# Optimal designs in regression problems with a general convex loss function

Abstract SUMMARY The polynomial regression model in which the errors are independent and identically distributed, is considered, and attention is focused on the estimate of β 8 , 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 β 8 , when the design D is used, is donoted by β( D ). Then D * is said to dominate D if, for every continuous convex function c , Ec{β(D * )})≤= Ec{β(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. © Oxford University Press http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Biometrika Oxford University Press

## Optimal designs in regression problems with a general convex loss function

Abstract

Abstract SUMMARY The polynomial regression model in which the errors are independent and identically distributed, is considered, and attention is focused on the estimate of β 8 , 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 β 8 , when the design D is used, is donoted by β( D ). Then D * is said to dominate D if, for every continuous convex function c , Ec{β(D * )})≤= Ec{β(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. © Oxford University Press

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