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.