Parametric density estimation by minimizing nonextensive entropy

In this paper, we consider parametric density estimation based on minimizing an empirical version of the Havrda-Charv´at-Tsallis ([15], [25]) nonextensive entropy. The resulting estimator, called the Maximum Lq-Likelihood estimator (MLqE), is indexed by a single distortion parameter q, which controls the trade-off between bias and variance. The method has two notable special cases. If q tends to 1, the MLqE is the Maximum Likelihood Estimator (MLE). When q = 1/2, the MLqE is a minimum Hellinger distance type of estimator with the perk of avoiding nonparametric techniques and the difficulties of bandwith selection. The MLqE is studied using asymptotic analysis, simulations and real-world data, showing that it conciliates two apparently contrasting needs: efficiency and robustness, conditional to a proper choice of q. When the sample size is small or moderate, the MLqE trades bias for variance, resulting in a reduced mean squared error compared to the MLE. At the same time, the MLqE exhibits strong robustness at expense of a slightly reduced efficiency in presence of observations discordant with the assumed model. To compute the MLq estimates, a fast and easy-to-implement algorithm based on a reweighting strategy is also described.