Exponentiated exponential family: An alternative to gamma and Weibull distributions.

*(English)*Zbl 0997.62076Summary: We study the properties of a new family of distributions, namely exponentiated exponential distributions, by R.C. Gupta, R.D. Gupta and P.L. Gupta [Commun. Stat., Theory Methods 27, No. 4, 887-904 (1998)]. The exponentiated exponential family has two parameters (scale and shape) similar to a Weibull or a gamma family. It is observed, that many properties of this new family are quite similar to those of a Weibull or gamma family, therefore this distribution can be used as a possible alternative to a Weibull or gamma distribution.

We present two real life data sets, where it is observed that in one data set exponentiated exponential distribution has a better fit compared to the Weibull or gamma distribution and in the other data set Weibull has a better fit than the exponentiated exponential or gamma distribution. Some numerical experiments are performed to see how the maximum likelihood estimators and their asymptotic results work for finite sample sizes.

We present two real life data sets, where it is observed that in one data set exponentiated exponential distribution has a better fit compared to the Weibull or gamma distribution and in the other data set Weibull has a better fit than the exponentiated exponential or gamma distribution. Some numerical experiments are performed to see how the maximum likelihood estimators and their asymptotic results work for finite sample sizes.

##### MSC:

62N02 | Estimation in survival analysis and censored data |

60E15 | Inequalities; stochastic orderings |

62E10 | Characterization and structure theory of statistical distributions |

##### Keywords:

Weibull distribution; likelihood ratio ordering; hazard rate ordering; stochastic ordering; Fisher information matrix; maximum likelihood estimator; gamma distribution
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\textit{R. D. Gupta} and \textit{D. Kundu}, Biom. J. 43, No. 1, 117--130 (2001; Zbl 0997.62076)

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