In this article, we investigate goodness-of-fit tests based on the empirical distribution function for the composite Gumbel (extreme value) hypothesis. The parameters of the Gumbel distribution are estimated using numerical methods through the maximum likelihood approach. We employ a Monte Carlo method to obtain critical values and determine the actual sizes of the tests for different sample sizes. Furthermore, we conduct a comprehensive simulation study to compare the power values of the tests against various alternatives and different sample sizes. Finally, for illustrative purposes, we analyse a real data set.
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