$n$ samples $X_{i}$ for $0\leq i<n$ are sampled from a $N(\mu, 1)$ , however, the sampling process is NOT i.i.d, there is a correlation among sampling process, the correlation matrix is $C$, where all the off diagonal elements are $r$.
Given these $n$ samples, how can I estimate $\mu$? How to estimate the confidence interval of this estimate ?
If there is no correlation, i.e., the samples are all iid, then the estimate should be easier:
$E(\mu) = \sum(X_{i})/n$
My guess is that , this problem has some connection with linear regression:
given a vector $m=[\mu, \mu, \mu ...]$ of size $n$, and a vector $x=[X_{0}, X_{1}, ... X_{n}]$
$ m = C*x$
Not sure if this is correct though, can anyone share some thoughts?