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

For estimating an unknown parameter θ, we introduce and motivate the use of balanced loss functions of the form L_{r, w, d0}(q, d)=wr(d₀, d)+ (1-w) r(q, d){L_{rho, omega, delta_0}(theta, delta)=omega rho(delta_0, delta)+ (1-omega) rho(theta, delta)}, as well as the weighted version q(q) L_{r, w, d0}(q, d){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 q, w ? [0,1){theta, omega in[0,1)}, and q(·) is a positive weight function. we develop Bayesian estimators under L_{r, w, d0}{L_{rho, omega, delta_0}} with ω > 0 by relating such estimators to Bayesian solutions under L_{r, w, d0}{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.