# On the asymptotic maximum likelihood estimation based on extremes

This work deals with the comparison of two estimation methods, the classical maximum likelihood estimation method and an asymptotic maximum likelihood estimation method (AEVML), when a censored sample of k upper order statistics is used. Three distribution functions were employed as underlying distributions: Lognormal, Generalized Pareto, and Truncated Weibull. These three distributions belong to the domain of attraction of the Gumbel limiting distribution exp(-exp(-x)) Using the joint density function of the set of upper order statistics, normalizing constants depending on the underlying distribution parameters, and normalizing constants proposed by I. Weissman (1978), under the parametric or nonparametric approach, the estimation of the underlying distribution parameters is achieved. The estimated distribution function parameters are defined in closed form, and estimates of parameters, large quantiles, and confidence intervals of these quantiles are determined Differences between the estimated parameters were found when the entire sample and the sample with the upper order statistics were generated in Monte Carlo simulation processes Although no global statements can be proposed, the bias produced during the simulations is adjusted through a multiple linear model Some examples in applying the methodology of the AEVML method are presented, and they indicate the bias can be significant when a censored sample of k upper order statistics is used in determining the estimates of the underlying distribution. The mean residual function is used to detect misspecification of the distribution function