Estimation of the mean squared error of predictors of small area linear parameters under a logistic mixed model

An accuracy measure (mean squared error, MSE) is necessary when small area estimators of linear parameters are provided. Even in the case when such estimators arise from the assumption of relatively simple models for the variable of interest, as linear mixed models, the analytic form of the MSE is not suitable to be calculated explicitly. Some good and widely used approximations are available for those models. For generalized linear mixed models, a rough approximation can be obtained by a linearization of the model and application of Prasad–Rao approximation for linear mixed models. Resampling methods, although computationally demanding, represent a conceptually simple alternative. Under a logistic mixed linear model for the characteristic of interest, the Prasad–Rao-type formula is compared with a bootstrap estimator obtained by a wild bootstrap designed for estimating under finite populations. A simulation study is developed in order to study the performance of both methods for estimating a small area proportion.