Dual extremal problems and their applications to minimax estimation problems

Abstract

ContentsIntroductionChapter I. Duality of non-convex extremal problems §1.0. Definitions §1.1. Conjugate functions §1.2. Characterization of convexity in terms of smoothness §1.3. Minimax theorems §1.4. A function conjugate to the maximum of a family of quadratic forms §1.5. Duality of non-convex extremal problems in the case of smoothness of the dual problemChapter II. Minimax estimation of parameters in linear models of observation with unspecified statistics of the second order §2.1. Formulation of the problem §2.2. Duality theorem §2.3. Minimax estimators and the least squares method §2.4. Linearity of minimax estimatorsChapter III. Applications to the theory of ill-posed problems §3.1. Formulation of the problem §3.2. Duality theorem §3.3. The Kuks-Ol'man estimator §3.4. Non-symmetric constraints on the regression parameters Bibliography