  Maximum likelihood estimation is a common method of estimating the parameters
of the probability distribution from a given sample. This paper aims to
introduce the maximum likelihood estimation in the framework of sublinear
expectation. We find the maximum likelihood estimator for the parameters of the
maximal distribution via the solution of the associated minimax problem, which
coincides with the optimal unbiased estimation given by Jin and Peng
\cite{JP21}. A general estimation method for samples with dependent structure
is also provided. This result provides a theoretical foundation for the
estimator of upper and lower variances, which is widely used in the G-VaR
prediction model in finance.
