Corrected confidence sets for sequentially designed experiments: Examples

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.