Hierarchical Estimation for Exchangeable Units: Continuous and Discrete Mixture Approaches

What is sometimes termed ensemble estimation or “borrowing strength from the ensemble” refers to inferences for collections of similar units, i = 1, . . . , n (schools, health agencies, etc.) (Burr and Doss, 2005; George et al., 1993; Morris, 1983; Rao, 1975). Among possible examples are collections of clinical trials, surgical death rates for hospitals, exam pass rates for schools, goal averages for basketball players (Hsiao, 1997), or teenage pregnancy rates in health areas (Deely and Smith, 1998). Fixed effects models for such collections are problematic (Marshall and Spiegelhalter, 1998), whereas hierarchical random effects approaches pool information across units to obtain more reliable estimates for each unit, and help identify units with unusually high or low values.