### Abstract

Recently, Ristić and Nadarajah [A new lifetime distribution. J Stat Comput Simul. 2014;84:135–150] introduced the Poisson generated family of distributions and investigated the properties of a special case named the exponentiated-exponential Poisson distribution. In this paper, we study general mathematical properties of the Poisson-X family in the context of the T-X family of distributions pioneered by Alzaatreh et al. [A new method for generating families of continuous distributions. Metron. 2013;71:63–79], which include quantile, shapes of the density and hazard rate functions, asymptotics and Shannon entropy. We obtain a useful linear representation of the family density and explicit expressions for the ordinary and incomplete moments, mean deviations and generating function. One special lifetime model called the Poisson power-Cauchy is defined and some of its properties are investigated. This model can have flexible hazard rate shapes such as increasing, decreasing, bathtub and upside-down bathtub. The method of maximum likelihood is used to estimate the model parameters. We illustrate the flexibility of the new distribution by means of three applications to real life data sets.

Original language | English |
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Pages (from-to) | 1-21 |

Number of pages | 21 |

Journal | Journal of Statistical Computation and Simulation |

DOIs | |

Publication status | Accepted/In press - Jan 26 2016 |

### Fingerprint

### Keywords

- Cauchy distribution
- generalized Poisson distribution
- Poisson distribution
- Poisson power Cauchy distribution
- Poisson-X family
- power Cauchy distribution
- T-X family

### ASJC Scopus subject areas

- Applied Mathematics
- Statistics and Probability
- Modelling and Simulation
- Statistics, Probability and Uncertainty

### Cite this

*Journal of Statistical Computation and Simulation*, 1-21. https://doi.org/10.1080/00949655.2016.1138224

**The Poisson-X family of distributions.** / Tahir, M. H.; Zubair, M.; Cordeiro, Gauss M.; Alzaatreh, Ayman; Mansoor, M.

Research output: Contribution to journal › Article

*Journal of Statistical Computation and Simulation*, pp. 1-21. https://doi.org/10.1080/00949655.2016.1138224

}

TY - JOUR

T1 - The Poisson-X family of distributions

AU - Tahir, M. H.

AU - Zubair, M.

AU - Cordeiro, Gauss M.

AU - Alzaatreh, Ayman

AU - Mansoor, M.

PY - 2016/1/26

Y1 - 2016/1/26

N2 - Recently, Ristić and Nadarajah [A new lifetime distribution. J Stat Comput Simul. 2014;84:135–150] introduced the Poisson generated family of distributions and investigated the properties of a special case named the exponentiated-exponential Poisson distribution. In this paper, we study general mathematical properties of the Poisson-X family in the context of the T-X family of distributions pioneered by Alzaatreh et al. [A new method for generating families of continuous distributions. Metron. 2013;71:63–79], which include quantile, shapes of the density and hazard rate functions, asymptotics and Shannon entropy. We obtain a useful linear representation of the family density and explicit expressions for the ordinary and incomplete moments, mean deviations and generating function. One special lifetime model called the Poisson power-Cauchy is defined and some of its properties are investigated. This model can have flexible hazard rate shapes such as increasing, decreasing, bathtub and upside-down bathtub. The method of maximum likelihood is used to estimate the model parameters. We illustrate the flexibility of the new distribution by means of three applications to real life data sets.

AB - Recently, Ristić and Nadarajah [A new lifetime distribution. J Stat Comput Simul. 2014;84:135–150] introduced the Poisson generated family of distributions and investigated the properties of a special case named the exponentiated-exponential Poisson distribution. In this paper, we study general mathematical properties of the Poisson-X family in the context of the T-X family of distributions pioneered by Alzaatreh et al. [A new method for generating families of continuous distributions. Metron. 2013;71:63–79], which include quantile, shapes of the density and hazard rate functions, asymptotics and Shannon entropy. We obtain a useful linear representation of the family density and explicit expressions for the ordinary and incomplete moments, mean deviations and generating function. One special lifetime model called the Poisson power-Cauchy is defined and some of its properties are investigated. This model can have flexible hazard rate shapes such as increasing, decreasing, bathtub and upside-down bathtub. The method of maximum likelihood is used to estimate the model parameters. We illustrate the flexibility of the new distribution by means of three applications to real life data sets.

KW - Cauchy distribution

KW - generalized Poisson distribution

KW - Poisson distribution

KW - Poisson power Cauchy distribution

KW - Poisson-X family

KW - power Cauchy distribution

KW - T-X family

UR - http://www.scopus.com/inward/record.url?scp=84958543338&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84958543338&partnerID=8YFLogxK

U2 - 10.1080/00949655.2016.1138224

DO - 10.1080/00949655.2016.1138224

M3 - Article

SP - 1

EP - 21

JO - Journal of Statistical Computation and Simulation

JF - Journal of Statistical Computation and Simulation

SN - 0094-9655

ER -