Using MLE to find the right distribution


I know that MLE can be used to find the best fitting parameter assuming a that the distribution is known, but is there a way (using Excel or SAS) to find what distribution best fits the data (together with the best parameter of the distribution), in other words, the distribution is not known in advance.

Thank you


TS Contributor
Not really. The logic behind Maximum Likelihood is that we consider the model to be true and the parameters unknown.

There are distributions that are special cases of other distributions, e.g. an exponential distribution is a special case of the Weibull distribution, which is a special case of the gamma distribution, or the uniform distribution is a special case of the beta distribution. So you could use the more general distribution and see if the paremeters are consistent with the more restricted distribution, but that way you can typically cover only a very limited set of distributions.

If you only care about the effects of covariates and not the entire distribution you can use techniques like GMM or quasi-likelihood to make less assumptions about the distribution.
I was looking around and found out that SAS has a procedure called SEVERITY which actually fits the data to the distributions in its system and then chooses the best fit based on AIC criteria.

So I guess that to find the best distribution is really a computational-heavy method, and I guess that (hope at least) that this procedure used the techniques you mentioned.

Thank you