$\newcommand{\E}{\operatorname{E}}$I have a problem which remind me of inconsistent system of equations from linear algebra, but I cannot see what the problem is. There are $X_1,X_2,\ldots,X_n$ random variables (the data). I want to estimate some observable $A$ which is a function of $n$ unknown quantities which I hope to estimate from the data. Assume these quantities are called $\theta_1,\theta_2,\dots,\theta_n$: $$ A = f(\theta_1, \theta_2, \ldots, \theta_n ) $$ I have obtained some estimators $\hat \theta_i$ such that the expectation $\E \hat \theta_i (A,X_1,X_2,\ldots,X_n) = \theta_i$ depend on $A$ and all observations for all $i$. The problem is that when I plug these into the equation for $A$ and I want to solve for $A$, the observable $A$ cancels. I.e. there is some function $g$ that does not depend on $A$ but \begin{align} A & = f(\E \hat \theta_1 (A,X_1,X_2,\ldots,X_n), \ldots, \E \hat \theta_n (A,X_1,X_2,\ldots,X_n) ) \\ & = g(\E\hat\theta_1(X_1,X_2,\ldots,X_n), \ldots, \E\hat\theta_n(X_1,X_2,\ldots,X_n) ) + A \end{align} That is a problem because it implies that $$ 0 = g(\E\hat\theta_1(X_1,X_2,\ldots,X_n), \ldots, \E\hat\theta_n(X_1,X_2,\ldots,X_n) ), $$ and instead of solving for $A$, $A$ is eliminated from the equations altogether! What is going on?
EDIT: To clarify what I am asking. Most of the time we are able to solve for $A$ in sets of equations. But every once in a while it doesn't work and $A$ cancels (like in my case). What is the name of sets of equations which has this property? How can one identify that a set of nonlinear equations cannot be solved for $A$? I would like more intimate understanding of the mechanisms which imply we cannot solve for $A$.
You have a function $(X_1,\ldots,X_n) \mapsto \widehat{\theta\,}_i (A, X_1, X_2, \ldots, X_n).$ A problem is that the value of this function appears to depend not just on the observable data $X_1, \ldots, X_n,$ but also on $A.$ But $A$ is not observable. An estimator must be observable.