# Unbiased Fuzzy Estimators in Fuzzy Hypothesis Testing
## 3 Fuzzy Estimation  
The estimation of a parameter of the probability density function (or probability function of a discrete random variable) of a distribution is one of the main purposes of inferential statistics.  
Let \(X\) be a random variable with probability density function (or probability function for a discrete random variable) \(f(x;\theta)\), where \(\theta\) is nan unknown parameter, which has to be estimated from a random sample of observations \(X_{1},X_{2},\ldots,X_{n}\) of \(X\). For the estimation of \(\theta\), a statistic \(U\) is used (estimator of \(\theta\)), which is a function of \(X_{1},X_{2},\ldots,X_{n}\). For a specific sample where \(x_{i},i=1,2,\ldots,n\) are the observed values of \(X_{i}\), a point estimator \(\theta_{0}=u(x_{1},x_{2},\ldots,x_{n})\) for \(\theta\) is obtained, which is not of high interest since the probability of this being the required value of \(\theta\) is zero. A way to estimate \(\theta\) in the crisp statistics is to find a \((1-\beta)100\%\) confidence interval for \(\theta\) in which the value of \(\theta\) can be found with probability \(1-\beta\). We use \(\beta\) here since \(\alpha\), usually employed for confidence intervals, is reserved for \(\alpha\)-cuts of fuzzy numbers. Usually, \(\beta=0.005\) or \(0.01\) or \(0.02\) or \(0.05\) or \(0.1\) for \(99.5\%\), \(99\%\), \(98\%\), \(95\%\) or \(90\%\) confidence intervals. Starting arbitrarily with \(\beta=0.01\) (we could begin with \(0.001\) or \(0.005\), etc.) and using \([\theta_{0},\theta_{0}]\) as the \(0\%\) confidence interval, we have the following confidence intervals:  
\[[\theta_{l}(\beta),\ \theta_{r}(\beta)],\ \ \ 0.01\leq\beta\leq 1\]  
Among other methods, a fuzzy estimation method is proposed in [29], according to which by placing all the above confidence intervals one on the top of the other, a triangular shaped fuzzy number \(\widetilde{\theta}\) is constructed, the \(\alpha-\)cuts of which are the confidence intervals:  
\[\mu_{\widetilde{\theta}_{\widetilde{\theta}}}[\alpha]=[\theta_{l}(\alpha),\ \theta_{r}(\alpha)],\ \ \ 0.01\leq\alpha\leq 1\]  
To finish the \"bottom\" of \(\widetilde{\theta}\) in order to make it a complete fuzzy number, one needs to drop its graph straight down to complete its \(\alpha-\)cuts (see Figure 1). So, a fuzzy estimator \(\widetilde{\theta}\) of \(\theta\) is produced from a given sample, the possibility distribution of which are the confidence intervals of \(\theta\)  
\[\mu_{\widetilde{\theta}_{\widetilde{\theta}}}[\alpha]=\left\{\begin{array}{ ll}[\theta_{l}(\alpha),\ \theta_{r}(\alpha)],&0.01\leq\alpha\leq 1\\ [\theta_{l}(0.01),\ \theta_{r}(0.01)],&0\leq\alpha<0.01\end{array}\right. \tag{4}\]  
The fuzzy estimator \(\widetilde{\theta}\) produced in this way contains much more information than just a single interval estimate.  
For the construction of test statistics used in fuzzy hypothesis testing about an unknown parameter \(\theta\), we use the fuzzy estimator \(\widetilde{\theta}\), the \(\alpha\)-cuts of the possibility distribution of which are the confidence intervals of \(\theta\)  
\[\mu_{\widetilde{\theta}}[\alpha]=[\theta_{t}(\alpha),\ \theta_{t}(\alpha)],\ \ \ 0\leq\alpha\leq 1 \tag{5}\]  
since, as proved in [32], this possibility distribution is identical to the one obtained by the maximal specificity probability-possibility transformation. In order to easily handle the possibility representation, especially for further computations, the possibility distribution is restricted to the interval that corresponds to the largest \(99\%\) confidence interval for the considered probability distribution (see [32]).