Limiting the Shrinkage for the Exceptional by Objective Robust Bayesian Analysis: the "Clemente Problem"
This paper addresses the “Clemente problem” in statistical inference—excessive shrinkage of outliers (e.g., elite athletes)—arising from the joint use of squared-error loss and light-tailed conjugate priors. To resolve this, we propose a Cauchy–Beta2 joint heavy-tailed prior framework, yielding a closed-form, objective, and robust location prior that unifies empirical Bayes (EB) and full Bayes (FB) approaches. Our method integrates heavy-tailed priors (e.g., Cauchy, double exponential), robust loss functions, and finite-translation estimators. Theoretically and empirically, the proposed paradigm substantially mitigates outlier shrinkage bias and achieves lower mean squared error than James–Stein estimation. Crucially, robust modeling choices exert a far greater impact on inference than the distinction between EB and FB methodologies. This work establishes the first analytically tractable, fully objective, and hyperparameter-free robust Bayesian inference framework—offering a principled alternative to conventional shrinkage estimation.