Title: On prediction and shrinkage estimation for balanced loss functions
Authors: Eric Marchand - Universite de Sherbrooke (Canada) [presenting]
William Strawderman - Rutgers University (United States)
Abstract: The estimation of a multivariate mean $\theta$ is considered under natural modifications of balanced loss function of the form: (i) $\omega \, \rho(\|\delta-\delta_0\|^2) + (1-\omega) \, \rho(\|\delta-\theta\|^2) $, and (ii) $\ell \left( \omega \, \|\delta-\delta_0\|^2 + (1-\omega) \, \|\delta-\theta\|^2 \right)\,$, where $\delta_0$ is a target estimator of $\gamma(\theta)$. After briefly reviewing known results for original balanced loss with identity $\rho$ or $\ell$, we provide, for increasing and concave $\rho$ and $\ell$ which also satisfy a completely monotone property, Baranchik-type estimators of $\theta$ which dominate the benchmark $\delta_0(X)=X$ for $X$ either distributed as multivariate normal or as a scale mixture of normals. Implications are given with respect to model robustness and simultaneous dominance with respect to either $\rho$ or $\ell$. Finally, we present a framework for predictive density estimation under balanced loss functions, we describe Bayesian representations and discuss frequentist performance of various predictive density estimators.