How to invert/infer a parameter in nonlinear conditional expectation function I wouldn't be surprised if this question has already been asked, as it sounds like a standard bookwork result. However, I'm not sure I know the language to describe it, and when I type in the the title it doesn't find in similar questions. Feel free to delete it if you can point me to a duplicate.
Suppose I have a random variable $Y \sim \psi(w)$ that's drawn from a distribution $\psi$ parameterised by a parameter $w$ (some arbitrary distribution), and all I know is the conditional expectation $E(Y \mid w)$ which we can label as the function $f(w)$. $f$ is not linear.
If I have a sample of N realised variates of $Y$: $\{y_i\}_{i=1}^{N}$, I can see two ways to estimate $w$.

*

*Take the mean of $y_i$ and invert that:
$$\hat{w} = f^{-1}\left( \frac{1}{N} \sum_{i=1}^N y_i \right)$$

*Take the mean of the inverses of $y_i$:
$$\tilde{w} = \frac{1}{N} \sum_{i=1}^N f^{-1}\left(  y_i \right)$$
Which of the two is correct? They can't be the same, because $f$ is nonlinear (so by Jensen's inequality they must be differen, correct me if I'm wrong).
My first guess would be 1 because for large N, $\frac{1}{N} \sum_{i=1}^N y_i \approx E(Y \mid w)$  and $f^{-1}(E(Y \mid w)) = w$ but I'm not sure if that's a valid argument to make and if there is a similar argument for 2. Is there some general result for the expectation of the second estimator?

If the function is invertable and the result is not too complex, you could use your approach #2 with a small adjustment: Don't just take the mean value but look at the distribution of computed results. It will have a mode (expected value) and you can directly look at the confidence bounds.
If the function is not invertable, it might be easy to get the correct value using a non-linear optimization. You could do a grid-search, a genetic algorithm or any other optimization to approach a value for omega that reliably predicts the values of y you have.
