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.
