Maximum Lq-Likelihood Estimation: A Study of Entropy Behavior for the Pareto-Exponential Distribution with Application

Shannon’s entropy is inherent in studies of probability distributions. The maximum Lq-likelihood estimation (MLqE) is based on q-entropy and it has been widely explored, especially in cases investigating the efficiency and comparability of Maximum Likelihood Estimation (MLE) method, as well as an alternative method for estimating parameters for heavy-tailed distributions. By means of Monte Carlo Simulation, in different scenarios, we studied Entropy and MLqE for Pareto-exponential distributions. We conclude that from the parametric relationship existing in the distribution, the MLqE shows better results when the distortion parameter is about 1 and that the Entropy values are smaller when shape parameter is smaller than scale parameter. From the estimation process, we observed that the selection of the scale parameter may improve the estimation of the shape parameter and produce better results regarding the gain in lower variability in MLqE and Entropy values. In terms of accuracy, we evaluated the ratio between mean squared error for the two methods and concluded that for small and moderate sample sizes the MLqE is more accurate than MLE in estimating tail probabilities when the distortion parameter is associated with the sample size and converges slowly to 1. Then, with an application, we perform analysis for the estimation of q and study of Entropy.