Model averaging has received much attention in the past two decades, which integrates available information by averaging over potential models. Although various model averaging methods have been developed, there is little literature on the theoretical properties of model averaging from the perspective of stability, and the majority of these methods constrain model weights to a simplex. The aim of this paper is to introduce stability from statistical learning theory into model averaging. Thus, we define the stability, asymptotic empirical risk minimization, generalization and consistency of model averaging, and study the relationship among them. Similar to the existing results in literature, we find that stability can ensure that the model averaging estimator has good generalization performance and consistency under reasonable conditions, where consistency means that the model averaging estimator can asymptotically minimize the mean squared prediction error. We also propose an $L_2$-penalty model averaging method without limiting model weights, and prove that it has stability and consistency. In order to overcome selection uncertainty of the $L_2$-penalty parameter, we use cross-validation to select a candidate set of $L_2$-penalty parameters, and then perform a weighted average of the estimators of model weights based on cross-validation errors. We demonstrate the usefulness of the proposed method with a Monte Carlo simulation and application to a prediction task on the wage1 dataset.

Divide and conquer algorithm is a common strategy applied in big data. Model averaging has the natural divide-and-conquer feature, but its theory has not been developed in big data scenarios. The goal of this paper is to fill this gap. We propose two divide-and-conquer-type model averaging estimators for linear models with distributed data. Under some regularity conditions, we show that the weights from Mallows model averaging criterion converge in L2 to the theoretically optimal weights minimizing the risk of the model averaging estimator. We also give the bounds of the in-sample and out-of-sample mean squared errors and prove the asymptotic optimality for the proposed model averaging estimators. Our conclusions hold even when the dimensions and the number of candidate models are divergent. Simulation results and a real airline data analysis illustrate that the proposed model averaging methods perform better than the commonly used model selection and model averaging methods in distributed data cases. Our approaches contribute to model averaging theory in distributed data and parallel computations, and can be applied in big data analysis to save time and reduce the computational burden.

Input uncertainty is an aspect of simulation model risk that arises when the driving input distributions are derived or “fit” to real-world, historical data. Although there has been significant progress on quantifying and hedging against input uncertainty, there has been no direct attempt to reduce it via better input modeling. The meaning of “better” depends on the context and the objective: Our context is when (a) there are one or more families of parametric distributions that are plausible choices; (b) the real-world historical data are not expected to perfectly conform to any of them; and (c) our primary goal is to obtain higher-fidelity simulation output rather than to discover the “true” distribution. In this paper, we show that frequentist model averaging can be an effective way to create input models that better represent the true, unknown input distribution, thereby reducing model risk. Input model averaging builds from standard input modeling practice, is not computationally burdensome, requires no change in how the simulation is executed nor any follow-up experiments, and is available on the Comprehensive R Archive Network (CRAN). We provide theoretical and empirical support for our approach.