Admissibility as a touchstone (Q1820521)

Consider the problem of simultaneously estimating the means \(\theta_ i\) of independent normal random variables \(X_ i\) with unit variance and loss function \(L(\theta,a)=\sum_{i}\lambda_ i(\theta_ i-a_ i)^ 2\) with \(\lambda_ i>0\). In the finite dimensional case, it is known that an estimator which is admissible with one set of weights (the \(\lambda_ i)\) is admissible for all sets of weights. In this paper, dimensionality is infinite, the \(\theta_ i\) are square summable and the \(\lambda_ i\) are summable. The estimator \(a_ i\equiv 1\) is shown to be admissible for \(\lambda_ i=e^{-ai}\) \((a>1/2)\) and inadmissible for \(\lambda_ i=1/i^{1+c}\) \((c>0)\).