Symmetric adaptive smoothing regimens for estimation of the spatial relative risk function

The spatial relative risk function is now regarded as a standard tool for visualising spatially tagged case-control data. This function is usually estimated using the ratio of kernel density estimates. In many applications, spatially adaptive bandwidths are essential to handle the extensive inhomogeneity in the distribution of the data. Earlier methods have employed separate, asymmetrical smoothing regimens for case and control density estimates. However, we show that this can lead to potentially misleading methodological artefacts in the resulting estimates of the log-relative risk function. We develop a symmetric adaptive smoothing scheme that addresses this problem. We study the asymptotic properties of the new log-relative risk estimator, and examine its finite sample performance through an extensive simulation study based on a number of problems adapted from real life applications. The results are encouraging.