New Bootstrap Bias Correction with Application to MSE Estimation under a Basic Small Area Model
Solange Correa¹, Danny Pfeffermann²
¹Department of Methods and Quality, Brazilian Institute of Geography and Statistics, Rio de Janeiro, RJ, Brazil; ²Department of Statistics, Hebrew University, Jerusalem, Israel; ³Southampton Statistical Sciences Research Institute, University of Southampton, Southampton, United Kingdom

In this paper, a general approach for correcting the bias of a MSE estimator and for obtaining estimates of the accuracy of the bias-corrected MSE estimator is proposed. The method, entitled Empirical Bias Correction (EBC), is based on the bootstrap resampling technique and involves identifying the functional relationship between estimates obtained from the original and bootstrap samples and the true values, drawn from a plausible parameter space. This implies that the relationship between the error of the estimator under study and its original and bootstrap estimates is extracted from the data themselves, rather than arbitrarily imposed. The bootstrap samples are used to study the behaviour of the bias and, consequently, the bias correction itself.

The study focuses on bias adjustment of the naïve MSE estimators of small area means under the Fay-Herriot (1979) model. An extensive Monte Carlo study is conducted to assess the performance of the EBC MSE estimators and also compare them with other estimators in common use in the literature, namely: the Prasad and Rao (1990) (PR) and the Datta, Rao and Smith (2005) (DRS) MSE estimators. Both PR and DRS MSE estimators are based on Taylor approximations of the true MSE under normality assumption of the model error terms. The classical additive, multiplicative and exponential (Hall and Maiti, 2006) bias-corrected MSE estimators are also assessed. The performance of the various estimators are evaluated under different scenarios defined by combinations of two different distributions of the model error terms (normal and location exponential distributions) and two different estimators of the variance of area random effects (Prasad-Rao and Fay-Herriot estimators). The results show that the EBC approach is effective in providing bias corrected estimators in all the scenarios considered.

Correa, S.T. (2008). Bias Correction in Multilevel Modelling of Survey Data with Applications to Small Area Estimation. Ph.D. Thesis. University of Southampton: U.K.

Fay, R. E. and Herriot, R. A. (1979). Estimates of income for small places: an application of James-Stein procedure to census data. Journal of the American Statistical Association, 74, 269-277.

Datta, G. S., Rao, J. N. K. and Smith, D. D. (2005). On measuring the variability of small area estimators under a basic area level model. Biometrika, 92, 183-196.

Hall, P. and Maiti, T. (2006). Nonparametric estimation of mean squared prediction error in nested-error regression models. Annals of Statistics, 34, 1733-1750.

Prasad, N. G. N. and Rao, J. N. K. (1990). The estimation of mean squared errors of small area estimators. Journal of American Statistical Association, 85, 163-171.

Keywords: Bias correction; Empirical best predictor; Bootstrap; Fay-Herriot model

Biography: Dr. Solange Correa has been methodologist at the Brazilian Institute of Geography and Statistics for 14 years. She finished her Ph.D. on “Bias corrections in multilevel modelling of survey data with applications to small area estimation” in 2008 at the University of Southampton, U.K., under joint supervision of Professor Danny Pfeffermann and Professor Chris Skinner.