Objective Bayesian Inference for Differential Effects in Unequal-Variance Two-Sample Normal Models

In this paper, we develop a unified objective Bayesian framework for inference on the differential effect in the unequal-variance two-sample normal model. Although general theories of objective priors are well established, the higher-order matching properties of priors for this specific parameter have not been fully characterized. Using a model-specific orthogonal parametrization, we derive reference priors and first- and second-order probability matching priors. The main methodological contribution is to show that the proposed second-order matching prior simultaneously satisfies posterior-quantile, alternative-coverage, highest posterior density, cumulative distribution function, and conditional-likelihood-ratio matching criteria. In contrast, the reference priors considered in this study satisfy only the first-order matching criterion. We also establish general conditions for posterior propriety under a broad class of noninformative priors. Simulation studies show that the proposed second-order matching prior generally provides frequentist coverage closer to the nominal levels than the reference priors, including in small-sample and unequal-variance settings. These results provide a theoretically justified and practically useful default prior for objective Bayesian inference on differential effects.