Asymptotics for least product relative error estimation and empirical likelihood with longitudinal data

A multiplicative regression model with longitudinal data is introduced, and a least product relative errors estimate is constructed based on relative errors. Generally, the least squares criterion and least absolute deviation criterion based on absolute errors are the most widely used criteria in the regression analysis. However, when response variables have different measurement scales, relative errors may be superior to absolute errors. Thence, we develop a least product relative errors estimator of parameter based on relative errors, and obtain their asymptotic properties where some nuisance parameters such as correlation structure of error terms are included. In addition, block empirical likelihood technique is employed to construct the confidence regions of the corresponding unknown regression parameter, avoiding density estimation. Simulation results confirm that the proposed methods perform well.