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Apologies in advance if this question is not completely well defined. Suppose that I am estimating a nonparametric model for a conditional expectation function $\mathbb E[Y_i | X_i]$ using some i.i.d. dataset. Suppose, however, that I do not directly observe $Y_i$, but instead, for each sample size, $n$, observe $\psi_{i,n} = Y_i + o_P(1)$, i.e. I observe $Y_i$ with noise, but the degree of corruption in each observation converges to 0 when I have a large sample. Let $\hat g_n$ be some abstract estimation procedure which takes $n$ observations of $(X_i,Y_i$ and spits out an estimator for the conditional expectation function of $Y_i$ given $X_i$. Moreover, suppose that given the population distribution for $(X_i,Y_i)$, $\hat g_n$ satisfies some rate condition, say, $r_n(\hat g_n(x) - \mathbb E[Y_i | X_i = x]) \overset p\to 0$ for all $x$ in the support of the $X_i$.

I an interested in conditions under which $\hat g_n$ converges to $\mathbb E[Y_i | X_i = x]$ if we plugged in $\psi_{i,n}$ into our estimator $\hat g_n$ instead of $Y_i$. In particular, does the $o_P(1)$ assumption on the error suffice, or do we need stronger conditions on the discrepancy between $\psi_{i,n}$ and $Y_i$ to get results. I'm guessing, for instance, noise that is $o_P(q)$ but is not even integrable can be highly problematic if $\hat g_n$ does not have a principled way to deal with outliers. Is there any literature I could look at that addresses questions like this?

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You can prove convergence rates for your problem using standard M-estimator analysis. The rate-of-convergence theorem of chapter 3 in van der Vaart, Wellner, 1998 will handle it.

It is also given in Lectures 12/13 of https://www.stat.berkeley.edu/~aditya/resources/FullNotes210BSpring2018.pdf

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