Bandwidth Matrix Selection

Changes in the bandwidth $h_n$ of the Kernel function, can severely effect the resulting estimate; choosing a large bandwidth will produce biased estimates that hide localised features whereas smaller bandwidths will increase the estimates variability by introducing sharp modulations. The simplest choice of bandwidth is as some function of $n$, for example $k_n = 1/n$, so that as the number of samples increases the kernel gets smaller. A more complex method minimizes the integrated mean squared error between $\hat{f}_n$ and $f_n$ with respect to $k_n$. In Figure 6, three bandwidth selection methods are compared.