Shrinkage estimators of the spatial relative risk function

The spatial relative risk function describes differences in the geographical distribution of two types of points, such as locations of cases and controls in an epidemiological study. It is defined as the ratio of the two underlying densities. Estimation of spatial relative risk is typically done using kernel estimates of these densities, but this procedure is often challenging in practice because of the high degree of spatial inhomogeneity in the distributions. This makes it difficult to obtain estimates of the relative risk that are stable in areas of sparse data while retaining necessary detail elsewhere, and consequently difficult to distinguish true risk hotspots from stochastic bumps in the risk function. We study shrinkage estimators of the spatial relative risk function to address these problems. In particular, we propose a new lasso-type estimator that shrinks a standard kernel estimator of the log-relative risk function towards zero. The shrinkage tuning parameter can be adjusted to help quantify the degree of evidence for the existence of risk hotspots, or selected to optimize a cross-validation criterion. The performance of the lasso estimator is encouraging both on a simulation study and on real-world examples.