Universal Domination and Stochastic Domination: Estimation Simultaneously Under a Broad Class of Loss Functions

Abstract

In Wald's statistical decision theory, the criterion of domination (or uniform betterness) is defined with respect to a specific loss. In practice, however, the exact form of a loss function is difficult to specify. Hence, it is important to study the domination criterion simultaneously under a class of loss functions. In this paper we focus on estimation problems. We mainly investigate the possibility of domination simultaneously under the class of loss functions $L(|\theta - \delta|)$, where $L$ is an arbitrary nondecreasing function. As usual, $\theta$ and $\delta$ (both in $p$-dimensional Euclidean space $R^p$) are, respectively, the unknown parameter of nature and the statistician's estimate. Domination simultaneously under this class of losses is called universal domination under Euclidean error. Several theoretical questions are resolved in this paper. In particular the criterion of universal domination is shown to be equivalent to the criterion of stochastic domination that compares the estimators by the stochastic ordering of their Euclidean distances from the estimators to the true parameter. Concrete results about universal domination relating to the usual estimator are also established. In particular when $X - \theta$ has a $p$-variate $t$ distribution, and $p = 1, 2$, there exists no estimator for $\theta$ that universally dominates $X$; however, for $p \geq 3$, estimators (of the type of James-Stein positive part estimators) that universally dominate $X$ are specified. When $X$ has a $p$-variate normal distribution with mean $\theta$ and identity covariance matrix, we show that for any dimension $p$, no James-Stein positive part estimators universally dominate $X$. However, under slightly smaller classes of losses, some James-Stein positive part estimators are shown to simultaneously dominate $X$. These hitherto unstudied losses are bounded and fairly practical.