Developing A Practical Measure: An Asymmetric Mean Squared Prediction Error for Small Area Estimation

The mean squared prediction error (MSPE) is widely used in small area estimation (SAE), as well as in other fields of statistics. Despite its popularity, the MSPE is not always practical in that it treats positive error, or over-prediction, and negative error, or under-prediction, equally. In practice, however, the consequences of these two types of errors are often different. This problem has long been known in statistics; however, a practical solution has not received much attention in SAE, which remains being dominated by the MSPE. A closely related object is the so-called best predictor (BP), which is equal to a conditional expectation under the MSPE criterion. We develop an asymmetric MSPE (AMSPE) measure that assigns different weights to the two different types of prediction errors. As a result, a different BP is developed, as well as its empirical version (EBP) that is implemented in practice. It is shown that, under the normality assumption, the AMSPE-based BP has a simple and elegant expression as the MSPE-based BP plus the conditional standard deviation multiplied by a deterministic constant, which only depends on the weight. As a natural measure of uncertainty, a second-order unbiased estimator of the area-specific AMSPE of the EBP is also developed. The developments are detailed under the area-level model for SAE. Theoretical properties of the proposed EBP and its AMSPE estimator are studied; their empirical performances are evaluated and compared with alternative methods. An iterative procedure for determining the weight in the AMSPE is developed and its global linear convergence is established. A real-data example is discussed. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.