# Topp-Leone Modified Weibull Model: Theory and Applications to Medical and Engineering Data
## 2 Topp-Leone Modified Weibull Distribution  
In this section, we introduce the Topp-Leone modified Weibull \((TLMW)\) distribution by taking \(\overline{G}(x)\) in Equation (3) to the cdf of \(TL\) in Equation (5). Hence, the cdf and pdf of \(TLMW\) can indeed be expressed as  
\[F_{TLMW}(x,\alpha,\beta,\theta,\lambda)=\left[1-e^{-2(ax+\beta x^{\theta})} \right]^{\lambda} \tag{9}\]  
and  
\[f_{TLMW}(x;\alpha,\beta,\theta,\lambda)=2\lambda\left(\alpha+\theta\beta x^{ \theta-1}\right)e^{-2(\alpha x+\beta x^{\theta})}\left[1-e^{-2(ax+\beta x^{ \theta})}\right]^{\lambda-1}, \tag{10}\]  
By writing \(X\sim TLMW(\varphi)\) where \(\varphi=(\alpha,\beta,\theta,\lambda)\) if the pdf of \(X\) can be written as (10). The hrf is defined by  
\[h_{TLMW}(x,\varphi)=\frac{2\lambda\left(\alpha+\theta\beta x^{\theta-1}\right) e^{-2(ax+\beta x^{\theta})}\left[1-e^{-2(ax+\beta x^{\theta})}\right]^{ \lambda-1}}{1-\left[1-e^{-2(ax+\beta x^{\theta})}\right]^{\lambda}}. \tag{11}\]  
Figures 1 and 2 shows the pdf and hrf for different values of \(\lambda\) and \(\theta\) and at \(\alpha=\beta=0.5\). The plots shows the pdf can be decreasing, uni-modal and right skewness but the hrf can be decreasing, increasing, J-shaped, U-shaped and bathtub.  
Figure 1: Plots of the pdf for TLMW model at \(\alpha=\beta=0.5\).  
_A Useful Expansion for the Density Function_  
In this part, the pdf representation of the \(TLMW\) distribution are discussed. The mathematical relation and the series representation are given below. If \(b\) is a positive real non-negative integer and \(|z|<1\) then  
\[(1-z)^{b-1}=\sum_{i=0}^{\infty}(-1)^{i}{b-1\choose i}z^{i}. \tag{12}\]  
Inserting Equation (12) into Equation (10) then,  
\[f_{TLMW}(x;\varphi)=2\lambda\sum_{i=0}^{\infty}(-1)^{i}{\lambda-1\choose i} \Big{(}\alpha+\theta\beta x^{\theta-1}\Big{)}e^{-2(i+1)(\alpha x+\beta x^{ \theta})}. \tag{13}\]  
The series expansion of  
\[e^{-2(i+1)\beta x^{\theta}}=\sum_{j=0}^{\infty}\frac{\big{[}-2(i+1)\beta x^{ \theta}\big{]}^{j}}{j!}, \tag{14}\]  
by substituting Equation (14) into Equation (13) then,  
\[f_{TLMW}(x;\varphi) = 2\lambda\sum_{i,j=0}^{\infty}\frac{(-1)^{i+j}}{j!}[2(i+1)\beta]^{ j}{\lambda-1\choose i}\Big{(}\alpha x^{\theta j}+\theta\beta x^{\theta(j+1)-1} \Big{)}e^{-2(i+1)(\alpha x)} \tag{15}\] \[= 2\lambda\sum_{i,j=0}^{\infty}\pi(i,j)\Big{(}\alpha x^{\theta j}+ \theta\beta x^{\theta(j+1)-1}\Big{)}e^{-2(i+1)(\alpha x)},\]  
where  
\[\pi(i,j)=\frac{(-1)^{i+j}}{j!}[2(i+1)\beta]^{j}{\lambda-1\choose i}.\]  
The \(TLMW\) distribution is a very flexible model which contains the well-known models listed below as special models:  
Figure 2: Plots of the hrf for TLMW model at \(\alpha=\beta=0.5\).  
* If \(\alpha=0\), the \(TLMW\) distribution reduces to the \(TL\) Weibull (TLW) distribution.
* If \(\theta=2\), then Equation (9) the \(TLMW\) distribution reduces to the \(TL\) Linear failure rate (TLLFR) model.
* If \(\beta=0\), the TLMW distribution reduces to the \(TL\) exponential (TLE) distribution.
* If \(\theta=2\) and \(\alpha=0\), then Equation (9) the TLMW distribution reduces to the \(TL\) Rayleigh (TLR) distribution.