Beyond Normality: Gain‐Probability Analysis for Symmetric Scale Mixture of Normal Distributions

Abstract

Gain‐Probability (G‐P) analysis quantifies the probability that a randomly selected individual from one group scores higher or lower than an individual from another group, by varying magnitudes. While G‐P methods have been developed under normality and various skewed distributions, symmetric heavy‐tailed settings remain largely unexplored, despite their prevalence in finance, environmental science, and other applied domains. We extend the G‐P framework to the broad family of scale mixtures of normal (SMN) distributions, including the Student's t, slash, variance gamma (VG), and Pearson Type VII distributions. Analytical expressions for G‐P under SMN are derived for both independent and matched data, and parameter estimation is performed using the expectation maximisation (EM) algorithm. Simulation studies show that the proposed estimators are accurate, robust to heavy tails, and improve with sample size, with performance most sensitive to group separation and noise level. An application to daily returns of US and Chinese equity indices demonstrates how G‐P analysis captures distributional tail effects that are overlooked by traditional tests. The results support G‐P analysis under SMN as a practical, interpretable alternative to significance testing, enabling robust inference for symmetric heavy‐tailed data in diverse applied settings.