Parametric Bootstrap Methods for Bias Correction in Linear Mixed Models

The empirical best linear unbiased predictor (EBLUP) in the linear mixed model (LMM) is useful for the small area estimation, and the estimation of the mean squared error (MSE) of EBLUP is important as a measure of uncertainty of EBLUP. To obtain a second-order unbiased estimator of the MSE, the second-order bias correction has been derived mainly based on Taylor series expansions. However, this approach is harder to implement in complicated models with more unknown parameters like variance components, since we need to compute asymptotic bias, variance and covariance for estimators of unknown parameters as well as partial derivatives of some quantities. The same difficulty occurs in construction of confidence intervals based on EBLUP with second-order correction and in derivation of second-order bias correction terms in the Akaike Information Criterion (AIC) and the conditional AIC. To avoid such difficulty in derivation of second-order bias correction in these problems, the parametric bootstrap methods are suggested in this paper, and their second-order justifications are established. Finally, performances of the suggested procedures are numerically investigated in comparison with some existing procedures given in the literature.