Bayesian versus frequentist measures of error in small area estimation

This paper compares analytically and empirically the frequentist and Bayesian measures of error in small area estimation. The model postulated is the nested error regression model which allows for random small area effects to represent the joint effect of small area characteristics that are not accounted for by the fixed regressor variables. When the variance components are known, then, under a uniform prior for the regression coefficients and normality of the error terms, the frequentist and the Bayesian approaches yield the same predictors and prediction mean‐squared errors (MSEs) (defined accordingly). When the variance components are unknown, it is common practice to replace the unknown variances by sample estimates in the expressions for the optimal predictors, so that the resulting empirical predictors remain the same under the two approaches. The use of this paradigm requires, however, modifications to the expressions of the prediction MSE to account for the extra variability induced by the need to estimate the variance components. The main focus of this paper is to review and compare the modifications to prediction MSEs proposed in the literature under the two approaches, with special emphasis on the orders of the bias of the resulting approximations to the true MSEs. Some new approximations based on Monte Carlo simulation are also proposed and compared with the existing methods. The advantage of these approximations is their simplicity and generality. Finite sample frequentist properties of the various methods are explored by a simulation study. The main conclusions of this study are that the use of second‐order bias corrections generally yields better results in terms of the bias of the MSE approximations and the coverage properties of confidence intervals for the small area means. The Bayesian methods are found to have good frequentist properties, but they can be inferior to the frequentist methods. The second‐order approximations under both approaches have, however, larger variances than the corresponding first‐order approximations which in most cases result in higher MSEs of the MSE approximations.