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.


1 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?

  • $\begingroup$ Thank you for the answer. Actually we did chi-square goodness-of-fit test. The idea is that while the model may fail in GoF test, it may still correctly predict the mean. The approach I described is used in the paper Baye, Michael R., and John Morgan. "Price Dispersion in the Lab and on the Internet: Theory and Evidence." RAND Journal of Economics (2004): 449-466. academia.edu/download/40458541/… $\endgroup$
    – bazenkov
    Nov 18, 2020 at 10:28

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