# Rates of convergence for estimating population mean squared error

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?