# Topp-Leone Modified Weibull Model: Theory and Applications to Medical and Engineering Data
## 4 Three Different Estimation Methods
### Maximum Likelihood Method  
In this subsection, parameters estimates are derived using ML method. The log likelihood function for \(TLMW\) distribution with the vector of parameters \(\varphi=(\alpha,\beta,\theta,\lambda)\) is  
\[\log\ L = n\log(2\lambda)+\sum_{i=1}^{n}\log\Bigl{(}\alpha+\theta\beta x_{i}^{ \theta-1}\Bigr{)}-2\sum_{i=1}^{n}(\alpha x_{i}+\beta x_{i}^{\theta}) \tag{21}\] \[+(\lambda-1)\sum_{i=1}^{n}\log\Bigl{(}1-e^{-2(\alpha x_{i}+\beta x _{i}^{\theta})}\Bigr{)}.\]  
The corresponding scoring function is supplied via  
\[U_{n}(\varphi)=\biggl{[}\frac{\partial L}{\partial\alpha},\frac{\partial L}{ \partial\beta},\frac{\partial L}{\partial\theta},\frac{\partial L}{\partial \lambda}\biggr{]}^{T}.\]  
The components of the score vector are described as  
\[\frac{\partial\log\ L}{\partial\alpha}=\sum_{i=1}^{n}\frac{1}{\alpha+\theta \beta x_{i}^{\theta-1}}-2\sum_{i=1}^{n}x_{i}+(\lambda-1)\sum_{i=1}^{n}\frac{2 x_{i}e^{-2(\alpha x_{i}+\beta x_{i}^{\theta})}}{1-e^{-2(\alpha x_{i}+\beta x_{i}^{ \theta})}}, \tag{22}\]  
\[\frac{\partial\log\ L}{\partial\beta}=\sum_{i=1}^{n}\frac{\theta x_{i}^{ \theta-1}}{\alpha+\theta\beta x_{i}^{\theta-1}}-2\sum_{i=1}^{n}x_{i}^{\theta} +(\lambda-1)\sum_{i=1}^{n}\frac{2x_{i}^{\theta}e^{-2(\alpha x_{i}+\beta x_{i} ^{\theta})}}{1-e^{-2(\alpha x_{i}+\beta x_{i}^{\theta})}}, \tag{23}\]  
\[\frac{\partial\log\ L}{\partial\theta}=\sum_{i=1}^{n}\frac{\beta x_{i}^{ \theta-1}(1+\theta\ln(x_{i}))}{\alpha+\theta\beta x_{i}^{\theta-1}}-2\beta \sum_{i=1}^{n}x_{i}^{\theta}\ln(x_{i})+(\lambda-1)\sum_{i=1}^{n}\frac{2 \beta x_{i}^{\theta}\ln(x_{i})e^{-2(\alpha x_{i}+\beta x_{i}^{\theta})}}{1-e^ {-2(\alpha x_{i}+\beta x_{i}^{\theta})}}, \tag{24}\]  
and  
\[\frac{\partial\log\ L}{\partial\lambda}=\frac{n}{\lambda}+\sum_{i=1}^{n}\log \Bigl{(}1-e^{-2(\alpha x_{i}+\beta x_{i}^{\theta})}\Bigr{)}. \tag{25}\]  
The \(ML\) estimates (MLEs) of \(\varphi\), symbolize \(\widehat{\varphi}\) can indeed be produced by optimizing the log-likelihood function with regard to the involved parameters. In this approach, R-software packages such as nlm() and maxLik() might be useful (see [37; 38]). Alternatively, the parameters can be found by concurrently solving four non-linear equations using an iterative process such as the Quasi Newton-Raphson method. Because the MLEs of the unknown parameters \(\varphi\) cannot be constructed in closed form, determining the exact distributions of the MLEs is difficult. As a result, specific confidence ranges can't founded for the parameters. The large sample approximation must be used. It is known that the asymptotic distribution of the MLE \(\hat{\varphi}\) is \((\hat{\varphi}-\varphi)\to N(0,I^{-1}(\varphi))\), see [39], where \(I^{-1}(\varphi)\), the inverse of the observed information matrix of the unknown parameters \(\varphi=(\alpha,\beta,\theta,\lambda)\) is,  
\[I^{-1}(\varphi)=\biggl{[}\frac{\partial^{2}l}{\partial\varphi^{2}}\biggr{]}_{( \alpha,\beta,\theta,\lambda)=(\hat{a},\hat{\beta},\hat{\theta},\hat{\lambda})} ^{-1},\]  
and whose elements are given in Appendix A.