This is a test and example case for GenericLikelihoodModel in statsmodels.
see https://www.statsmodels.org/dev/generated/statsmodels.miscmodels.tmodel.TLinearModel.html for the TLinearModel and the source code for how to implement generic maximum likelihood models.
Some examples are in the unit tests https://github.com/statsmodels/statsmodels/blob/master/statsmodels/miscmodels/tests/test_tmodel.py
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