I have $X_1, X_2, ..., X_n$ correlated observations, I know the joint distribution, e.g., $f_{X_i,X_j}$. I have the nth observation missing, what is the best way to estimate the value of $X_n$.
I know that I can do n-1 nonlinear ML estimations for $X_n$. and thus will have a vector of estimates say $\hat{X}_n = \{\hat{X}_n(i)~ |~ \forall i\in \{1,...,n-1\}\}$, it is obvious that taking only one of the estimates is not enough, however, I am not sure how to combine these estimates.
Alternatively, I can use a weighted linear combination of the observations to estimate $X_n$, but this might be worse than the above. I am wondering if anybody has seen this problem, because its seems a valid problem, references or direction are appreciated.