Sumca: simple, unified, Monte-Carlo-assisted approach to second-order unbiased mean-squared prediction error estimation

Abstract

We propose a simple, unified, Monte-Carlo-assisted approach (called ‘Sumca’) to second-order unbiased estimation of the mean-squared prediction error (MSPE) of a small area predictor. The MSPE estimator proposed is easy to derive, has a simple expression and applies to a broad range of predictors that include the traditional empirical best linear unbiased predictor, empirical best predictor and post-model-selection empirical best linear unbiased predictor and empirical best predictor as special cases. Furthermore, the leading term of the MSPE estimator proposed is guaranteed positive; the lower order term corresponds to a bias correction, which can be evaluated via a Monte Carlo method. The computational burden for the Monte Carlo evaluation is much less, compared with other Monte-Carlo-based methods that have been used for producing second-order unbiased MSPE estimators, such as the double bootstrap and Monte Carlo jackknife. The Sumca estimator also has a nice stability feature. Theoretical and empirical results demonstrate properties and advantages of the Sumca estimator.