Geometric Model Averaging

This paper proposes a geometric model averaging (GMA) method in the framework of general likelihood estimation. Instead of using the classical weighted arithmetic mean, we consider a weighted geometric mean of the conditional probability density functions estimated from different candidate models, allowing both parameter uncertainty and model misspecification. The optimal weight is the minimizer of the Kullback-Leibler (KL) divergence of the true density to the weighted density. The geometric form of the weighted density makes it possible to find an approximately unbiased estimator of the KL divergence, and hence the optimal weight can be estimated consequently. Also, the geometric mean is naturally more applicable to density tail estimation than arithmetic mean due to its special form. The proposed GMA method provides a unified framework for a wide range of problems and is connected to many existing model averaging methods. When all the candidate models are misspecified, the weight of GMA is asymptotically optimal in the sense of achieving the lowest possible KL divergence. When correct candidate models are included, estimation consistency is established. Results of a comprehensive simulation study and two real data analyses show favorable performance of GMA relative to alternative model selection and model averaging methods.