Precisão e acurácia dos estimadores de máxima verossimilhança dos parâmetros da distribuição Gumbel não estacionária

Abstract

The Gumbel distribution is often used in the extreme events modelling. For this purpose it is necessary that its parameters are estimated. The estimator most used for this is the maximum likelihood estimator. The maximum likelihood estimators have good asymptotic properties. In case of linear trend in the data series, the maximum likelihood estimators may produce biased estimates for the parameters of the Gumbel distribution. To overcome this situation, Gumbel model with trend can be used. This model is, basically, the (stationary) Gumbel model to no trend data, with the insertion of the trend in its position parameter. As in the stationary Gumbel model, the parameters of the trend Gumbel model can be estimate by means maximum likelihood estimators. Depending on sample size and on how big is the trend presented by the data, the stationary Gumbel model can be more appropriate than the trend Gumbel model, or vice versa. In this respect, it is important to know how model must be used. The objective of this study is, therefore, to evaluate the accuracy and the precision of the maximum likelihood estimators of the Gumbel model with linear trend’s parameters. To do this, have been simulated 1,000 data samples on 240 different scenarios and have been calculated the mean squared errors and the relative mean biases of each simulated scenario. The results lead to the following conclusions: if the trend was bigger than 0.1% of the Gumbel model’s position parameter value, the trend Gumbel model must be fitted. In the case of the sample size be equal or bigger than 50 and there is suspect of trend, the trend Gumbel model must be fitted.