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

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.