Suppose I have an i.i.d. sample $\{(Y_i, X_i)\}_{i=1}^n$ on which I am trying to estimate a conditional expectation model:

$$Y = g(X) + \varepsilon,\quad \mathbb E[\varepsilon | X] = 0$$

There is a lot of literature on how different estimators of $g$ will imply different rates of convergence to the true conditional expectation function for different regularity assumptions. However, for my application, I am not interested in $g$ directly but instead care about a single summary statistic about $g$, namely

$$\theta = \mathbb E[(g(X) - \mathbb E[Y])^2] \quad (\text{alternatively}, \eta = \mathbb E[\varepsilon ^2])$$

My question is the following: are there settings in which $\theta$ or $\eta$ can be estimated at a faster rate than $g$ itself?


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