The Alpha power Gompertz distribution: characterization, properties, and applications (Q2023851)

The authors investigate properties of the random variable with parameters \(\alpha\),\(\beta\),\(\lambda>0\) and distribution function given by \[ F_{\alpha,\beta,\lambda}(x)=\frac{1}{\alpha-1}\left(\alpha^{1-\exp\left\{-\frac{\lambda}{\beta}\left[\exp\left\{\beta x\right\}-1\right]\right\}}-1\right)\,, \] for \(x>0\) in the case where \(\alpha\not=1\). In the case where \(\alpha=1\), the distribution function is given by \[ F_{1,\beta,\lambda}(x)=1-\exp\left\{-\frac{\lambda}{\beta}\left[\exp\left\{\beta x\right\}-1\right]\right\}\,, \] for \(x>0\). Properties of this distribution investigated in the present paper include quantiles, moments, entropy, hazard rate, order statistics, and maximum likelihood estimation of parameters. The authors also show how this distribution can be represented as a mixture of Gompertz distributions, and conclude with numerical examples to illustrate the application of this distribution to data.