# Is there a known MLE for the numerator df of a sample of F statistics?

Suppose you have observations $f_1, f_2, \ldots, f_n$ which are drawn i.i.d. from a central F distribution, with unknown numerator degrees of freedom, $n_1$ and known denominator degrees of freedom, $n_2$. What is the preferred method for computing/estimating the MLE of $n_1$? If the likelihood is known to be concave as a function of $n_1$ (I would guess it is, but am not certain), I could just use a numerical method like golden section search. Is there anything better?

I am aware of Spruill's method for ML estimation of the non-centrality parameter when the $f_i$ are noncentral, and $n_1$ and $n_2$ are known. It appears to use the transformation $$w_i = \frac{f_i}{n_2/n_1 + f_i}$$ Is this a standard transformation for F distributed RVs that might be applicable to this problem, or just a red herring?

• The $F$ distribution has some nice monotonicity properties associated with it. If you want, you can look at this question and answer. I'm not quite sure the paper there will be immediately relevant to your problem, though. – cardinal Oct 10 '11 at 17:35

It doesn't look like there could be a closed form solution for it, but it would be easy to derive numerically. An advantage of having a single parameter is that you can easily plot the likelihood. How you go about finding the MLE depends on whether you're doing it just once for a particular data set or whether you plan to continually do it. If I were doing it just once, I'd first plot the log likelihood and then use some generic optimizer, like optimize in R [see manual page].