Bounded Length Confidence Intervals for the Zero of a Regression Function

The problem of determining a bounded length confidence interval for the zero of a regression function R(·) is discussed. In case R(·) = F(·) - p, F a distribution function, 0 ≥ p ≥ 1, a closed stopping rule is given for the up-down method of experimentation. For a larger class of regression functions a closed stopping rule is given for Robbins-Monro type of experimentation. The stopping rule for the Robbins-Monro process depends on prior knowledge of an upper and a lower bound on the zero of R(·). It is shown that given suitable assumptions about the random variables used in experimentation finite confidence intervals for the zero of R(·) may be found, such confidence intervals providing an upper and a lower bound on the zero of R(·) with prespecified level of confidence.