Optimal Design: An Introduction to the Theory for Parameter Estimation, Volume 10Prior to the 1970's a substantial literature had accumulated on the theory of optimal design, particularly of optimal linear regression design. To a certain extent the study of the subject had been piecemeal, different criteria of optimality having been studied separately. Also to a certain extent the topic was regarded as being largely of theoretical interest and as having little value for the practising statistician. However during this decade two significant developments occurred. It was observed that the various different optimality criteria had several mathematical properties in common; and general algorithms for constructing optimal design measures were developed. From the first of these there emerged a general theory of remarkable simplicity and the second at least raised the possibility that the theory would have more practical value. With respect to the second point there does remain a limiting factor as far as designs that are optimal for parameter estimation are concerned, and this is that the theory assumes that the model be collected is known a priori. This of course underlying data to is seldom the case in practice and it often happens that designs which are optimal for parameter estimation allow no possibility of model validation. For this reason the theory of design for parameter estimation may well have to be combined with a theory of model validation before its practical potential is fully realized. Nevertheless discussion in this monograph is limited to the theory of design optimal for parameter estimation. |
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Optimal Design: An Introduction to the Theory for Parameter Estimation S. Silvey Limited preview - 2013 |
Optimal Design: An Introduction to the Theory for Parameter Estimation S. Silvey No preview available - 2013 |
Common terms and phrases
algorithms approximate theory choose concave function consider control variable convergence convex combinations convex cone convex function convex hull convex set criterion function D-optimal defined denote design measure corresponding design measure putting design point design problem duality theorem ellipsoid example exists exp 0₁ Fedorov finite Fisher's information matrix Gâteaux derivative Gƒ(x h₂ Hence HH¹ induced design space interest iteration Lagrangian linear regression linear regression design log det M₁ M₁₁ M₂ maximize log measure ʼn minimum-ellipsoid problem N-observation design n₁ non-linear non-negative definite non-singular information matrix Note observations optimal design measure optimal measure P₁(b parameter positive definite possible practical prior knowledge probability distribution Proof Pukelsheim 1980 random variable random vector Section sequential designs sequentially constructed design singular information matrix step-lengths subset Suppose take the value Theorem 3.7 u₁ variance matrix vector with distribution verify W-algorithm x¹(M xx¹