E-optimal designs for polynomial regression without intercept |
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| Institution: | 1. National Sun Yat-sen University, Taiwan;2. Fakultät für Mathematik, Otto-von-Guericke Universität, Postfach 4120, D-39016 Magdeburg, Germany;2. Sino-Africa Joint Research Center, Chinese Academy of Sciences, Kunming, 650223, China;3. Institute for Biotechnology Research, Jomo Kenyatta University of Agriculture and Technology, Nairobi 00200, Kenya;4. Veterinary Research Institute, Kenya Agriculture and Livestock Research Organization, Nairobi 00200, Kenya;5. Directorate of Veterinary Services, State Department of Livestock, Ministry of Agriculture, Livestock and Fisheries, Nairobi 00625, Kenya;11. State Key Laboratory for Conservation and Utilization of Bio-Resources in Yunnan, Yunnan University, Kunming 650091, China |
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| Abstract: | We give all E-optimal designs for the mean parameter vector in polynomial regression of degree d without intercept in one real variable. The derivation is based on interplays between E-optimal design problems in the present and in certain heteroscedastic polynomial setups with intercept. Thereby the optimal supports are found to be related to the alternation points of the Chebyshev polynomials of the first kind, but the structure of optimal designs essentially depends on the regression degree being odd or even. In the former case the E-optimal designs are precisely the (infinitely many) scalar optimal designs, where the scalar parameter system refers to the Chebyshev coefficients, while for even d there is exactly one optimal design. In both cases explicit formulae for the corresponding optimal weights are obtained. Remarks on extending the results to E-optimality for subparameters of the mean vector (in heteroscdastic setups) are also given. |
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