Loss Functions for Estimation of Extremes in Disease Mapping

Hierarchical probability models are commonly used to estimate small-area disease-morbidity or disease-mortality rates. From the resulting estimates it is often desirable to identify small areas (e.g., counties) with unusually high or low disease risk after accounting for known risk factors. Traditional estimates of the unexplained risk are based on the squared-error loss function; such estimates have good ensemble properties but may be suboptimal for some features of the distribution of risk parameters. We explore the use of alternative loss functions to derive improved estimates of extreme values. A disease mapping application is used to illustrate the approach. A simulation study is used to compare the different loss functions.