,y\leq x}\min\{\mu_{\widetilde{U}}(x),\mu_{\widetilde{CV}_{ 2}}(y)\} \tag{25}\]  
So if the core of \(\widetilde{U}\) is at the left of \(u_{c2}\), (25) gives the following:  
\[\varphi_{R}(X_{1},X_{2},\ldots,X_{n})=1-\mu_{\widetilde{U}}^{R}(u_{c2})=1-y_{2} \tag{26}\]  
where \(y_{2}\) is the ordinate of the point of intersection of the right part \(\mu_{\widetilde{U}}^{R}\) of the possibility distribution of \(\widetilde{U}\) with the vertical at \(u_{c2}\). The possibility of acceptance of \(H_{0}\) is as follows:  
\[\varphi_{A}(X_{1},X_{2},\ldots,X_{n})=\text{Ness}(\widetilde{U}>\widetilde{CV }_{2})=1-\sup_{x,y,x\leq y}\min\{\mu_{\widetilde{U}}(x),\mu_{\widetilde{CV}_{ 2}}(y)\} \tag{27}\]  
So if the core of \(\widetilde{U}\) is at the right of \(u_{c2}\), (27) gives the following:  
\[\varphi_{A}(X_{1},X_{2},\ldots,X_{n})=1-\mu_{\widetilde{U}}^{L}(u_{c2})=1-y_{2} \tag{28}\]  
where \(y_{2}\) is the ordinate of the point of intersection of the left part \(\mu_{\widetilde{U}}^{L}\) of the possibility distribution of \(\widetilde{U}\) with the vertical at \(u_{c2}\). (c) For the _two-sided test_ (alternative hypothesis \(H_{1}:\theta\
eq\theta_{0}\)), the possibility of rejection of \(H_{0}\) is equal to the following:  
\[\varphi_{R}(X_{1},X_{2},\ldots,X_{n}) = \text{Ness}(\widetilde{U}>\widetilde{CV}_{1}\text{ \ or \ }\widetilde{U}<\widetilde{CV}_{2}) \tag{29}\] \[= \text{max}\Big{\{}\widetilde{U}>\widetilde{CV}_{1},\ \widetilde{U}< \widetilde{CV}_{2}\Big{\}}\]  
where (\(u_{c3}\) and \(u_{c4}\) are the critical values of the two-sided defined in (3)). \[\mu_{\widetilde{CV}_{1}}(u)=\left\{ \begin{array}{ll}1,&u=u_{c3}\\ 0,&\text{elsewhere}\end{array} \right.\text{ and \ }\mu_{\widetilde{CV}_{2}}(u)=\left\{ \begin{array}{ll}1,&u=u_{c4}\\ 0,&\text{elsewhere}\end{array} \right. \tag{30}\]  
So if the core of \(\widetilde{U}\) is at the right of \(u_{c3}\), (33) gives the following:  
\[\varphi_{R}(X_{1},X_{2},\ldots,X_{n})=1-y_{1} \tag{31}\]where \(y_{1}=\mu_{\widetilde{U}}^{L}(u_{c3})\) and if it is at the left of \(u_{c4}\), it gives the following:  
\[\varphi_{R}(X_{1},X_{2},\ldots,X_{n})=1-y_{2} \tag{32}\]  
where \(y_{2}=\mu_{\widetilde{U}}^{R}(u_{c4})\). The possibility of acceptance of \(H_{0}\) is as follows:  
\[\varphi_{A}(X_{1},X_{2},\ldots,X_{n})=\text{Ness}(\widetilde{U}<\widetilde{CV }_{1}\text{ and }\widetilde{U}>\widetilde{CV}_{2}) \tag{33}\]  
So if the core of \(\widetilde{U}\) is between \(u_{c3}\) and \(u_{c4}\), (34) gives the following:  
\[\varphi_{A}(X_{1},X_{2},\ldots,X_{n})=\min\{1-y_{1},1-y_{2}\} \tag{34}\]  
where \(y_{1}=\mu_{\widetilde{U}}^{R}(u_{c3})\) and \(y_{2}=\mu_{\widetilde{U}}^{L}(u_{c4})\) (the ordinates of the point of intersection of the right part \(\mu_{\widetilde{U}}^{R}\) of the possibility distribution of \(\widetilde{U}\) with the vertical at \(u_{c3}\) and of its left part \(\mu_{\widetilde{U}}^{L}\) with the vertical at \(u_{c4}\)). If the possibility of the rejection or acceptance of \(H_{0}\) is very low (for example lower than 0.2), then we _cannot make a decision_ on rejecting or accepting \(H_{0}\). Tests on the Mean of a Normal Distribution with Known Variance or of Any Distribution from a Large Sample  
We test at the significance level \(\gamma\) the null hypothesis \(H_{0}:\mu=\mu_{0}\) for the mean \(\mu\) of a random variable \(X\), which follows the normal distribution with known variance \(\sigma\), using a random sample of observations of \(X\) of size \(n\). In the crisp case, we test \(H_{0}\), using the statistic (see [1]). \[Z=\frac{\overline{X}-\mu_{0}}{\sigma/\sqrt{n}} \tag{35}\]  
where \(\overline{X}\) is the statistic of the sample mean.