# Topp-Leone Modified Weibull Model: Theory and Applications to Medical and Engineering Data
## 1 Introduction and Motivation  
Over the last four decades or more, a major amount of statistical research has developed a number of innovative statistical distributions. This criterion was inspired by theoretical concepts, real-world applications, or both. The most recent advancement is to build a larger range of statistical distributions by adding one (or more) shape parameters to a baseline distribution, which tends to improve the diversity of the statistical models developed. Particularly appropriate to data sets for researching reliability experiments, medical, clinical trials, finance, infant mortality rate, and engineering applications, among other things. The Weibull distribution is a well-liked distribution that may be used to predict how long a product will last, such as a ball bearing, a car part, or electrical insulation see [1; 2; 3; 4; 5; 6; 7; 8; 9; 10]. It is also frequently employed in biological and medical applications. This distribution, however, lacks a bathtub or upside-down shaped hrf, making it ineffective for simulating the lifetime of some systems. To overcome this shortcoming, refs. [11; 12; 13] studied the modified Weibull (MW) distribution that can be used to describe various reliability models. It has three parameters, \(\alpha\) and \(\beta\) are two scale parameters and \(\theta\) is the shape parameter. The distribution function (cdf), the probability density function (pdf), survival function (sf) and hazard rate function (hrf) of the MW distribution due to [13] are provided via  
\[G(x,\alpha,\beta,\theta)=1-e^{-(ax+\beta x^{\theta})},x>0,\alpha,\beta,\theta>0, \tag{1}\]  
\[g(x,\alpha,\beta,\theta)=\left(\alpha+\theta\beta x^{\theta-1}\right)e^{-(ax+ \beta x^{\theta})},x>0,\alpha,\beta,\theta>0, \tag{2}\]  
\[\overline{G}(x,\alpha,\beta,\theta)=e^{-(ax+\beta x^{\theta})},x>0,\alpha, \beta,\theta>0, \tag{3}\]  
and  
\[h(x,\alpha,\beta,\theta)=\left(\alpha+\theta\beta x^{\theta-1}\right)x>0, \alpha,\beta,\theta>0. \tag{4}\]  
Recently, many authors have proposed many generalizations of the MW distribution to improve its flexibility. For example, generalized MW by [14], beta MW by [15], exponentiated MW by [16], new MW by [17], transmuted MW by [18], McDonald MW by [19], Kumaraswamy MW [20], gamma MW by [21], Burr XII MW by [22], log-logistic MW by [23], half logistic MW by [24], beta exponentiated MW by [25] and sine MW by [26]. Topp-Leone (\(TL\)) distribution was proposed by [27]. Recently, statisticians have been showing interest on Topp-Leone (\(TL\)) distribution as a generator. This distribution has one parameter and its domain is restricted to \((0,1)\). Furthermore, this distribution is not particularly adaptable. It is an extension of the triangular distribution and can be used as an alternative in circumstances where a bounded distribution, such as the uniform, triangular, trapezoid, or beta, is acceptable. Ref. [28] introduced \(TL\)-generated (\(TL\)-\(G\)) family of distributions and the cdf of the \(TL\)-\(G\) can be defined as  
\[F(x;\lambda)=\left(1-\overline{G}^{2}(x)\right)^{\lambda}=G^{\lambda}(x)[2-G( x)]^{\lambda},x\in R,\lambda>0, \tag{5}\]  
the associated pdf is available below  
\[f(x;\lambda)=2\lambda g(x)\overline{G}(x)(1-\overline{G}^{2}(x))^{\lambda-1}. \tag{6}\]  
The sf and hrf of the \(TL\)-\(G\) are provided via  
\[\overline{F}(x;\lambda)=1-(1-\overline{G}^{2}(x))^{\lambda}, \tag{7}\]  
and  
\[h((x;\lambda)=\frac{2\lambda g(x)\overline{G}(x)(1-\overline{G}^{2}(x))^{ \lambda-1}}{1-(1-\overline{G}^{2}(x))^{\lambda}}, \tag{8}\]  
respectively. Many researchers have recently suggested numerous generalizations of the \(TL\)-\(G\) to increase its versatility.