On Improved Loss Estimation for Shrinkage Estimators

Abstract

Let $X$ be a random vector with distribution $P_{\theta}$ where $\theta$ is an unknown parameter. When estimating $\theta$ by some estimator $\varphi(X)$ under a loss function $L(\theta,\varphi)$, classical decision theory advocates that such a decision rule should be used if it has suitable properties with respect to the frequentist risk $R(\theta,\varphi)$. However, after having observed $X=x$, instances arise in practice in which $\varphi$ is to be accompanied by an assessment of its loss, $L(\theta,\varphi(x))$, which is unobservable since $\theta$ is unknown. A common approach to this assessment is to consider estimation of $L(\theta,\varphi(x))$ by an estimator $\delta$, called a loss estimator. We present an expository development of loss estimation with substantial emphasis on the setting where the distributional context is normal and its extension to the case where the underlying distribution is spherically symmetric. Our overview covers improved loss estimators for least squares but primarily focuses on shrinkage estimators. Bayes estimation is also considered and comparisons are made with unbiased estimation.