**6. Applications**

In this section we present two real data applications to illustrate the better performance of the GTPN model over other well known models in the literature. For these comparisons we also consider the Weibull (WEI) and the Generalized Lindley (GL, Zakerzadeh [20]) models. The density function of the Weibull distribution is given by

$$f(\mathbf{x}; \sigma, \lambda) = \frac{\sigma}{\lambda} \mathbf{x}^{\sigma - 1} e^{-\mathbf{x}^{\sigma}/\lambda},$$

with *x* > 0, *σ* > 0 and *λ* > 0, whereas for the GL model is given by

$$f(\mathbf{x}; \boldsymbol{\alpha}, \boldsymbol{\theta}, \boldsymbol{\gamma}) = \frac{\theta^2 (\theta \mathbf{x})^{\alpha - 1} (\boldsymbol{\alpha} + \boldsymbol{\gamma} \mathbf{x}) e^{-\theta \mathbf{x}}}{(\boldsymbol{\gamma} + \boldsymbol{\theta}) \boldsymbol{\Gamma} (\boldsymbol{\alpha} + \mathbf{1})},$$

,

with *x* > 0, *θ* > 0 *α* > 0 and *γ* > 0.
