On the equivalence of spectral theory and Bayesian analysis in minimax linear estimation
We consider the linear model \[ {\underset n\times {1} y} ={\underset n\times {k} X} {\underset k\times {1}\beta} +{\underset n\times {1} \varepsilon}, \quad E\varepsilon =0, \quad E\varepsilon \varepsilon'= I_n \] under the circular constraints \(\beta\in {\mathfrak B}= \{\beta \in\mathbb{R}^k: \beta' \beta\leq 1\}\). More general linear models can be brought into this form by reparametrization and transformation [see, e.g., the first author, ibid., 17-42 (1996; Zbl 0844.62011)]. The \(B'B\)-minimax linear estimator \((B'B\)-MILE) is obtained by minimizing \[ \sup_{\beta \in{\mathfrak B}} E\biggl(\bigl\| B(Cy+d -\beta) \bigr\|^2 \biggr), \] where \(B\in \mathbb{R}_{m\times k}\), \(\text{rank} (B)=m\), \(C\in \mathbb{R}_{k\times n}\), \(d\in \mathbb{R}_{k\times 1}\). Only \(m=1\) gives an elementary formula, the ridge estimator. The second author [J. Stat. Plann. Inference 13, 297-318 (1986; Zbl 0602.62054)] has shown that \(d=0\) is the optimal choice, since \({\mathfrak B}\) is symmetric. In this paper, we deal with the case \(m=k\). The more general case \(m\leq k\) will be dealt with in a subsequent paper.
scientific article; zbMATH DE number 885564