  In this paper, the maximum L$q$-likelihood estimator (ML$q$E), a new
parameter estimator based on nonextensive entropy [Kibernetika 3 (1967) 30--35]
is introduced. The properties of the 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. When $q$ is properly
chosen for small and moderate sample sizes, the ML$q$E can successfully trade
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.
