My question refers to the book "Nonparametric Econometrics - Theory and Practice" by Li & Racine. Here, the variance for a kernel density estimator using the pointwise perspective (for fixed x) is derived as followed: \begin{align} var(\hat{f}_n(x))&=var\Big(\frac{1}{nh}\sum^n_{i=1}k(\frac{X_i-x}{h})\Big)\\ &=\frac{1}{n^2h^2}var\Big(\sum^n_{i=1}k(\frac{X_i-x}{h})\Big)\\ &=\frac{1}{nh^2}var\Big(k(\frac{X_1-x}{h})\Big)\\ &=\frac{1}{nh^2}(E(k(\frac{X_i-x}{h})^2)-E(k(\frac{X_1-x}{h}))^2)\\ &=\frac{1}{nh^2}\Big(h\int f(x+h*u) k^2(u)du-(h\int f(x+hu)*k(u)du)^2\Big)\\ &=\frac{1}{nh^2}\Big(h\int (f(x)+f^{(1)}(x)hu) k^2(u)du-O(h^2)\Big)\\ &=\frac{1}{nh}\Big(f(x)\int k^2(u)du+O(h\int|u|k^2(u)du)-O(h)\Big)\\ &=\frac{1}{nh}(\kappa f(x)+O(h)) \end{align} , here k is a kernel function with classical assumptions, $X_i,x_1$ realizations, f the true density, h a bandwidth and n the sample size, besides $\kappa=\int k^2(u)du$. What I cannot understand are the last three equalities, i.e. why $\int f^{(1)}(x)hu*k^2(u)du$ results in the bounded term with $O(h\int|u|k^2(u)du)$. The boundedness is obvious since the first derivative is some constant at given x.
- How does one obtain the particular value for the Big O upper bound (especially in the form where the absolute value of u is used)?
- And how are the two Big O terms subtracted from each other to obtain the final equality with O(h)?
I appreciate any help!