Improved Empirical Likelihood Inference for Multiplicative Regression with Independent and Longitudinal Data

Multiplicative regression model has been proven to be an excellent model for analyzing data with positive responses. When constructing the confidence regions of the regression parameters, one had to either directly estimate the asymptotic covariance matrix involving the estimation of the unknown density function of the model error, then the normal approximation can be conducted, or resort to the time-consuming resampling methods to avoid the difficulty of estimating the covariance matrix. Recently, an empirical likelihood (EL) approach has been proposed and has comparable performances when the sample size is moderate or large. However, all these methods become inefficient and unsatisfactory when the sample size is small. This paper proposed the adjusted empirical likelihood (AEL) approach to improve the performance of the EL combined with the least absolute relative error and the least product relative error criteria for independent data and longitudinal data with small sample sizes, respectively. It is shown that the adjusted empirical log-likelihood ratio is asymptotically Chi-squared distributed. Simulation studies indicate that the proposed AEL method performs better than EL method for small sample sizes and is as efficient as EL method when the sample size is moderate.