# Unbiased Fuzzy Estimators in Fuzzy Hypothesis Testing
## 3 Fuzzy Estimation
### Estimation of the Variance of a Normal Variable  
If the random variable \(X\) follows normal distribution, then the \((1-\beta)100\%\) confidence interval of the variance \(\sigma^{2}\) of \(X\) derived from a random sample of observations of \(X\) of size \(n\) and sample variance \(s^{2}\) is as follows [1]:  
\[\left[\frac{(n-1)s^{2}}{\chi_{\frac{\alpha}{2};n-1}^{2}}\,,\ \ \frac{(n-1)s^{2}}{\chi_{1-\frac{\beta}{2};n-1}^{2}}\right] \tag{10}\]  
where (\(F_{n-1}^{-1}\) the inverse distribution function of the \(\chi_{n-1}^{2}\) distribution)  
\[\chi_{1-x;n-1}^{2} = F_{n-1}^{-1}(\alpha)\] \[\chi_{x;n-1}^{2} = F_{n-1}^{-1}(1-\alpha) \tag{11}\]  
Figure 2: The possibility distribution of the fuzzy estimator \(\widetilde{\mu}\) of the mean \(\mu\) of a random variable \(X\), which follows normal distribution with variance \(\sigma^{2}=2.3\) derived from a sample of \(n=100\) observations with sample mean \(\overline{x}=5\). So according to (5), the \(\alpha\)-cuts of the possibility distribution of the fuzzy estimator of the variance \(\sigma^{2}\) are as follows:  
\[\mu_{\widetilde{\sigma_{\mathrm{B}}^{2}}}[\alpha]=\ \left[\frac{(n-1)s^{2}}{\chi_{ \frac{2}{2}n-1}^{2}}\,\ \ \frac{(n-1)s^{2}}{\chi_{1-\frac{2}{2}n-1}^{2}}\right],\ \ \alpha\in[0,1] \tag{12}\]  
Therefore, the \(\alpha\)-cuts of the fuzzy estimator of the standard deviation \(\sigma\) of \(X\) are as follows:  
\[\mu_{\widetilde{\sigma_{\mathrm{B}}}}[\alpha]=\ \left[s\sqrt{\frac{(n-1)}{\chi_{ \frac{2}{2}n-1}^{2}}}\,\ \ s\sqrt{\frac{(n-1)}{\chi_{1-\frac{2}{2}n-1}^{2}}}\right],\ \ \alpha\in[0,1] \tag{13}\]  
It can be shown that the above fuzzy estimator \(\widetilde{\sigma_{\mathrm{B}}^{2}}\) is biased because its core is not at \(s^{2}\). A fuzzy estimator of a parameter is defined as biased if its core is not at the crisp point estimator of the parameter. In [29], an unbiased fuzzy estimator \(\widetilde{\sigma_{\mathrm{B}}^{2}}\) of the variance \(\sigma^{2}\) of a normal random variable is obtained from a sample of size \(n\) and variance \(s^{2}\) by putting one above the other for the following confidence intervals:  
\[\left[\frac{(n-1)s^{2}}{L(\lambda)},\ \ \frac{(n-1)s^{2}}{R(\lambda)}\right],\ \ \ 0\leq\lambda\leq 1 \tag{14}\]  
where  
\[L(\lambda) = (1-\lambda)\chi_{0.005n-1}^{2}+(n-1)\lambda \tag{15}\] \[R(\lambda) = (1-\lambda)\chi_{1-0.005n-1}^{2}+(n-1)\lambda \tag{16}\]  
and (\(F_{n-1}^{-1}\) the inverse distribution function of the \(\chi_{n-1}^{2}\) distribution)  
\[\chi_{1-0.005n-1}^{2}=F_{n-1}^{-1}(0.005)\ \ \ \mathrm{and}\ \ \ \chi_{0.005n-1}^{2}=F_{n-1}^{-1}(0.995)\]  
For a value of \(\lambda\), the corresponding value of \(\alpha\) is as follows [29]:  
\[\chi_{1-x;n-1}^{2} = F_{n-1}^{-1}(\alpha)\]0  
where \(\chi_{n-1}^{2}(x)\) the probability density function of the \(\chi_{n-1}^{2}\) distribution. The range of this one to one and onto function is \([0.01,1]\). So, the \(\alpha-\)cuts of the correct unbiased fuzzy estimator \(\widetilde{\sigma_{\mathrm{B}}^{2}}\) are as follows:  
\[\mu_{\widetilde{\sigma_{\mathrm{B}}^{2}}}[\alpha]=\left\{ \begin{array}{ll} \ \ \left[\frac{(n-1)s^{2}}{L(\lambda)},\ \ \frac{(n-1)s^{2}}{R(\lambda)}\right],&\ \ \ \lambda=\phi^{-1}(\alpha)\ \ \ \ \ \ 0.01\leq\alpha\leq 1\\ \ \ \ \left[\frac{(n-1)s^{2}}{L(0.01)},\ \ \frac{(n-1)s^{2}}{R(0.01)} \right],&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0\leq\alpha<0.01\end{array} \right. \tag{18}\]  
where the function \(\alpha=\phi(\lambda)\) and its inverse \(\lambda=\phi^{-1}(\alpha)\) (it gives \(\lambda\) as a function of \(\alpha\)) are implemented numerically. In the implementation of \(\widetilde{\sigma_{\mathrm{B}}^{2}}\) of [29], the \(\lambda\)-cuts defined by (14) (instead of its \(\alpha\)-cuts of (18)) are used for the construction of \(\mu_{\widetilde{\sigma_{\mathrm{B}}^{2}}}\) and the statistics generated by it.