Data Fusion Using Weighted Likelihood

This article proposes to perform data fusion by using an adaptive weighted likelihood function when data sets are available from related populations. The main objective of data fusion is to integrate information from different sources to improve the quality of inference when the sample size from the target population is small or moderate. The weighted likelihood function is employed simply as an instrument to facilitate the data fusion process. The weighted likelihood method has informationtheoretic justification and embraces the widely used classical likelihood method which utilizes only on the data set from the target population. The degree of information integration in the proposed data fusion process is determined by the likelihood weights which should be chosen in a reasonable and adaptive way. The major challenge in the proposed data fusion process is then to choose likelihood weights adaptively and effectively when the deterministic relationships among all related parameters are unknown. We propose adaptive likelihood weights based on the estimated likelihood ratio. We show that the data fusion involving all relevant data sets could significantly improve the mean squared error (MSE) of the classical maximum likelihood estimator which only uses data set from the target population. It also increases the power for hypothesis testing. The proposed estimator is shown to be consistent and asymptotically normally distributed in the framework of generalized linear models. The advantage of the proposed weighted likelihood estimator for linear models is illustrated numerically by a simulation study. A real data example is also provided.