A generalization of the exponential-Poisson distribution.

*(English)*Zbl 1176.62005Summary: The two-parameter distribution known as exponential-Poisson (EP) distribution, which has a decreasing failure rate, was introduced by C. Kus [Comput. Stat. Data Anal. 51, No. 9, 4497–4509 (2007; Zbl 1162.62309)]. We generalize the EP distribution and show that the failure rate of the new distribution can be decreasing or increasing. The failure rate can also be upside-down bathtub shaped. A comprehensive mathematical treatment of the new distribution is provided. We provide closed-form expressions for the density, cumulative distribution, survival and failure rate functions; we also obtain the density of the \(i\) th order statistic.

We derive the \(r\) th raw moment of the new distribution and also the moments of order statistics. Moreover, we discuss estimation by maximum likelihood and obtain an expression for Fisher’s information matrix. Furthermore, expressions for the Rényi and Shannon entropies are given and an application using a real data set is presented. Finally, simulation results on maximum likelihood estimation are presented.

We derive the \(r\) th raw moment of the new distribution and also the moments of order statistics. Moreover, we discuss estimation by maximum likelihood and obtain an expression for Fisher’s information matrix. Furthermore, expressions for the Rényi and Shannon entropies are given and an application using a real data set is presented. Finally, simulation results on maximum likelihood estimation are presented.

##### MSC:

62E10 | Characterization and structure theory of statistical distributions |

62N05 | Reliability and life testing |

62F10 | Point estimation |

62G30 | Order statistics; empirical distribution functions |

62N02 | Estimation in survival analysis and censored data |

62B10 | Statistical aspects of information-theoretic topics |

65C60 | Computational problems in statistics (MSC2010) |

##### Software:

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\textit{W. Barreto-Souza} and \textit{F. Cribari-Neto}, Stat. Probab. Lett. 79, No. 24, 2493--2500 (2009; Zbl 1176.62005)

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