Confidence distributions and hypothesis testing

Abstract

The traditional frequentist approach to hypothesis testing has recently come under extensive debate, raising several critical concerns. Additionally, practical applications often blend the decision-theoretical framework pioneered by Neyman and Pearson with the inductive inferential process relied on the p-value, as advocated by Fisher. The combination of the two methods has led to interpreting the p-value as both an observed error rate and a measure of empirical evidence for the hypothesis. Unfortunately, both interpretations pose difficulties. In this context, we propose that resorting to confidence distributions can offer a valuable solution to address many of these critical issues. Rather than suggesting an automatic procedure, we present a natural approach to tackle the problem within a broader inferential context. Through the use of confidence distributions, we show the possibility of defining two statistical measures of evidence that align with different types of hypotheses under examination. These measures, unlike the p-value, exhibit coherence, simplicity of interpretation, and ease of computation, as exemplified by various illustrative examples spanning diverse fields. Furthermore, we provide theoretical results that establish connections between our proposal, other measures of evidence given in the literature, and standard testing concepts such as size, optimality, and the p-value.