P-value and Postrerior Probability for the Interval Hypothesis Testing

Abstract

This article examines the behavior of frequentist $p$-values and Bayesian posterior probabilities for testing interval hypotheses in the Pareto distribution.
Here, we consider the hypothesis $H_{0\epsilon}:|\theta-\theta_0|\leq \epsilon$ against $H_{1\epsilon}:|\theta-\theta_0|>\epsilon$, where $\theta$ is the shape parameter of the Pareto distribution. Our analysis reveals a fundamental divergence between classical and Bayesian approaches: while the $p$-value for the interval hypothesis converges to the point-null $p$-value as $\epsilon \to 0$, the Bayesian posterior probability exhibits no such convergence. This discrepancy persists across various prior distributions, including non-informative, exponential, and conjugate priors, and remains significant even for small values of $\epsilon$. We demonstrate that common approximation methods for reconciling these measures fail for the Pareto distribution, emphasizing the distribution-dependent nature of this problem. The results highlight important limitations in using point-null approximations for interval hypothesis testing, particularly within the Bayesian framework. Our findings contribute to the ongoing discussion about reconciling frequentist and Bayesian evidence and provide guidance for practitioners working with power-law distributions in various application domains.