Three-Parameters Gumbel Distribution: Properties and Application

Abstract

In this research, we introduced a new three-parameter Gumbel distribution by adding a parameter to the traditional Gumbel distribution using the Marshall-Olkin method. This new distribution enhances flexibility and provides more efficient estimators for various data types, including normal, skewed, andextreme data. We derived the probability density function, cumulative distribution function, and other statistical properties of the new distribution. The parameters are estimated using the Maximum Likelihood Estimation (MLE) method, and thoroughly investigated the properties of the estimators, focusing on their asymptotic bias, consistency, and mean square error (MSE). Through simulation studies and real data applications, we demonstrate the superiority of the new distribution over existing models, evidenced by smaller Akaike Information Criterion (AIC) values and more efficient parameter estimates. We recommend the new distribution for future analyses, particularly for large sample sizes, and suggest further research to refine the location parameter for improved efficiency.