Given a sequence of independent identically distributed random variables $X_1,\ldots,X_m \sim \chi^2_n / n$ is there literature on estimates for the degrees of freedom $n$?
In an attempt to find the MLE I calculated the roots of the log likelihood function assuming continuous values of $n$. This tells us that $n$ should satisfy $$2\frac{1}{n} – \psi(n) = \log \left( \sqrt{2}\prod_{i=1}^m x_i^{\frac{1}{m}} \right) $$ where $\psi$ denotes the digamma function. This estimate is not quite as insightful as I'd hoped. An easy estimate is simply noting that $\operatorname{Var}(\chi^2_n / n) = 2/n$. However, I don't know whether this will actually be a good estimate in practice.
Best Answer
The expected value of this distribution is $1$ and the variance is $2/n.$ If one writes $$ \frac 2 n = \frac 1 m \sum_{i=1}^m (X_i-1)^2 $$ and solves for $n,$ one gets an estimate $$ \widehat n = \frac{2m}{\sum_{i=1}^n (X_i-1)^2}. $$ Take that as a first approximation to the solution of the equation that sets the log-likelihood equal to $0,$ and then use Newton's method to get a better approximation. Iterate as needed, but if $m$ is large, I suspect the second approximation may be fairly good.
I have used $1$ rather than the sample mean when estimating the variance, because you know the population mean is $1.$