I am struggling for 2 hours now and decided to give up. I want to compute the variance of the KDE $$\hat f_H(x) = n^{-1}\sum_{i=1}^n\det(H^{-1})K(H^{-1}(x - X_i)).$$ My steps:
I got to the point where
$$\begin{align*}\frac{1}{n\det(H)^2}\mathbb E[K(H^{-1}(x - X_i))^2] &= ... \\ &= ...\\ &= \frac{1}{n\det(H)^2}\int K(u)^2f(x+Hu)\:\mathrm du.\end{align*}$$ A first order Taylor approximation of $f(x + Hu)$ yields (?) $$f(x + Hu) = f(x) + \nabla f(x)'Hu + O(\Vert Hu\Vert^2)$$ but then I got stuck because I cant get rid of the big oh (I tried its definition and definition of $\Vert\cdot\Vert$: $\mathrm{trace}(\int K(u)^2 uu'\:\mathrm du)H'H $ but the term in the trace may not exist. I just have assumptions about $\int K(u)uu'\:\mathrm du = \nu I$ and $\int K(u)^2\:\mathrm du = R(K)$.)
I also have $$\begin{align*}\frac{1}{n\det(H)^2}\mathbb E[K(H^{-1}(x - X_i))]^2 &= ... \\ &= ... \\ &= f(x)^2 + f(x)O(\Vert H\Vert^2) + O(\Vert H\Vert^2)^2 \\ &= f(x) + O(\Vert H\Vert^2)\end{align*}$$ (which is hopefully correct). According to this page https://bookdown.org/egarpor/NP-UC3M/kde-ii-asymp.html the result should be $$\frac{R(K)}{n\det(H)}f(x) + o\left(\frac{1}{n\det(H)}\right).$$ Note that the page defines $H$ slightly different but that should be neglible.
