Improving MLE via a non-extensive information measure

Although the maximum likelihood estimator enjoys asymptotic
optimality properties, its finite-sample performance for a small or moderate sample
size can be much improved when the Log-Likelihood is replaced by a
L$q$-Likelihood, which is motivated from a non-extensive measure of
information (in contrast to the additive Kullback-Leiber information).
The properties of the resulting estimator, ML$q$E, are studied via
asymptotic analysis and computer simulations. The behavior of the
ML$q$E is characterized by the degree of distortion $q$ applied to the
assumed model to amplify or diminish the density value. When $q$ is
properly chosen for small and moderate sample sizes, the ML$q$E
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 ML$q$E is established. The
advantage of the new estimation method is more clearly seen for higher
dimensional estimations. The talk is based on joint work with Davide
Ferrari.