  We use the Maximum $q$-log-likelihood estimation for Least informative
distributions (LID) in order to estimate the parameters in probability density
functions (PDFs) efficiently and robustly when data include outlier(s). LIDs
are derived by using convex combinations of two PDFs,
$f_\epsilon=(1-\epsilon)f_0+\epsilon f_1$. A convex combination of two PDFs is
considered as a contamination $f_1$ as outlier(s) to underlying $f_0$
distributions and $f_\epsilon$ is a contaminated distribution. The optimal
criterion is obtained by minimizing the change of Maximum q-log-likelihood
function when the data have slightly more contamination. In this paper, we make
a comparison among ordinary Maximum likelihood, Maximum q-likelihood
estimations, LIDs based on $\log_q$ and Huber M-estimation. Akaike and Bayesian
information criterions (AIC and BIC) based on $\log_q$ and LID are proposed to
assess the fitting performance of functions. Real data sets are applied to test
the fitting performance of estimating functions that include shape, scale and
location parameters.
