Assume that I have a random sample $X_1,\dots,X_m$ from some distribution $F$, and I want to estimate the first two moments of $F$. Obviously, any sane person would use the sample moments, but I was wondering about the following situation:
Instead of using the sample moments, I employ kernel density estimation, for example with a Gaussian kernel, which leaves me with some distribution, in this case a mixutre of normal distributions due to teh Gaussian kernel. Now, my question is: Are there any results on conditions under which the mean and variance of the "KDE-distribution" converge to the true distribution $F$? How does this depend on kernel, bandwidth, or properties of the sample?
I wasn't able to find research on this as usually, questions such as "What is the expectation of the KD-estimate at a point $x$?" seem to be addressed, rather than the moments of the resulting distribution.