# Bootstrap Resampling Vs KDE Resampling

Let $\xi\in\mathbb{R}^{m}$ be a random vector with joint desity function $f$, and let $\widehat{\xi}_{1},\ldots,\widehat{\xi}_{N}$ be a sample of $\xi$. We have that the kernel density estimator (KDE) of $f$ is $$\widehat{f}_{h}(x):=\frac{1}{Nh}\sum_{i=1}^{N}\mathcal{K}\left(\frac{x-\widehat{\xi}_{i}}{h}\right).$$ where $h> 0$ and $\mathcal{K}$ is a probability density function such that $\int x \mathcal{K}(x)dx=0$ and $\int x^{2} \mathcal{K}(x)dx=1$.

In the literature on this topic is customary to say tha $h$ must be chosen so as to minimize the MISE or ISE (see here) and there are many ways to choose it, suppose we already chose it.

Since $\widehat{f}_{h}(x)$ is an estimate of $f$ then it is expected that if I generate a sample from $\widehat{f}_{h}(x)$ this will be similar to samples $f$, then generate samples of $\widehat{f}_{h}(x)$ is a way to resampling the random variable $\xi$. Called this method KDE resampling.

The question: Where do I find papers or books that talk about KDE resampling?

Which is better: Boostrap resampling or KDE resampling?

• Not sure, but I think the point of KDE resampling is to obscure (become asymptotically independent of) the original data in order to improve privacy. If there is no need to obscure the original data, then I think KDE resampling may not have any merit. Have you seen otherwise? Bootstrap resampling does not have obscuration of the original data as an objective; rather to estimate variability of a statistical estimator. – Mark L. Stone May 14 '18 at 16:27
• @MarkL.Stone In most cases where a resampling is used is to estimate a parameter, that is, if $F$ is the distribution function that we do not know, we want to estimate $\alpha(F)$, a way of estimating this is bootstrap estimator $\alpha(\widehat{F}_{N})$, where $\widehat{F}_{N}$ is the empirical distribution. We recall $\alpha(\widehat{F}_{N})$ is estimated generating samples of $\widehat{F}_{N}$ (this is boostrap). The same idea can be applied to $\widehat{F}_{h}$ instead $\widehat{F}_{N}$ where $\widehat{F}_{h}$ is the comulative probability function of $\widehat{f}_{h}$ (kernel estimation). – Diego Fonseca May 14 '18 at 16:50
• @MarkL.Stone I'm not sure but I think that's the idea of smoothing boostrap. – Diego Fonseca May 14 '18 at 16:50