# Computing Linear Mini-Max Estimators

• Kurt Helmes
• C. Srinivasan
Part of the Theory and Decision Library book series (TDLC, volume 18)

## Abstract

Consider a vector of data y = θ + ε, y ∈ℝn, where ε = (εi)1 ≤i≤n is an er-ror vector which satisfies the “usual” conditions, and θ∈Θ ⊂ ℝn. The problem is to find/compute a vector m for which (m, y) minimizes the maximal risk among all linear estimators, i.e. m is a solution of $$\begin{array}{*{20}{c}} {\min }&{\max {\rm E}} \\ {m \in {\mathbb{R}^n}}&{\theta \in \Theta } \end{array}\left[ {\left\langle {\ell ,\theta } \right\rangle - {{\left\langle {m,\theta } \right\rangle }^2}} \right],\ell \in {\mathbb{R}^n}$$ given. A solution of this mini-max problem is determined by the solution of the fractional optimization problem $$\mathop {\max }\limits_{\theta \in \Theta } \frac{{{{\left\langle {l,\theta } \right\rangle }^2}}}{{1 + {{\left\| \theta \right\|}^2}}}$$ We present an efficient method to solve the fractional programming problem for the special case when θ is a symmetric, bounded set described by linear inequalities, i.e. Θ = {θ | Aθ ≤ b}. We also report on studies where the new method has been compared with direct approaches to the non-linear optimization problem.

## Keywords

Fractional Programming Problem Maximal Risk Quadratic Loss Function Minimax Risk Linear Regression Estimation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. [1]
Bertsekas, D.P. (1995). Nonlinear Programming. Athena Scientific, Belmont.Google Scholar
2. [2]
Drygas, H. (1991). On an extension of the Girko inequality in linear minimax estimation. Proceeding of the Probastat ′91 Conference, Bratislava/CSFR.Google Scholar
3. [3]
Drygas, H. (1993). Spectral methods in linear minimax estimation. Kasseler Mathematische Schriften 4/93. To be published in the Proceedings of the Oldenburg Minimax Workshop (3./4.8.1992).Google Scholar
4. [4]
Drygas, H. and Lauter (1993). On the representation of the linear minimax estimator in the convex linear model. Kasseler Mathematische Schriften No. 7/93.Google Scholar
5. [5]
Gaffke, N. and Heiligers (1989). Bayes, admissible, and minimax linear estimators in linear models with restricted parameter space. Statistics 20, 487–508.
6. [6]
Gaffke, N. and Heiligers (1991). Note on a Paper by P. Alson. Statistics 22, 3–8.
7. [7]
Helmes, K. and Srinivasan (1992). Linear minimax estimation. Springer Verlay Lecture Notes in Economics and Mathematics, 389, 9–23.Google Scholar
8. [8]
Helmes, K. and Christopeit (1996). Linear estimation with ellipsoidal constraints. To appear in Acta Applicandae Mathematicae, vol. 43, No. 1, 3–15.
9. [9]
Hoffmann, K. (1979). Characterization of minimax linear estimators in linear regression. Math. Operationsforsch. u. Statistik, Ser. Statist., 20, 19–26.Google Scholar
10. [10]
Ibragimov, A.D. and Hasminski (1987). Estimation of linear functionals in Gaussian noise. Theory Prob. Appl., vol. 32, No. 1, 30–39.
11. [11]
Ibragimov, A.D. and Hasminski (1984). On non-parametric estimation of the value of a linear function in Gaussian white noise. Theory of Probability and Its Applications, Vol. XXIX, No. 1, 18–32.Google Scholar
12. [12]
Lauter, H. (1975). A minimax linear estimator for linear parameters under restrictions in form of inequalities. Math. Operationsforsch. u. Statist., 6, Heft 5, 689–695.
13. [13]
Nash, S.G. and Sofer (1996). Linear and Nonlinear Programming. McGraw-Hill, New York.Google Scholar
14. [14]
Pilz, J. (1986) . Minimax linear regression estimation with symmetric parameter restrictions. J. Statist. Plann. Inference 13, 297–318.Google Scholar
15. [15]
Pilz, J. (1991) . Bayesian estimation and experimental design in linear regression models. 2nd edition, Wiley, Chichester, New York.Google Scholar
16. [16]
Pinelis, I.F. (1988). On minimax risk. Theory Prob. Appl., vol. 33, 104–109.Google Scholar
17. [17]
Ritter, K. (1995). Asymptotic optimality of regular sequence design. To appear in Annals of Statistics. Google Scholar
18. [18]
Stahlecker, P. and Drygas (1992). Representation theorems in linear minimax estimation. Report No. V-85–92, University of Oldenburg.Google Scholar
19. [19]
Stahlecker, P., Janner and Schmidt (1991). Linear-affine Minimax-Schatzer unter Ungleichungsrestriktionen. Allg. Statist. Archiv, 75, 245–264.Google Scholar
20. [20]
Stahlecker, P. and Trenkler (1991). Linear and ellipsoidal restrictions in linear regression. Statistics 22, 163–176.