Loss functions for estimation of extrema with an application to disease mapping
It is often of interest to find the maximum or near maxima among a set of vector-valued parameters in a statistical model; in the case of disease mapping, for example, these correspond to relative-risk "hotspots" where public-health intervention may be needed. The general problem is one of estimating nonlinear functions of the ensemble of relative risks, but biased estimates result if posterior means are simply substituted into these nonlinear functions. The authors obtain better estimates of extrema from a new, weighted ranks squared error loss function. The derivation of these Bayes estimators assumes a hidden-Markov random-field model for relative risks, and their behaviour is illustrated with real and simulated data.