I am trying to fit informed prior distributions to data using MLE, and F occasionally provides a best fit (lowest AIC value).
I am starting with only very basic knowledge of probability theory, so I am not even confident that it is appropriate to fit an F in these cases. One of the interesting aspects of probability theory is that distributions often represent specific data generating processes - and are therefore useful for specific data types. However, I can not find any such justification for the F distribution. For example, the NIST handbook states:
Since the F distribution is typically used to develop hypothesis tests and confidence intervals and rarely for modeling applications, we omit any discussion of parameter estimation.
However, relationships between the F and Beta distributions suggest that F might be appropriate in general for ratios.
Are there theoretical (not computational) reasons to use or not use the F distribution as an informed prior distribution, or is having a continuous parameter with values > 0 sufficient justification?