Table 8 Some pmfs of the DE-X\(^{\theta }\) family based on different X distributions.
Distribution of x | pmf | Abbreviation |
|---|---|---|
Exponential | \(e^{-\alpha x^{\theta } \left[ 1-e^{- \beta x}\right] } - e^{-\alpha (x+1)^{\theta }\left[ 1-e^{- \beta (x+1)}\right] }\) | DEE |
Weibull | \(e^{-\alpha x^{\theta } \left[ 1-e^{- \left( \frac{x}{\beta }\right) ^{\eta }}\right] } - e^{-\alpha (x+1)^{\theta }\left[ 1-e^{- \left( \frac{x+1}{\beta }\right) ^{\eta }}\right] }\) | DEW |
Fréchet | \(e^{-\alpha x^{\theta } \left[ e^{- \left( \frac{x}{\beta }\right) ^{-\eta }}\right] } - e^{-\alpha (x+1)^{\theta }\left[ e^{- \left( \frac{x+1}{\beta }\right) ^{-\eta }}\right] }\) | DEFr |
Gamma | \(e^{-\alpha x^{\theta } \left[ \frac{\Gamma {(\eta ,\beta x)}}{\Gamma {(\eta )}}\right] } - e^{-\alpha (x+1)^{\theta }\left[ \frac{\Gamma {(\eta ,\beta (x+1))}}{\Gamma {(\eta )}}\right] }\) | DEG |
Lomax | \(e^{-\alpha x^{\theta } \left[ 1-\left( 1+ \beta x\right) ^{-\eta }\right] } - e^{-\alpha (x+1)^{\theta }\left[ 1-(1+ \beta (x+1))^{-\eta }\right] }\) | DEL |
Burr | \(e^{-\alpha x^{\theta } \left[ 1-\left( 1+ x^{\beta }\right) ^{-\eta }\right] } - e^{-\alpha (x+1)^{\theta }\left[ 1-(1+ (x+1)^{\beta })^{-\eta }\right] }\) | DEB |
