Briefly, we shall see the definition of a kernel density estimator in the multivariate case.
Suppose that the data is ddimensional 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
$latex \displaystyle \mathrm{MISE}\approx\frac{\left(\mu_{2}(K)\right)^{4}}{4}\left[\sum_{j=1}^{d}h_{j}^{4}\int f_{jj}^{2}(x)dx+\sum_{j\neq k}h_{j}^{2}h_{k}^{2}\int f_{jj}f_{kk}dx\right]+\frac{\left(\int K^{2}(x)dx\right)^{d}}{nh_{1}\cdots h_{d}} &fg=000000$
where $latex {f_{jj}}&fg=000000$ is the second partial derivative of $latex {f}&fg=000000$. The optimal bandwidth satisfies $latex {h_{i}=O(n^{1/(4+d)})}&fg=000000$ leading to a risk of order $latex {O(n^{4/(4+d)})}&fg=000000$ (for further details see Hardle (2004)).
The interesting effect of $latex {O(n^{4/(4+d)})}&fg=000000$ here is that the risk increase exponentially as the dimension grows. We call to this behavior the curse of dimensionality. This phenomena says that the data is more sparse as we increase the dimensionality. This table from Silverman (1986) shows the sample size required to ensure a relative mean squared error less than 0.1 at 0 when the density is multivariate normal and the optimal bandwidth is selected.
Dimension  Sample size 


1  4 
2  19 
3  67 
4  223 
5  768 
6  2790 
7  10,700 
8  43,700 
9  187,000 
10  842,000 
For this reason it is important to search methods for dimension reduction. One of these methods was proposed by Li (1991) in its article Sliced Inverse Regression for Dimension Reduction. I used this method to find another efficient estimator based in a Taylor approximation (see Solís Chacón, M et al. (2012) ). In a next post I going to talk a little about the details of those articles.
Sources:
 Hardle, W. (2004). Nonparametric and Semiparametric Models. Springer Series in Statistics. Springer.
 Li, K.C. (1991). Sliced Inverse Regression for Dimension Reduction. Journal of the American Statistical Association, 86(414), 316327. Retrieved from http://www.jstor.org/stable/2290563
 Tsybakov, A. (2009). Introduction to nonparametric estimation. Springer.
 Silverman, B. W. (1986). Density Estimation for Statistics and Data Analysis, volume 26. Chapman & Hall/CRC.
 Solís Chacón, M., Loubes, J.M., Clement, M. & Da Veiga, S. (2012). Efficient estimation of conditional covariance matrices for dimension reduction. Arxiv preprint arXiv: Retrieved from http://arxiv.org/abs/1110.3238