Admissibility as a Touchstone

Abstract

Consider the problem of estimating simultaneously the means $\theta_i$ of independent normal random variables $x_i$ with unit variance. Under the weighted quadratic loss $L(\theta, a) = \sum_i\lambda_i(\theta_i - a_i)^2$ 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 $\theta_i$ is estimated using observations other than $x_i$. 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 $\theta_i$ are square summable, the estimator $\delta_i(x) \equiv 1$ is admissible for $\lambda_i = e^{-ic}, c > \frac{1}{2}$, but inadmissible for $\lambda_i = 1/i^{1+c}, c > 0$. In the latter case, a dominating estimator $\pi = (\pi_1, \pi_2, \cdots)$ is of the form $\pi_i(x) = 1 - \varepsilon_i(x)$, where $\varepsilon_i$ links all the observations $x_1, x_2, \cdots$. Infinite models frequently arise in estimation problems for Gaussian processes. For example, in estimating the drift function $\theta$ of the Wiener process $W$ under the loss $L(\theta, a) = \int\lbrack\theta(t) - a(t)\rbrack^2 dt$, the transformation $x_i = \int \Phi_i dW$ with $\Phi_i$ an appropriate complete orthonormal sequence gives rise to a model which is equivalent to an infinite model with $\lambda_i = 1/i^2$.