In this work, we
consider parametric estimation based on minimizing an empirical version of the Havrda-Charvát-Tsallis nonextensive
entropy. The resulting estimator, the Maximum Lq-Likelihood Estimator (MLqE), depends on
the degree of distortion q applied to
the assumed model. If q tends to 1 as
the sample size increases, 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 bandwidth selection.
The behavior of the MLqE
is studied in three aspects: sample size, dimensionality of parameter space and
presence of outliers. Worthiness of the new methodology is collected from
multiple viewpoints: asymptotic analysis, numerical studies and exploration of
real-world data. When q is properly
chosen for small and moderate sample sizes (or large number of parameters), the
MLqE successfully
trades bias for precision, resulting in a substantial reduction of the mean
squared error. When the sample size is large and q tends to 1, a necessary and sufficient condition to ensure a
proper asymptotic normality and efficiency of MLqE is established. In the
presence of observations discordant with the assumed model, the MLqE affords
considerable robustness at expense of a slightly reduced efficiency. To compute
the MLq
estimates, a fast and easy-to-implement algorithm based on a re-weighting strategy
is also described.