I have N normally distributed variables, each with the same known standard deviation and mean. They are not independant, and are each correlated with the other with a Pearson's correlation coefficient of P, which we do not know.
If we take a single sample of each of the N variables, I would like to derive a MLE of the underlying correlation P. It stands to reason that we should be able to etimate the value P, given the clustering of the sampled values and how close they are to the sample mean (a high value of P would result in a high level of clustering etc). I imagine some form of MLE would work here, but I'm coming a little unstuck on how to derive a PDF. My current train of thought would be to derive a PDF for the RMSE of each of the N samples relative to the mean sample value (which should be highly dependant on P as stated), and interpret this as a function of P and then derive the MLE formualtion etc but this seems like it would be an incredibly hard PDF to derive. Is there something I'm missing? Is there a better way of doing this?
Without loss of generality one assumes that $X_i$ has mean $0$ and variance $1.$
The MLE is a complicated method (and you would have to assume the extra hypothesis that the random vector $(X_1,\ldots,X_n)$ is Gaussian) while the biased estimator of $P$ defined by $(\overline {X}_n)^2=(\frac{1}{n}(X_1+\cdots+X_n))^2$ is very simple: $$\mathbb{E}[(\overline {X}_n)^2]=\frac{1}{n}+(1-\frac{1}{n})P$$
Note that the law $(X_1,\ldots,X_n)$ does not exist if $P<-\frac{1}{n-1}.$