Universal Domination and Stochastic Domination: U-Admissibility and U-Inadmissibility of the Least Squares Estimator

Assume the standard linear model Xn×1 = An×p θp×1 + εn×1, where ε has an n-variate normal distribution with zero mean vector and identity covariance matrix. The least squares estimator for the coefficient θ is θ^ ≡ (A′A)−1A′X. It is well known that θ^ is dominated by James-Stein type estimators under the sum of squared error loss |θ−θ^|2 when p ≥ 3. In this article we discuss the possibility of improving upon θ^, simultaneously under the "universal" class of losses: {L(|θ - θ^|) : L (.) any nondecreasing function} An estimator that can be so improved is called universally inadmissible (U-inadmissible). Otherwise it is called U-admissible. We prove that θ^ is U-admissible for any p when A′A = I. Furthermore, if A′A ≠ I, then θ^ is U-inadmissible if p is "large enough." In a special case, p ≥ 4 is large enough. The results are surprising. Implications are discussed.