Adjusted Maximum Likelihood Method Based on Shrinkage Factor Bias Reduction for Multivariate Fay–Herriot Model

ABSTRACT One problem encountered in small‐area estimation is the zero estimate of the variance component in the Fay–Herriot model. This problem affects the accuracy of the empirical best linear unbiased prediction (EBLUP) estimator. The zero estimate of the variance component causes zero weight of the direct estimates in EBLUPs. Several methods have been investigated to solve this problem by focusing on reducing the bias of the variance component estimator. However, an unbiased estimator of the variance component might not yield an unbiased estimator of the small‐area mean. Therefore, Yoshimori and Lahiri [Journal of Multivariate Analysis, 124 , 281–294 (2014)] proposed a variance component estimation method by reducing the bias of the shrinkage factor of EBLUP for the Fay–Herriot model. In this study, we extend their method to propose two new adjusted likelihood methods based on shrinkage factor bias reduction for the multivariate Fay–Herriot model. Moreover, we perform a simulation study to investigate the performance of the proposed methods comparing them with existing methods such as the residual likelihood method and the adjusted residual likelihood method of Angkunsit and Suntornchost [Journal of Statistical Planning and Inference, 219 , 231–249 (2022)]. Simulation results show that the two proposed methods perform better than the existing methods. Finally, we apply the proposed methods to the bivariate Fay–Herriot model of the average household income and the average household expenditure in Thailand.