On Improved Loss Estimation for Shrinkage Estimators

TL;DR

This paper explores methods for estimating the loss of shrinkage estimators, especially in normal and spherically symmetric distributions, emphasizing improved loss estimators and comparisons with Bayesian and unbiased approaches.

Contribution

It provides an expository overview of loss estimation techniques for shrinkage estimators, highlighting improvements and extensions in normal and spherically symmetric cases.

Findings

Enhanced loss estimators for shrinkage methods are developed.

Comparisons show advantages over unbiased estimators.

Bayesian approaches are also discussed.

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.