Maximum lq-likelihood estimator of the heavy-tailed distribution parameter

Studying the extreme value theory (EVT) involves multiple main objectives, among them the estimation of the tail index parameter. Some estimation methods are used to estimate the tail index parameter like maximum likelihood estimation (MLE). Additionally, the Hill estimator is one type of maximum likelihood estimator, which is a more robust with a large sample than a small sample. This research proposes the construction of an alternative estimator for the parameter of the heavy-tailed distribution using the maximum lq-likelihood estimation (MLqE) approach in order to adapt the ML and Hill estimator with the small sample. Furthermore, the maximum lq-likelihood estimator asymptotic normality is established. Moreover, several simulation studies in order to compare the MLq estimator with the ML estimators are provided. In the excesses over high suitable threshold values the number of the largest observation k will lead to an efficient estimate of the Hill estimator. For this, selection of k in the Hill estimator was investigated using the method of the quantile type 8 which is effective with the hydrology data. The performance of the Hill estimator and the lq-Hill estimator is subsequently compared by employing real relies with the distribution of hydrology data.