ABSTRACT
D. STEPHEN COAD University of Sussex, Falmer, Brighton, England
1. INTRODUCTION Ford and Silvey (1980) proposed an adaptive design for estimating the point at which a regression function attains its minimum. In their model, there are potential observations of the form
where xk are design points chosen from the interval — 1 < x < 1, θ\ and θ2 > 0 are unknown parameters, and €\,€2, ... are i.i.d. standard normal random variables. Interest centered on estimating the value —Θλ/2Θ2 at which the regression function attains its minimum. After examining the asymptotic variance of the maximum likelihood estimator, Ford and Silvey proposed the following adaptive design: first take observations at x x = — 1 and x2 = +l; thereafter, if (xk, yk), k = 1 have been determined, let y~ or y„ denote the sum of yk for which k < n and xk = — 1 or xk = -hi, respectively; take the next observation at xn+\ = +1 if \y*\ < \y~\ and take the next observation at = —1 otherwise. After the experiment is run,
investigators may want confidence intervals, and a problem arises here. Owing to the adaptive nature of the design, it is not the case that the max imum likelihood estimators, say θηΛ and θη2, have a bivariate normal dis tribution. This problem was addressed by Ford et al. (1985) and Wu (1985). The former paper proposed an exact solution which seems overly conserva tive. The latter proposed an asymptotic solution which, at a practical level, ignores the adaptive nature of the design. This paper contains an asymptotic solution which does not ignore the adaptive nature of the design.
