Bayesian and Robust Bayesian analysis under a general class of balanced loss functions

Abstract

For estimating an unknown parameter θ, we introduce and motivate the use of balanced loss functions of the form ${L_{\rho, \omega, \delta_0}(\theta, \delta)=\omega \rho(\delta_0, \delta)+ (1-\omega) \rho(\theta, \delta)}$, as well as the weighted version ${q(\theta) L_{\rho, \omega, \delta_0}(\theta, \delta)}$, where ρ(θ, δ) is an arbitrary loss function, δ ₀ is a chosen a priori “target” estimator of ${\theta, \omega \in[0,1)}$, and q(·) is a positive weight function. we develop Bayesian estimators under ${L_{\rho, \omega, \delta_0}}$ with ω > 0 by relating such estimators to Bayesian solutions under ${L_{\rho, \omega, \delta_0}}$ with ω = 0. Illustrations are given for various choices of ρ, such as absolute value, entropy, linex, and squared error type losses. Finally, under various robust Bayesian analysis criteria including posterior regret gamma-minimaxity, conditional gamma-minimaxity, and most stable, we establish explicit connections between optimal actions derived under balanced and unbalanced losses.