Accurate identification of the extremes among an ensemble of parameters is an important practical problem in a variety of areas. Two applications considered in this dissertation are animal breeding and disease mapping. In these areas, the extremes correspond to animals that should be selected for future breeding or small areas with unusually high or low disease incidence;A common model for selection experiments in animal breeding, the animal model, contains parameters corresponding to individual animal breeding values. Estimates and/or ranks of these parameters are used to select animals with high genetic merit for breeding subsequent generations. The traditional method is to select animals based on best linear unbiased estimates of the breeding values after making the assumption that the variance component parameters are equal to their restricted maximum likelihood estimates. Other selection methods apply a decision-theoretic Bayesian approach to obtain the breeding value estimates that minimize the posterior expected value of suitable loss functions. We consider several different loss functions and present a simulation approach for comparing different selection rules. An example pertaining to data from a flour beetle breeding experiment is included;In disease mapping, random effects in a common Poisson-lognormal model correspond to, possibly spatially-correlated, small-area risk factors not explained by the covariates in the model. We present a detailed discussion of two variations of a statistical model that is commonly used for analyzing disease incidence/mortality data. Epidemiologists are interested in identifying regions that have high or low relative risk for disease incidence. In this context, we develop a new type of loss function, weighted ranks squared error loss, that incorporates information about the ranks of parameters. The loss function depends on the specification of a weight vector, c. Choosing c = 1 gives squared error loss as a special case. The weights can also be chosen to place emphasis on estimating parameters with high or low ranks, as is desirable when estimating disease incidence relative risks. The different models and a number of loss functions are compared on data pertaining to lip cancer rates in Scotland.