Cramer-Rao bounds and estimation of the parameters of the Gumbel distribution

Maximum Likelihood (ML) algorithms and Cramer-Rao (CR) bounds for the location and scale parameters of the Gumbel distribution are discussed. First we consider the case in which the scale parameter is known, obtaining the estimator of the location parameter by solving the likelihood equation and then evaluating its performance. We next consider the case where both the location parameter and the scale parameter are unknown and need to be estimated simultaneously from the reference samples. For this case, performance is analyzed by means of Monte Carlo simulation and compared with the asymptotic CR bound.>