  The maximum ${\log}_q$ likelihood estimation method is a generalization of
the known maximum $\log$ likelihood method to overcome the problem for modeling
non-identical observations (inliers and outliers). The parameter $q$ is a
tuning constant to manage the modeling capability. Weibull is a flexible and
popular distribution for problems in engineering. In this study, this method is
used to estimate the parameters of Weibull distribution when non-identical
observations exist. Since the main idea is based on modeling capability of
objective function
$\rho(x;\boldsymbol{\theta})=\log_q\big[f(x;\boldsymbol{\theta})\big]$, we
observe that the finiteness of score functions cannot play a role in the robust
estimation for inliers. The properties of Weibull distribution are examined. In
the numerical experiment, the parameters of Weibull distribution are estimated
by $\log_q$ and its special form, $\log$, likelihood methods if the different
designs of contamination into underlying Weibull distribution are applied. The
optimization is performed via genetic algorithm. The modeling competence of
$\rho(x;\boldsymbol{\theta})$ and insensitiveness to non-identical observations
are observed by Monte Carlo simulation. The value of $q$ can be chosen by use
of the mean squared error in simulation and the $p$-value of Kolmogorov-Smirnov
test statistic used for evaluation of fitting competence. Thus, we can overcome
the problem about determining of the value of $q$ for real data sets.
