Jointly optimal bandwidth selection for the planar kernel-smoothed density-ratio

The kernel-smoothed density-ratio or 'relative risk' function for planar point data is a useful tool for examining disease rates over a certain geographical region. Instrumental to the quality of the resulting risk surface estimate is the choice of bandwidth for computation of the required numerator and denominator densities. The challenge associated with finding some 'optimal' smoothing parameter for standalone implementation of the kernel estimator given observed data is compounded when we deal with the density-ratio per se. To date, only one method specifically designed for calculation of density-ratio optimal bandwidths has received any notable attention in the applied literature. However, this method exhibits significant variability in the estimated smoothing parameters. In this work, the first practical comparison of this selector with a little-known alternative technique is provided. The possibility of exploiting an asymptotic MISE formulation in an effort to control excess variability is also examined, and numerical results seem promising.