About the validity of two statements Let $f(x)$ be some smooth univariate density, and let the leave-one-out Nadaraya-Watson estimator $\widehat{f}_{-i}(x)$ be defined as follows:
$\widehat{f}_{-i}(x)=\frac{1}{(n-1)h}\sum_{j=1,j\neq i}^nK(\frac{X_j-x}{h})$, where $K(\cdot)$ is the kernel function and bandwidth $h\rightarrow 0$ at some specified speed so that we have $\underset{x\in J}{\sup} |\widehat{f}_{-i}(x)-f(x)|=o_{P}(n^{-1/4})$, where $J$ is a compact subset of the support of $X$ that excludes the boundary of the support. (An example for the unknown true density $f(x)$ could be the standard normal density, and an example for the known interval $J$ would be $J=[-50,50]$ )
I have the following two statements:
$\frac{1}{n}\sum_{i=1}^n|\mathbf{1}(X_i\in J)(\widehat{f}_{-i}(X_i)-f(X_i))|=o_{p}(n^{-1/2})$
$|\frac{1}{n}\sum_{i=1}^n \widehat{A}(X_i)\mathbf{1}(X_i\in J)(\widehat{f}_{-i}(X_i)-f(X_i))|=o_{p}(n^{-1/2})$, where $\widehat{A}(x)$ is a consistent estimator of some function $A(x)$, and is uniformly bounded on the support of $X$ with probability 1, and $\mathbf{1}(\cdot)$ is the indicator function.
Which of these two statements is valid or more likely to be valid? You can add additional assumptions if needed. Intuitive or rigorous justification, related reference are all welcome, thanks!
 A: The first statement is  much stronger, because the absolute value is inside the sum.  The impact of including $\hat A()$ will depend on  what it  is. If it's deterministic or consistently  estimated, it shouldn't  make much difference, but  if you took $$\hat A(X_i)= \mathrm{sign}(\hat f_{-i}(X_i)-f(X_i))$$ you'd be effectively moving  the absolute value inside the sum.  I'm assuming $J$ is an arbitrary compact subset of the support of $X$, otherwise you could take $J$ to be a single point (or empty).
I would say the first  statement is  unlikely to be true,  since I would not expect the individual terms to be $o_p(n^{-1/2})$ (or even $O_p(n^{-1/2})$, and there is no cancellation.
The second statement is  more plausible; it would easily be true if $(\hat f_{-i}(X_i)-f(X_i))$ were independent and mean zero. You have bounds on the maximum bias, but you'd need to also control the  proportion of observations $X_i$ in regions where the bias was near the maximum.  If $f$ had a pointy mode (where $\hat f$ will be biased down) or there were regions where the density wasn't bounded away from zero (so $\hat f_{-i}(X_i)$ will be biased down) then you can't rely on the bias cancelling (because you can choose $J$ to cover just those regions).
