we see that gtn in ( is the average of collection of normal pdfswhere each normal distribution is centered at the data point xi and has covariance matrix id major question is how to choose the bandwidth so as to best approximate the unknown pdf choosing very small will result in "spikyestimatewhereas large will produce an over-smoothed estimate that may not identify important peaks that are present in the unknown pdf figure illustrates this phenomenon in this case the data are comprised of points uniformly drawn from the unit square the true pdf is thus on [ ] and elsewhere figure two two-dimensional gaussian kdeswith (leftand (rightlet us write the gaussian kde in ( as xi ph gtn ( sn = sd ( where kzk rd ( / ( pis the pdf of the -dimensional standard normal distribution by choosing different probability density ph in ( )satisfying ph(xph(-xfor all xwe can obtain wide variety of kernel density estimates simple pdf ph isfor examplethe uniform pdf on [- ] - if [- ] ph( otherwise ph(zfigure shows the graph of the corresponding kdeusing the same data as in figure and with bandwidth we observe qualitatively similar behavior for the gaussian and uniform kdes as rulethe choice of the function ph is less important than the choice of the bandwidth in determining the quality of the estimate the important issue of bandwidth selection has been extensively studied for onedimensional data to explain the ideaswe use our usual setup and let { xn be the observed (one-dimensionaldata from the unknown pdf firstwe define the loss function as (xg( )) lossf ( ) ( )( (