Annals of Data Science

, Volume 4, Issue 4, pp 457–482 | Cite as

\(\alpha \) Logarithmic Transformed Family of Distributions with Application

  • Sanku Dey
  • Mazen Nassar
  • Devendra KumarEmail author


In this paper, a new three-parameter distribution, called \(\alpha \) logarithmic transformed generalized exponential distribution (\(\alpha LTGE\)) is proposed. Various properties of the proposed distribution, including explicit expressions for the moments, quantiles, moment generating function, mean deviation about the mean and median, mean residual life, Bonferroni curve, Lorenz curve, Gini index, Rényi entropy, stochastic ordering and order statistics are derived. It appears to be a distribution capable of allowing monotonically increasing, decreasing, bathtub and upside-down bathtub shaped hazard rates depending on its parameters. The maximum likelihood estimators of the unknown parameters cannot be obtained in explicit forms, and they have to be obtained by solving non-linear equations only. The asymptotic confidence intervals for the parameters are also obtained based on asymptotic variance covariance matrix. Finally, two empirical applications of the new model to real data are presented for illustrative purposes.


Generalized exponential distribution Conditional moments Stochastic ordering Maximum likelihood estimators 


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Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of StatisticsSt. Anthony’s CollegeShillongIndia
  2. 2.Department of Statistics, Faculty of CommerceZagazig UniversityZagazigEgypt
  3. 3.Department of StatisticsCentral University of HaryanaMahendragarhIndia

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