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.