Test whether the theoretical model estimated from the sample predicts the same mean

Question

I have a sample of experimentally observed data and a parametric distribution model which is expected to explain the data. I estimated the model parameters by the maximum likelihood. Now I need to test whether the sample mean does not significantly differ from the 'theoretical' mean predicted by the model.

The naive approach is to compute the 'theoretical' mean and perform a one-sample t-test. But this approach seems flawed because the model was estimated from the same sample, hence the 'theoretical' mean actually depends on the data.

What techniques may be appropriate for the task, and what theory lies behind this formulation? This question seems similar, but it doesn't have a definite answer.

Answer 1

In general, estimating parameters of a distribution and then using these parameters to estimate the mean does not yield an unbiased estimator of the mean. A systematic difference between the statistical mean (knwon to be the unbiased estimator with the smallest MSE) and the estimator obtained with your method should thus be expected.

This is similar to the "method of moments" apporach, in which moments are (undiasedly) estimated and then other parameters are computed therefrom. There are examples, where this approach can even yield impossible values (e.g. for the uniform distribution).

If you are interested in the appropriateness of your model, why do you not try a goodness-of-fit test?