Metrika (2003) 58: 193–208 DOI 10.1007/s001840200237 > Springer-Verlag 2003 Designs for estimating an extremal point of quadratic regression models in a hyperball Viatcheslav B. Melas1, Andrey Pepelyshev1, Russell C. H. Cheng2 1 St. Petersburg State University, Department of Mathematics, Bibliotechnaya sq., 2, St. Petersburg,198904, Russia (e-mail: v.melas@pobox.spbu.ru) 2 Department of Mathematics, University of Southampton, Highfield, Southampton, SO17 1BJ, United Kingdom Received February 2002 Abstract. This paper is devoted to studying optimal designs for estimating an extremal point of a multivariate quadratic regression model in the unit hyper- ball. The problem of estimating an extremal point is reduced to that of esti- mating certain parameters of a corresponding nonlinear (in parameters) regression model. For this reduced problem truncated locally D-optimal designs are found in an explicit form. The result is a generalization of the results of Fedorov and Mu ¨ ller (1997) for onedimensional quadratic regression function in the unit segment. Key words: Estimating of an extremum point, quadratic regression model, truncated locally D-optimal designs, equivalence theorems 1 Introduction The present paper considers optimal designs for estimating an extremal point of a multivariate quadratic regression model in the unit hyperball. The extremal point in this case is an explicitly given function of parameters of the model. The estimation problem, by a re-parametrization, can be reduced to the problem of estimating certain parameters of a nonlinear (in parameters) regression model. For this reduced problem we derive truncated locally D- optimal design in an explicit form. In practice one often needs to find optimal conditions for a process per- formance. In the pioneer paper (Box, Wilson, 1951) an iteration procedure for solving such problems was suggested. This procedure is based on the approx- imation of unknown dependence by a linear or quadratic function of several variables. However, studying of such problems from the optimal designs point of view begins only in the last years (see papers cited below). Probably the reason of such a delay is connected with that the optimal design theory (see