## Multivariate kernel density estimation

Briefly, we shall see the definition of a kernel density estimator in the multivariate case. Suppose that the data is d-dimensional so that $latex {X_{i}=(X_{i1},\ldots,X_{id})}&fg=000000$. We will use the product kernel $latex \displaystyle \hat{f}_{h}(x)=\frac{1}{nh_{1}\cdots h_{d}}\left\{ \prod_{j=1}^{d}K\left(\frac{x_{j}-X_{ij}}{h_{j}}\right)\right\} . &fg=000000$ The risk is given by $latex \displaystyle \mathrm{MISE}\approx\frac{\left(\mu_{2}(K)\right)^{4}}{4}\left[\sum_{j=1}^{d}h_{j}^{4}\int f_{jj}^{2}(x)dx+\sum_{j\neq k}h_{j}^{2}h_{k}^{2}\int f_{jj}f_{kk}dx\right]+\frac{\left(\int K^{2}(x)dx\right)^{d}}{nh_{1}\cdots h_{d}} &fg=000000$

## Choosing the smoothing parameter

Two popular methods to find the bandwidth $latex {h}&fg=000000$ for the nonparametric density estimator are the plug-in method and the method cross-validation. The first one we will focus in the “quick and dirty” plug-in method introduced by Silverman (1986). In cross-validation we will minimize a modified version of the quadratic risk of $latex {\hat{f}_{h}}&fg=000000$. The […]

## A global measure of risk for kernel estimators in Nikolski classes

Photos of Sergey Nikolskii from The Russian Academy of Sciences The MSE  gives an error of the estimator $latex {\hat{p}_{n}}&fg=000000$ at an arbitrary point $latex {x_{0}}&fg=000000$, but it is worth to study a global risk for $latex {\hat{p} _{n}}&fg=000000$. The mean integrated squared error (MISE) is an important global measure, $latex \displaystyle \mathrm{MISE}\triangleq\mathop{\mathbb E}_{p}\int\left(\hat{p} _{n}(x)-p(x)\right)^{2}dx &fg=000000$ […]

## Kernel density estimation

I will make a summary of ideas about nonparametric estimation, including some basics results to develop more advanced theory later. In the first post  we talk something about the density estimation and the nonparametric regression. Later, in posts about histogram (I,II,III,IV) , we saw how the histogram is a nonparametric estimator and we studied its […]

## Density Estimation by Histograms (Part I)

We are going to introduce the histogram as a simple nonparametric density estimator.  I will divide this presentation in several posts for simplicity reasons. Let us $latex {X_1,\ldots,X_n}&fg=000000$ with pdf $latex {f}&fg=000000$. The histogram is the simplest nonparametric estimator of $latex {f}&fg=000000$.

## Importance of nonparametric statistics in regression.

I would like to start this blog with some basic ideas about density estimation and nonparametric regression. The study of the probability density function (pdf) is called nonparametric estimation. This kind of estimation can serve as a block building in nonparametric regression. The typical regression problem is setting as follows. Assume that we have a […]