A sequential design for estimating a nonlinear parametric function Kamel Rekab a, * , Mohamed Tahir b a Department of Mathematics, Florida Institute of Technology, Melbourne, FL 32901, USA b U.A.E. University, Al Ain, United Arab Emirates Abstract A fully-sequential design for estimating a nonlinear function of the parameters in the simple linear regression model is proposed and its asymptotic behavior is investigated both theoretically and by simulation. The design requires that the observations be taken at x ¼1 and specifies whether the next observation is to be taken at x ¼1 or 1. It is shown that, under this design, the mean number of observations taken at x ¼ 1, m k , converges with probability one to an optimal value as k !1, where k denotes the total number of design points. The simulation study indicates that m k converges in L 2 to the optimal value with the order of Oðk 2 Þ. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Asymptotically optimal fixed design; Fisher information matrix; Least squares estimator; Fully-sequential design; Second order approximation; Simple linear regression model 1. Introduction A major difficulty in designing an experiment to estimate a nonlinear parametric function is that the performance of the design depends on the un- known parameters in the model. To utilize the information fully, the experi- ment must be conducted sequentially, in the sense that the choice of the next design point is determined by the estimates of the parameters obtained from the observations made to date. Ford and Silvey [2] constructed a sequential design, in the sense mentioned above, to estimate a particular function of the Applied Mathematics and Computation 138 (2003) 113–120 www.elsevier.com/locate/amc * Corresponding author. E-mail address: rekab@cs.fit.edu (K. Rekab). 0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII:S0096-3003(02)00113-3