Minimax regression designs for approximately linear models with autocorrelated errors |
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| Institution: | 1. Department of Mathematical Sciences, University of Alberta, Edmonton, Alta., Canada T6G 2G1;2. Department of Mathematical Sciences, Lakehead University, Thunder Bay, Ont., Canada P7B 5E1;1. School of Business, Western Sydney University, Parramatta, NSW, 2150 Australia;2. School of Marketing, University of New South Wales, Sydney, NSW, 2052, Australia;3. Faculty of Business and Economics, University of Auckland, Auckland, New Zealand;1. Department of Epidemiology and Public Health, Tokyo Dental College, Tokyo, Japan;2. Tokyo Dental College Suidobashi Hospital, Tokyo, Japan;1. Department of Mathematics, Covenant University, Canaanland, Ota, Nigeria;2. Department of Electrical and Information Engineering, Covenant University, Canaanland, Ota, Nigeria;3. Department of Mathematics, University of Lagos, Akoka, Lagos, Nigeria;1. Health Services Management, School of Public Health, Social Determinants of Health Research Center, Yasuj University of Medical Sciences, Yasuj, Iran;2. Research Center for Health Sciences, Institute of Health, Department of Environmental Health, School of Health, Shiraz University of Medical Sciences, Shiraz, Iran;3. Behvarz Training Center, Torbat Heydariyeh University of Medical Sciences, Torbat Heydariyeh, Iran;4. Medical Toxicology and Drug Abuse Research Center (MTDRC), Birjand University of Medical Sciences (BUMS), Birjand, Iran;5. Department of Environmental Health Engineering, School of Public Health, Social Development & Health Promotion Research Center, Gonabad University of Medical Sciences, Gonabad, Iran;6. Department of Environmental Engineering, School of Engineering and Technology, Murdoch University, Western Australia, Australia;7. Social Determinates of Health Research Center, Saveh University of Medical Sciences, Saveh, Iran |
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| Abstract: | We study the construction of regression designs, when the random errors are autocorrelated. Our model of dependence assumes that the spectral density g(ω) of the error process is of the form g(ω) = (1 − α)g0(ω) + αg1(ω), where g0(ω) is uniform (corresponding to uncorrelated errors), α ϵ 0, 1) is fixed, and g1(ω) is arbitrary. We consider regression responses which are exactly, or only approximately, linear in the parameters. Our main results are that a design which is asymptotically (minimax) optimal for uncorrelated errors retains its optimality under autocorrelation if the design points are a random sample, or a random permutation, of points from this distribution. Our results are then a partial extension of those of Wu (Ann. Statist. 9 (1981), 1168–1177), on the robustness of randomized experimental designs, to the field of regression design. |
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