## 4 Methods of Estimation and Test
### Cramer-Von Mises and Anderson-Darling Methods  
In recent years, some authors have used classical GOF statistics such as Cramer-von Mises and Anderson-Darling statistics to derive the estimators of the unknown parameters. Ref. [26] used these methods for the binomial-exponential 2 distributions, while [27] evaluated the process capability index for normal distribution. Aside from these methods and using this approach, we propose the use of the Pearson Chi-square statistic \(X^{2}\) where the estimators noted \(CSMM\) are obtained by calculating the minimum of \(X^{2}\) with respect to the unknown parameters.  
Ref. [28] and later [29] showed that the Cramer-von Mises estimates based on the distance of Cramer-von Mises GOF statistics are the least biased compared to the other estimators. For calculation purposes, ref. [29] gives the formula \(C(\Psi)\) for this statistic, where the estimators ensure its minimum concerning the unknown parameters, where  
\[C(\Psi)=\frac{1}{12n}+\sum_{i=1}^{n}\left\{1-\left(1+\frac{\theta x_{i}^{\beta} \left(\theta+2+\theta x_{i}^{\beta}\right)}{2+\theta}\right)e^{-\theta x_{i}^{ \beta}}-\frac{2i-1}{2n}\right\}^{2}. \tag{24}\]  
They also can be gained as the solution of the subsequent equations:  
\[\frac{\partial C(\Psi)}{\partial\theta}=\sum_{i=1}^{n}\left\{1-\left(1+\frac{ \theta x_{i}^{\beta}\left(\theta+2+\theta x_{i}^{\beta}\right)}{2+\theta} \right)e^{-\theta x_{i}^{\beta}}-\frac{2i-1}{2n}\right\}\varphi_{1}(x_{i}, \Psi)=0\]  
\[\frac{\partial C(\Psi)}{\partial\beta}=\sum_{i=1}^{n}\left\{1-\left(1+\frac{ \theta x_{i}^{\beta}\left(\theta+2+\theta x_{i}^{\beta}\right)}{2+\theta} \right)e^{-\theta x_{i}^{\beta}}-\frac{2i-1}{2n}\right\}\varphi_{2}(x_{i}, \Psi)=0.\]  
The Anderson-Darling estimation method is based on the classical Anderson and Darling \((AD)\) goodness-of-fit statistic test proposed by [30]. This statistic is essentially used to fit data to a theoretical hypothesized model \(F_{0}(x)\). When the parameters are unknown, they can be estimated by \(\hat{\theta}_{ADE}\), \(\hat{\beta}_{ADE}\) which minimize the \(AD\) statistic given in the form  
\[ADE(\Psi)=-n-n^{-1}\sum_{i=1}^{n}(2i-1)\{\log F(x_{i})+\log(1-F(x_{-i+n+1}))\},\]  
which is equivalent to canceling the first derivatives of this function with respect to \(\beta\) and \(\theta\). For more details on this method, see [29]. In their paper, ref. [31] showed that the \(ADE\) estimation method gives the most efficient estimators. As is known, the Anderson-Darling statistic gives more weight to the tails of the distribution, so for right- or left-tailed distributions, the \(ADE\) right tail (\(ADE_{R-T}\)) and \(ADE\) left tail (\(ADE_{L-T}\)) can be considered for estimating the unknown parameters, as in [27; 31; 32].  
The \(ADE_{R-T}\) estimators of \(\theta\) and \(\beta\) denoted by \(\hat{\theta}_{ADE_{(R-T)}}\), \(\hat{\beta}_{ADE_{(R-T)}}\), respectively, are found by minimizing the equation given by  
\[ADE_{(R-T)}(\Psi)=\frac{n}{2}-2\sum_{i=1}^{n}\log F(x_{i})-\frac{1}{n}\sum_{i =1}^{n}(2i-1)\log\{1-F(x_{-i+n+1})\}\]  
or  
\[\frac{\partial}{\partial\theta}\Big{[}ADE_{(R-T)}(\Psi)\Big{]}=0,\ \frac{\partial}{\partial\beta}\Big{[}ADE_{(R-T)}(\Psi)\Big{]}=0\]  
Similarly, the \(ADE_{L-T}\) estimators denoted by \(\hat{\theta}_{ADE_{(L-T)}}\), \(\hat{\beta}_{ADE_{(L-T)}}\) are acquired by minimizing the equation  
\[ADE_{(L-T)}(\Psi)=-\frac{3}{2}n+2\sum_{i=1}^{n}F(x_{i})-\frac{1}{n}\sum_{i=1}^ {n}(2i-1)\log\{F(x_{i})\}\]  
or  
\[\frac{\partial}{\partial\theta}\Big{[}ADE_{(L-T)}(\Psi)\Big{]}=0,\ \frac{\partial}{\partial\beta}\Big{[}ADE_{(L-T)}(\Psi)\Big{]}=0\]  
with respect to the unknown distribution parameters. For more about the Anderson-Darling estimators, see [27; 31; 32]. Conversely, ref. [33] demonstrated that the \(ADE\) minimum-distance estimation method is a highly competitive method compared to the ML estimation. In addition, ref. [24] confirmed the superiority of this method.