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Biometrika 1968 55(1):53-66; doi:10.1093/biomet/55.1.53
© 1968 by Biometrika Trust
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Optimal designs in regression problems with a general convex loss function

P. J. LAYCOCK and S. D. SILVEY

University of Glasgow

The polynomial regression model in which the errors are independent and identically distributed, is considered, and attention is focused on the estimate of ß8, the coefficient, of the term of highest degree. A design D is a set of n values in [–1, 1] of the concomitant, variable x, and the least-squares estimator of ß8, when the design D is used, is donoted by ß(D). Then D* is said to dominate D if, for every continuous convex function c,

E[c{ß(D*)})≤= E[c{ß(D)}.

Does there exist a design D which dominates all others in this sense ? In general the answer is no, though for certain values of n and any symmetric error distribution the answer is yes. A crucial property of Tchebycheff designs emerges from discussion of this problem and this supports their use in practice. In addition two useful methods for comparing different designs are suggested.


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