On the Estimation of Mean Squared Prediction Error in Small Area Estimation
We critically examine the exact mean squared prediction error (MSPE) of an empirical best linear unbiased predictor (EBLUP) for certain special cases of the well known Fay Herriot model widely used in small area estimation. This study is useful in understanding the impact of the neglected higher order terms in standard second order aymptotics in small area estimation problems. We argue that in certain cases the naive MSPE estimator, which is generally considered to have a serious underestimation problem as it ignores the uncertainty due to estimation of the variance component, may be adequate and may even overestimate the true MSPE. We propose a new weighted jackknife MSPE estimator as an alternative to the Taylor linearization method. Our proposed method is equivalent to other second order unbiased MSPE estimators in terms of the standard second order asymptotics. However, in our simulation study the proposed jackknife MSPE estimators appear to be better than the rival Taylor linearization method in terms of bias, especially when the regularity conditions that govern the second order unbiasedness property are violated. We observe that for a special case of the Fay Herriot model, our jackknife MSPE estimator is equivalent, in a higher order asymptotic sense, to a hierarchical Bayes solution under at priors on the hyperparameters.