### Abstract

This paper provides a comparison of the three-parameter exponentiated Weibull (EW) and generalized gamma (GG) distributions. The connection between these two different families is that the hazard functions of both have the four standard shapes (increasing, decreasing, bathtub, and arc shaped), and in fact, the shape of the hazard is the same for identical values of the three parameters. For a given EW distribution, we define a matching GG using simulation and also by matching the 5 ^{th}, 50 ^{th}, and 95 ^{th} percentiles. We compare EW and matching GG distributions graphically and using the Kullback-Leibler distance. We find that the survival functions for the EW and matching GG are graphically indistinguishable, and only the hazard functions can sometimes be seen to be slightly different. The Kullback-Leibler distances are very small and decrease with increasing sample size. We conclude that the similarity between the two distributions is striking, and therefore, the EW represents a convenient alternative to the GG with the identical richness of hazard behavior. More importantly, these results suggest that having the four basic hazard shapes may to some extent be an important structural characteristic of any family of distributions.

Original language | English (US) |
---|---|

Pages (from-to) | 3772-3780 |

Number of pages | 9 |

Journal | Statistics in Medicine |

Volume | 33 |

Issue number | 21 |

DOIs | |

State | Published - Sep 20 2014 |

### Fingerprint

### Keywords

- Exponentiated weibull distribution
- Generalized gamma distribution
- Parametric survival

### ASJC Scopus subject areas

- Epidemiology
- Statistics and Probability
- Medicine(all)

### Cite this

*Statistics in Medicine*,

*33*(21), 3772-3780. https://doi.org/10.1002/sim.6159

**A comparison of the generalized gamma and exponentiated Weibull distributions.** / Cox, Christopher; Matheson, Matthew.

Research output: Contribution to journal › Article

*Statistics in Medicine*, vol. 33, no. 21, pp. 3772-3780. https://doi.org/10.1002/sim.6159

}

TY - JOUR

T1 - A comparison of the generalized gamma and exponentiated Weibull distributions

AU - Cox, Christopher

AU - Matheson, Matthew

PY - 2014/9/20

Y1 - 2014/9/20

N2 - This paper provides a comparison of the three-parameter exponentiated Weibull (EW) and generalized gamma (GG) distributions. The connection between these two different families is that the hazard functions of both have the four standard shapes (increasing, decreasing, bathtub, and arc shaped), and in fact, the shape of the hazard is the same for identical values of the three parameters. For a given EW distribution, we define a matching GG using simulation and also by matching the 5 th, 50 th, and 95 th percentiles. We compare EW and matching GG distributions graphically and using the Kullback-Leibler distance. We find that the survival functions for the EW and matching GG are graphically indistinguishable, and only the hazard functions can sometimes be seen to be slightly different. The Kullback-Leibler distances are very small and decrease with increasing sample size. We conclude that the similarity between the two distributions is striking, and therefore, the EW represents a convenient alternative to the GG with the identical richness of hazard behavior. More importantly, these results suggest that having the four basic hazard shapes may to some extent be an important structural characteristic of any family of distributions.

AB - This paper provides a comparison of the three-parameter exponentiated Weibull (EW) and generalized gamma (GG) distributions. The connection between these two different families is that the hazard functions of both have the four standard shapes (increasing, decreasing, bathtub, and arc shaped), and in fact, the shape of the hazard is the same for identical values of the three parameters. For a given EW distribution, we define a matching GG using simulation and also by matching the 5 th, 50 th, and 95 th percentiles. We compare EW and matching GG distributions graphically and using the Kullback-Leibler distance. We find that the survival functions for the EW and matching GG are graphically indistinguishable, and only the hazard functions can sometimes be seen to be slightly different. The Kullback-Leibler distances are very small and decrease with increasing sample size. We conclude that the similarity between the two distributions is striking, and therefore, the EW represents a convenient alternative to the GG with the identical richness of hazard behavior. More importantly, these results suggest that having the four basic hazard shapes may to some extent be an important structural characteristic of any family of distributions.

KW - Exponentiated weibull distribution

KW - Generalized gamma distribution

KW - Parametric survival

UR - http://www.scopus.com/inward/record.url?scp=84905562615&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84905562615&partnerID=8YFLogxK

U2 - 10.1002/sim.6159

DO - 10.1002/sim.6159

M3 - Article

C2 - 24700647

AN - SCOPUS:84905562615

VL - 33

SP - 3772

EP - 3780

JO - Statistics in Medicine

JF - Statistics in Medicine

SN - 0277-6715

IS - 21

ER -