The Likelihood Pivot: Performing Inference with Confidence

Abstract

Maximum likelihood estimation is a popular statistical method. To account for possible model misspecification, the sandwich estimate of variance can be used to generate asymptotically correct confidence intervals. Several ad hoc attempts have been made to correct for small sample bias in the sandwich, but none of these are a generally accepted correction. After analyzing the original proof of the sandwich estimate of variance, we derive a pivot-based method that uses fewer approximating assumptions than the sandwich method and consequently performs better than the sandwich and its adjustments at small sample sizes. The pivot-based confidence intervals prove better than the sandwich alternatives in multiple simulation studies in both fully parametric and semiparametric scenarios. Further, asymptotic efficiency is proven which shows that the pivot-based confidence intervals perform as well as sandwich-based confidence intervals for large sample sizes. All of these results are shown for both one-dimensional parameters of interest and then for multi-dimensional parameters of interest. This pivot is also explored in a Bayesian setting. We show how to use the pivot in a pseudo-Bayesian analysis. Analyzing this pivot and a monotonic transformation of this pivot show that not all pivots provide equal pseudo-Bayesian confidence interval coverage. From this work a principle is derived that further justifies the use of our pivot. A theoretical result is proven which can be used to determine which transformations of our pivot will also give good confidence interval coverage. A simulation study is performed that shows our pseudo-Bayesian inference provides better confidence interval coverage than standard Bayesian inference, when the likelihood has been misspecified. Finally, a dataset obtained from the Centers for Disease Control is used as a population. We estimate confidence interval coverage generated by our pivot and the sandwich alternatives on subsamples of the dataset for several different models and show that our pivot still performs favorably with real world populations.