Admissibility under the Frequentist′s Validity Constraint in Estimating the Loss of the Least-Squares Estimator

We consider the problem of estimating the sum of squared error loss L = |β−β̂|2 of the least-squares esitmator β̂ for β, the regression coefficient. The standard estimator L̂0 is the expected value of L. Here the error variance is assumed to be known. Previous results of Johnstone (1988. In Statistical Decision Theory and Related Topics IV (S. Gupta and J. Berger, Eds.), 1, 361-379, Springer-Verlag, New York) show that L̂0 is inadmissible under the loss (L̂−L)2 if the dimension of β̂ is five or more. However, since we are estimating the loss, a typical frequentist principle will lead to the usage of estimators which are frequentist valid. Johnston′s improved esitmator, however, violates this principle. In this paper, we prove that it is impossible to improve upon L̂0 among the class of frequentist valid estimators. The work parallels Hwang and Brown (1991, Ann. Statist.10 1964-1977) for the corresponding confidence set problems, although the argument is entirely different and much simpler.