A review of methods of parameter estimation for the extreme value type-1 distribution

Abstract

Four important methods of estimating the parameters of the extreme value type-1 (Gumbel) distribution, namely: (1) moments (MMM); (2) maximum likelihood (MML); (3) maximum entropy (MME); and (4) probability weighted moments (PWM), are considered, suitable solution procedures are recommended and related problems discussed. Approximate formulae for computing the variances-covariances of the estimators by each method are given in terms of the population value of the scale parameter, and these are shown to indicate very well the performance of the related method. A simulation study to evaluate the performance of these methods in terms of commonly used criteria, i.e. the bias, root mean square error and goodness-of-fit statistic is described. It was found that the MMM is not as good as the remaining three methods, of which the PWM is best in terms of the bias, and the MML is best in terms of the root mean square error and efficiency. The MME, by all the criteria, ranks second best and follows the MML more closely than the PWM.