I have a ( simple) question concerning the asymptotic notation ( in the case of the variance of a kde): Assumptions are: $$ h \to 0 \ as\ n\to \infty\ s.t\ nh \to \infty $$
At some point i get $$ {E}[\cdot]- \frac{1}{n} [f(x) -O(h²)]² $$ which should should be transformd to $$ {E}[\cdot]- O\left(\frac{1}{n}\right)$$ for some density function $f(x)$
My question would be if someone could tell me why the O-notation is $O(n^{-1})$ and not $O(\frac{h^4}{n})$
Thanks in advance.
With your hypothesis ($nh\to 0$ as $n\to\infty$), we know that $\;h=o\Bigl(\dfrac1n\Bigr)$, so if $f(x)\ne 0$, $\;f(x)-O(h^2)\sim f(x)$, whence $$\frac1n\bigl[f(x)-O(h^2)\bigr]\sim_\infty \frac{f(x)}n=O\Bigl(\frac1n\Bigr).$$