Asymptotic Comparison of Predictive Densities for Dependent Observations

This paper studies Bayesian predictive densities based on different priors and frequentist plug-in type predictive densities when the predicted variables are dependent on the observations. Average Kullback-Leibler divergence to the true predictive density is used to measure the performance of different inference procedures. The notion of second-order KL dominance is introduced, and an explicit condition for a prior to be second-order KL dominant is given using an asymptotic expansion. As an example, we show theoretically that for mixed effects models, the Bayesian predictive density with prior from a particular improper prior family dominates the performance of REML plug-in density, while the Jeffreys prior is not always superior to the REML approach. Simulation studies are included which show good agreement with the asymptotic results for moderate sample size. Some key words: Mixed effect models; Kullback-Leibler divergence; Jeffreys prior; Predictive density; Prediction fit. ∗Xuanyao He is Ph.D. student, Department of Statistics and Operations Research, the University of North Carolina at Chapel Hill, NC 27599 (E-mail: [email protected]). Richard L. Smith is Professor, Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599 (Email: [email protected]). Zhengyuan Zhu is Assistant Professor, Department of Statistics and Operations Research, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599 (E-mail: [email protected]). 1