Admissibility as a touchstone

Consider the problem of estimating simultaneously the means θi of independent normal random variables xi with unit variance. Under the weighted quadratic loss L(θ,a)=∑iλi(θi−ai)2 with positive weights it is well known that: (1) An estimator which is admissible under one set of weights is admissible under all weights. (2) Estimating individual coordinates by proper Bayes estimators results in an admissible estimator. (3) Estimating individual coordinates by admissible estimators may result in an inadmissible estimator, when the number of coordinates is large enough. A dominating estimator must link observations in the sense that at least one θi is estimated using observations other than xi. We consider an infinite model with a countable number of coordinates. In the infinite model admissibility does depend on the weights used and by linking coordinates it is possible to dominate even estimators which are proper Bayes for individual coordinates. Specifically, we show that when θi are square summable, the estimator δi(x)≡1 is admissible for λi=e−ic,c>12, but inadmissible for λi=1/i1+c,c>0. In the latter case, a dominating estimator π=(π1,π2,⋯) is of the form πi(x)=1−εi(x), where εi links all the observations x1,x2,⋯. Infinite models frequently arise in estimation problems for Gaussian processes. For example, in estimating the drift function θ of the Wiener process W under the loss L(θ,a)=∫[θ(t)−a(t)]2dt, the transformation xi=∫ΦidW with Φi an appropriate complete orthonormal sequence gives rise to a model which is equivalent to an infinite model with λi=1/i2.