A discrete analog of Gumbel distribution: properties, parameter estimation and applications

Abstract

A discrete version of the Gumbel distribution (Type-I Extreme Value distribution) has been derived by using the general approach of discretization of a continuous distribution. Important distributional and reliability properties have been explored. It has been shown that depending on the choice of parameters the proposed distribution can be positively or negatively skewed; possess long-tail(s). Log-concavity of the distribution and consequent results have been established. Estimation of parameters by method of maximum likelihood, method of moments, and method of proportions has been discussed. A method of checking model adequacy and regression type estimation based on empirical survival function has also been examined. A simulation study has been carried out to compare and check the efficacy of the three methods of estimations. The distribution has been applied to model three real count data sets from diverse application area namely, survival times in number of days, maximum annual floods data from Brazil and goal differences in English premier league, and the results show the relevance of the proposed distribution.

Keywords:

• Gumbel distribution
• homogeneous skewness
• long tail
• log-concavity
• simulation study
• Skellam distribution

Mathematics Subject Classification (2010):

• 60E05
• 62E15
• 62F10
• 62Q05

Acknowledgments

The authors would like to acknowledge the associate editor and the reviewers for their encouraging comments and suggestions. The research of Dhrubajyoti Chakravarty was done while he was a research scholar in Dibrugarh University.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Subrata Chakraborty http://orcid.org/0000-0002-6405-1486

Josmar Mazucheli http://orcid.org/0000-0001-6740-0445

Wesley Bertoli http://orcid.org/0000-0002-4671-1268

Additional information

Funding

The research of Wesley Bertoli is partially supported by the Federal University of Technology – Paraná and by a doctoral grant from Araucária Foundation [grant no. 018/2015]. Josmar Mazucheli gratefully acknowledges the partial financial support from Fundação Araucária [grant no. 064/2019].