Mettika" Metrika (1997) 46:147-157 Another View on Optimal Design for Estimating the Point of Extremnm in Quadratic Regression VALERY V. FEDOROV Mathematical SciencesSection, Oak Ridge National Laboratories, PO Box 2008, TN 37831-6367, USA WERNER G. MfdLLER t Department of Statistics, University of Economics, Augasse 2-6, 1090 Vienna, Austria Abstract: In this paper we illustrate how certain design problems can be simplifiedby reparamet- rization of the response function. This alternative viewpoint provides further insights than the more traditional approaches, like minimax, Bayesian or sequential techniques. It will also improve a practitioner's understanding of more general situations and their "classical" treatment. Key Words." Optimum design, turning point, reparametrization. 1 Introduction The problem of optimal design for estimating the point of extremum -f12/2f13 in the quadratic regression model Yi : fll + fl2X'i + f13X2 + ~i, i : 1,..., n (1) where y~ is an observation at point -1 _< x~ _< t, E[e~J = 0, E[e~eJ -= 1 and all observations are independent, has received considerable attention in the stat- istical literature. Suggested solutions to this nonlinear design problem (or the similar one with the restriction fit = 0) include sequential and batch sequen- tial procedures, see Ford and Silvey (1980) and Miiller and P6tscher (1992), and Bayesian or minimax approaches, see Mandal (1978), Chaloner (1989) i This paper was initiated when Werner G. Miiller visited the Department of Statistics and Actuarial Sciences of the University of Iowa and his research was partially supported by the Fulbright Commission Mutual Educational Exchange Grant, the Exportakademiestipendium der Bundeswirtschafts-kammerand the OeNB-WU-F6rderungspreis. 0026-1335/97/46:2/147-157 $2.50 © 1997 Physica-Verlag, Heidelberg