A FIXED-WIDTH INTERVAL FOR 1/BETA IN SIMPLE LINEAR-REGRESSION

Consider the regression model gamma(i)=beta x(i)+e(i) and the problem of constructing a confidence interval for 1/beta with beta is an element of(0, beta*) where beta*>0. Uniformity down to beta=0 is a major difficulty. In fact, any procedure based on a fixed sample size, will have either infinite expected width or zero confidence [Gleser and Hwang, Ann. Statist, 18 (1987) 1389-1399] confidence being the infimum of the coverage probability. Sequential sampling is used to construct fixed-width intervals of the form (1/($) over cap beta(tau)-h, 1/($) over cap beta(tau)+h), where tau is an integer valued stopping time, ($) over cap beta(tau),is the least-squares estimator for beta based on tau observations and h is the half-length of the interval. Stopping times tau(h) are derived so that these intervals have coverage probabilities converging to a set value gamma as h-->0. This convergence is uniform down to beta=0 Furthermore, the predictors xi may be chosen adaptively.