Model-averaged confidence distributions

Abstract

Model averaging is commonly used to allow for model uncertainty in parameter estimation. As well as providing a point estimate that is a natural compromise between the estimates from different models, it also provides confidence intervals with better coverage properties, compared to those based on a single best model. In recent years, the concept of a confidence distribution has been promoted as a frequentist analogue of a Bayesian posterior distribution. The confidence distribution for a parameter is a visual representation of the set of $$100(1-\alpha )\%$$ confidence intervals for all possible $$\alpha $$, and was first proposed over 60 years ago. The purpose of this paper is to promote the use of model-averaged confidence distributions. One of the advantages of doing so is the ability to see unusual shapes in the distribution, such as multi-modality. This allows a more comprehensive assessment of the uncertainty about the parameter of interest, in exactly the same way that a model-averaged posterior distribution can be more useful than a model-averaged credible interval. We show that the model-averaged tail-area (MATA) method for calculating a model-averaged confidence interval leads to the corresponding MATA confidence distribution being a mixture of the confidence distributions associated with the individual models, the mixing being determined by the model weights. We consider two ecological examples that illustrate the advantages of a model-averaged confidence distribution over a model-averaged confidence interval.