## 5 Methods
### Goodness-of-Fit and Model Selection Criteria
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Distribution** & **MLM Estimate** \\ \hline W2 & Log-likelihood function & \(\ln L=n\ln\frac{\hat{F}}{\mu}-\frac{1}{\mu}\sum_{i=1}^{n}x_{i}^{\alpha}+(a-1) \sum_{i=1}^{n}\ln x_{i}\) \\  & Likelihood equations & \(\frac{1}{\delta}-\frac{\sum_{i=1}^{n}x_{i}^{\alpha}\ln x_{i}}{\sum_{i=1}^{n} \sqrt{x_{i}}}+\frac{1}{\mu}\sum_{i=1}^{n}\ln x_{i}=0\) \\  & & \(b=\Big{[}\frac{1}{\mu}\sum_{i=1}^{n}x_{i}^{\alpha}\Big{]}^{1/\alpha}\) \\ \hline W3 & Log-likelihood function & \(\ln L=n\ln\frac{\hat{F}}{\mu}-\frac{1}{\mu}\sum_{i=1}^{n}(x_{i}-c)^{\alpha}+(a- 1)\sum_{i=1}^{n}\ln(x_{i}-c)\) \\  & Likelihood equations & \(\frac{1}{\delta}-\frac{\sum_{i=1}^{n}(x_{i}-c)^{\alpha}}{\sum_{i=1}^{n}(x_{i} -c)^{\alpha}}+\frac{1}{\mu}\sum_{i=1}^{n}\ln(x_{i}-c)=0\) \\  & & \(b=\Big{[}\frac{1}{\mu}\sum(x_{i}-c)^{\alpha}\Big{]}^{1/\alpha}\) \\  & & \(\frac{1}{\delta}-\frac{1}{\mu}\frac{\sum_{i=1}^{n}(x_{i}-c)^{\alpha}}{\sum_{i=1 }^{n}(x_{i}-c)^{\alpha}}\sum_{i=1}^{n}\frac{1}{\mu_{i}-c}=0\) \\ \hline EW & Log-likelihood function & \(\ln L=n\ln\big{(}\frac{\hat{F}}{\mu}\big{)}-\frac{1}{\hat{F}}\sum_{i=1}^{n}x_{i }^{\alpha}+(a-1)\sum_{i=1}^{n}\ln x_{i}+(\gamma-1)\sum_{i=1}^{n}\ln\big{[}1- \exp\big{(}-(\frac{\gamma}{\hat{F}})^{\alpha}\big{)}\big{]}\) \\  & Likelihood equations & \(\frac{\mu}{\delta}-\sum_{i=1}^{n}\big{(}\frac{\gamma}{\hat{F}}\big{)}^{\alpha} \ln(\frac{\gamma}{\hat{F}})+\sum_{i=1}^{n}\ln\big{(}\frac{\hat{F}}{\mu}\big{)}+( \gamma-1)\sum_{i=1}^{n}\frac{\exp\big{(}-(\frac{\gamma}{\hat{F}})^{\alpha} \big{)}\big{(}\big{)}\big{(}\hat{F}}\big{)}^{\alpha}\ln\big{(}\frac{\hat{F}}{ \mu}\big{)}\) \\  & & \(-\frac{\mu\delta}{\hat{F}}\sum_{i=1}^{n}\big{(}\frac{\gamma}{\hat{F}}\big{)}^{ \alpha}-\frac{\delta}{\delta}(\gamma-1)\sum_{i=1}^{n}\frac{\exp\big{(}-(\frac{ \gamma}{\hat{F}})^{\alpha}\big{)}\big{(}\hat{F}}\big{)}{\big{[}1-\exp\big{(}-( \frac{\gamma}{\hat{F}})^{\alpha}\big{)}\big{]}}=0\) \\  & & \(\gamma=-\frac{\mu\delta}{\sum_{i=1}^{n}\big{[}1-\exp\big{(}-(\frac{\gamma}{ \hat{F}})^{\alpha}\big{)}\big{)}\big{]}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Log-likelihood functions and likelihood equations of selected probability distributions. The application of the maximum likelihood method (MLM) for the parameter estimation allows us to use the information criteria--[PERSON]'s information criterion (_AIC_) and Bayesian information criterion (_BIC_)--to decide the GOF for the distributions. The _AIC_ and the _BIC_ are defined as follows [67; 68]  
\[AIC=-2\ln L+2k, \tag{12}\]  
\[BIC=-2\ln L+k\ln n \tag{13}\]  
where \(\ln L\) is the maximum value of the log-likelihood function for the estimated model, \(k\) is number of estimated parameters, and \(n\) is the sample size. Furthermore, the coefficient of determination (\(R^{2}\)) and the root mean square error (_RMSE_) are considered to decide on the best fitting model. The _RMSE_ determines the accuracy of the model by calculating the average of the square difference between the observed and the predicted probabilities of the theoretical distribution. The \(R^{2}\) is used to measure the linear relationship between the observed and the predicted probabilities of the theoretical distribution. The _RMSE_ and \(R^{2}\) are calculated by:  
\[RMSE=\left(\frac{1}{n}\sum_{i=1}^{n}\bigl{[}F_{n}(x_{i})-\hat{F}(x_{i})\bigr{]} ^{2}\right)^{\frac{1}{2}}, \tag{14}\]  
\[R^{2}=\frac{\sum_{i=1}^{n}\bigl{[}\hat{F}(x_{i})-\overline{F}\bigr{]}^{2}}{ \sum_{i=1}^{n}\big{[}\hat{F}(x_{i})-\overline{F}\bigr{]}^{2}+\sum_{i=1}^{n} \bigl{[}F_{n}(x_{i})-\hat{F}(x_{i})\bigr{]}^{2}} \tag{15}\]  
where \(\overline{F}=\frac{1}{n}\sum_{i=1}^{n}\hat{F}(x_{i})\). In general, lower values of _KS_, _AD_, _AIC_, _BIC_, _RMSE_, and a higher value of \(R^{2}\) indicate better fit of the theoretical distribution to the wind speed data as compared to the others.