A UNIFIED APPROACH TO SECOND ORDER OPTIMALITY CRITERIA IN NONLINEAR DESIGN OF EXPERIMENTS

HOLGER DETTE^{1} AND YURI GRIGORIEV^{2}*

1Fakultät für Mathematik, Ruhr-Universität Bochum, 44780 Bochum, Germany 2Fakultät für Angewandte Mathematik, Technische Universität, 630092 Novosibirsk, Russian Federation

(Received March 17, 1998; revised June 14, 1999)

Abstract. In the nonlinear regression model we consider the optimal design problem with a second order design D-criterion. Our purpose is to present a general approach to this problem, which includes the asymptotic second order bias and variance criterion of the least squares estimator and criteria using the volume of confidence regions based on different statistics. Under assumptions of regularity for these statistics a second order approximation of the volume of these regions is derived which is proposed as a quadratic optimality criterion. These criteria include volumes of confidence regions based on the u_{n}- representable statistics. An important difference between the criteria presented in this paper and the second order criteria commonly employed in the recent literature is that the former criteria are independent of the vector of residuals. Moreover, a refined version of the commonly applied criteria is obtained, which also includes effects of nonlinearity caused by third derivatives of the response function.

Key words and phrases: D-optimal design, nonlinear regression model, second order approximation, asymptotic expansion.