For a simple example, I am fitting data with a likelihood function generated by a normal distribution. The first model is the normal distribution with two-parameters. The second competing model is the extreme value distribution with fixed variance=0.5, so there is only one parameter: the mean.
We can then compare those two models by AIC or likelihood-based discrimination tests for non-nested models such as Clarke Test and Vuong Test. Because more parameters are penalized for overfitting, it is possible that the second model is a better fit.
However, in theory, the second model is not a good model because it is an "overspecified" model in some sense. Here my "overspecified" means that the variance parameter should theoretically be a parameter rather than a constant. We should not specify it before fitting with the data without a firm reason.
The second model also unfairly avoids the penalization for having two parameters.
What is the rigorous academic term for this "over-specification"?
I've thought about over-fitting and under-fitting. They are relevant but exactly the word I am looking for.