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OPTIMAL DESIGNS WITH RESPECT TO ELFVING'S PARTIAL

MINIMAX CRITERION IN POLYNOMIAL REGRESSION

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HOLGER DETTE^{1} AND WILLIAM J. STUDDEN^{2}

^{1} *Institut für Mathematische Stochastik, Universität Göttingen and Universität Dresden,*

Mommsenstr. 13, 01062 Dresden, Germany

^{2} *Department of Statistics, Purdue University, 1399 Mathematical Sciences Bldg.,*

West Lafayette, IN 47907-1399, U.S.A.
(Received December 21, 1992; revised April 26, 1993)

**Abstract.**
For the polynomial regression model on the
interval [*a*,*b*] the optimal design problem with respect to
Elfving's minimax criterion is considered. It is shown that the
minimax problem is related to the problem of determining optimal
designs for the estimation of the individual parameters. Sufficient
conditions are given guaranteeing that an optimal design for an
individual parameter in the polynomial regression is also minimax
optimal for a subset of the parameters. The results are applied to
polynomial regression on symmetric intervals [-*b*,*b*] (*b* __<__ 1)
and on nonnegative or nonpositive intervals where the conditions
reduce to very simple inequalities, involving the degree of the
underlying regression and the index of the maximum of the absolute
coefficients of the Chebyshev polynomial of the first kind on the
given interval. In the most cases the minimax optimal design can be
found explicitly.

*Key words and phrases*:
Approximate design theory,
scalar optimality, minimax criterion, polynomial regression.

**Source**
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