Abstract
The optimal design problem for the estimation of several linear combinationsc′ l ϑ (l=1, …,m) is considered in the usual linear regression modely=f′(x)ϑ (f(x) ∈ ℝk,ϑ ∈ ℝk). An optimal design minimizes a (weighted)p-norm of the variances of the least squares estimates for the different linear combinationsc′ l ϑ. A generalized Elfving theorem is used to derive the relation of the new optimality criterion to theE-optimal design problem. It is shown that theE-optimal design for the parameterϑ minimizes such a (weighted)p-norm whenever the vectorc=(c′ 1, …, c′k)′ is an inball vector of a symmetric convex and compact “Elfving set” inEquationSource % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWefv3ySLgznf% gDOjdaryqr1ngBPrginfgDObcv39gaiuqacqWFDeIucqWFaCVCdaah% aaWcbeqaaGqaciaa+Tgadaahaaadbeqaaiaa+jdaaaaaaaaa!4497!v\mathbb{R}^{k^2 } $$ .
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Dette, H. A new interpretation of optimality forE-optimal designs in linear regression models. Metrika 40, 37–50 (1993). https://doi.org/10.1007/BF02613663
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DOI: https://doi.org/10.1007/BF02613663