Some Contributions to Description and Validation of the Extreme Value Distribution

Abstract: This thesis focuses on the validation and description of the Gumbel distribution. Since this is a scale and location parameter distribution, the generalized least squares regression of the order statistics on the expected values can be used, without the necessity of iteration, to obtain the best linear unbiased estimates of the parameters. In order to implement this procedure, we need information about the expected values and variances-covariances of order statistics from the standard extreme value distribution. Numerical problems in determining these values and lack of exact values of means, variances for n > 100 and covariances for n > 30, are major challenges which we must deal with. In two papers, by applying the method of least squares, we present approximation algorithms to approximate the means, variances and covariances of the order statistics of the standard extreme value distribution. In both papers we compare the accuracy of our proposed models by using available tabulated values and values obtained from Monte Carlo methods. In the case where one or both of the parameters in the distribution are known or unknown, as in papers three to six, we present and compare goodness-of-fit tests based on different approaches. These papers tackle tests of the null hypothesis that a random sample comes from the extreme value distribution of type I (minima). The test procedure is to calculate an appropriate test statistic and reject null hypothesis if the value of the statistic used exceeds the percentage point at the type I error level.

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