MODELING A THREE PARAMETER GUMBEL DISTRIBUTION USING MARSHALL-OLKINS TECHNIQUE

Abstract

This research introduced a new three-parameter Gumbel distribution by adding a parameter to the traditional Gumbel distribution using the Marshall-Olkin method. We derived the probability density function, cumulative distribution function, and other statistical properties of the new distribution. The parame ters of the distribution are estimated using the Maximum Likelihood Estimation (MLE) method. The new distribution improved flexibility and provided more effi cient estimators for a broader range of data types, including normal, skewed, and extreme data. The properties of the estimators are thoroughly investigated, in cluding their asymptotic bias, consistency, and mean square error (MSE). Through simulation studies and real data applications, the research demonstrates the supe riority of the new distribution over existing models, evidenced by smaller Akaike Information Criterion (AIC) values and more efficient parameter estimates. The research recommends the new distribution for future analyses, particularly for large sample sizes, and suggests further research to refine the location parameter, study some characteristics like quartile deviation, order statistics, and character istic function, and apply different parameter estimation methods to improve the efficiency of a three-parameter Gumbel distribution.