Spectral methods in linear minimax estimation

Abstract

We consider the minimax-linear estimator in a linear regression model with circular constraints. Two necessary and sufficient conditions for the optimality of an estimator, the socalled left spectral equation and the right spectral equation (Girko spectral equation), are derived. For the special case of a simple maximal eigenvalue and a single eigenspace explicit estimation formulas are derived. These formulas also show some of the shortcomings of the minimax-linear estimator (MILE). Finally, the relation with Bayesian analysis and the Hoffmann-Liuter estimator is outlined. (1991): 62J05. Mathematics Subject Classification

Linear estimation in regression analysis using fuzzy prior information

“…Kuks/Olman (1972), L uter (1975), Hoffmann (1979), Toutenburg (1982), Pilz (1986), Trenkler/Stahlecker (1987), Gaffke/Heiligers (1989), Drygas (1993), Christopeit/Helmes (1996) and Girko (1996)) we observe that V is assumed to be known and that the prior knowledge on β is represented by a non-empty compact subset Β of K k . Thus, a linear minimax estimator b* = C*y is determined by minimizing the maximum value of the risk:…”

“…Here and throughout this section, C G 7£* xn and Z) G 7l 5 xn are always assumed to satisfy equation (6). As D…”

Quasi-Minimax Estimation in the Linear Model with Measurement Errors

“…Wilczyński (2007) obtained some results under the condition that the matrices X X A and T have the same eigenvectors. More contributions can be found in, e.g., Läuter (1975), Hoffmann (1979), Pilz (1986), Trenkler and Stahlecker (1987), Gaffke and Heiligers (1989), Pilz (1991), Drygas (1993), Rao and Toutenburg (1999), Blaker (2000), and Wilczyński (2005). Recently, Yang et al (2009) use the matrix condition number to obtain an explicit solution which can be achieved under the simpler assumptions.…”

On the equivalence of spectral theory and bayesian analysis in minimax linear estimation

“…More general linear models can be brought into this form by reparametrization and transformation (Drygas, 1991(Drygas, , 1996). The B~B-minimax linear estimator (B~B-MILE) is obtained by minimizing sup E(II B(Cu + d -p) 112), (3) /36~…”