On computing the maximum likelihood estimates for the q -Pareto distribution and application to earthquakes

Abstract. We define a generalized exponential distribution (G-ED) that overlaps with a generalized Pareto distribution (GPD). Then, we introduce the q-Pareto distribution (q-PD) with two-parameter family distribution (strict positive shape and scale parameters) via transforming the G-ED. We demonstrate that the q-PD can be used to model the exceedances over a threshold, and we focus to estimate the q-PD parameters. Often, the estimation is taken the maximum likelihood estimation (MLE) because have a consistent estimator with an asymptotic normal distribution and is asymptotically effective in many cases. We present an algorithm to compute the MLE for q-PD parameters with confidence intervals for each estimator. Numerical examples are given to illustrate the results obtained. Efficient numerically methods are developed for the MLE of the q-PD parameters to maximize the log-likelihood of the q-PD.