Statistics in kernel estimate

Question

Let $X_1,\dots,X_n$ a random sample with unknow density $f$. Take a estimator of $f$ as $$\hat{f}_h(x)=\frac{1}{nh}\sum_{i=1}^n K\Big(\frac{x-X_i}{h}\Big)$$ where $K(.)$ is any kernel.

I know that $$E[\hat{f}_h(x)]\rightarrow f(x),\text{when}\qquad h\rightarrow 0$$

How I find the median, percentiles and other statistics for kernel estimate?

It's driving me crazy, because I'm see some exercises where they just give the Kernel estimate, and say things like "Take $K(.)$ as Gaussian kernel and find a 90-th percentile estimator", "Take $K(.)$ as a triangle kernel and find the median estimator".

I do not understand this, since the estimated density converges to the true density, the estimators would not be the same as found in the real density?

Answer 1

Perhaps this is a simple-minded approach, but I'd approximate any quantile estimate with one based on the ordered data. Then use that approximation as an initial guess $q^{(0)}$ to iteratively solve the equation $\int_\infty^q \hat{f}(x)\: dx = p$, where $p$ is the proportion associated with the desired statistic, and $q$ is the desired estimate.