ABSTRACT
This chapter introduces the kernel method for the multivariate case and relates the multivariate histograms and scatter plots. The definition of the kernel estimator as a sum of 'bumps' centred at the observations is easily generalized to the multivariate case. The chapter discusses the important alternative method for multivariate data is the adaptive kernel approach. The kernel method for the multivariate case will be introduced along with some comparisons made with multivariate histograms and scatter plots. The advantage of these kernels over the Epanechnikov kernel is that the kernels, and hence the resulting density estimates, have higher differentiability properties. An attractive intuitive approach, suggested by Fukunaga is first to 'pre-whiten' the data by linearly transforming them to have unit covariance matrix; next to smooth using a radially symmetric kernel; and finally to transform back. The chapter derives the approximate expressions for the bias and the variance of the estimators, and these can be used to give some guidance concerning the appropriate choice of kernel and window width.
