Using confidence distribution sampling to visualize confidence sets

Abstract

This paper presents a new sampling-based methodology designed to fa- cilitate the visual analysis of the confidence sets generated by an inference function such as the likelihood. This methodology generates a sample of parameters from a confidence distribution. This distribution is designed so that its probabilities on the parameter space are equal to the asymptotic coverage probabilities of the targeted confidence sets. Plotting these samples provides a picture of the inference function surface around the point estimator optimizing the inference function. Once the sample is created, one can also picture the profile inference function confidence sets for various functions of the parameters, all without further numerical optimization. The result is similar to a Bayesian analysis based on samples from the posterior. One distinction is that we can target the samples to obtain a clearer picture of the confidence set boundary. We illustrate the methodology with four different inference functions. Although this methodology is related to Fisher's concept of fiducial inference, the fiducial-like confidence distribution we create here is chosen for its ability to recover the confidence sets generated by the inference function and for its ease in computation, nothing more. Unlike resampling methods such as the parametric bootstrap, our method uses only the original data set, as in Bayesian inference. We use illustrative examples to compare our sampling-based confidence sets with those based on numerical optimization, and to compare the confidence regions generated by different inference functions.