Bounded Length Confidence Intervals for the Zero of a Regression Function

Abstract

The problem of determining a bounded length confidence interval for the zero of a regression function $R(\cdot)$ is discussed. In case $R(\cdot) = F(\cdot) - p, F$ a distribution function, $0 \geq p \geq 1,$ 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 $R(\cdot)$. It is shown that given suitable assumptions about the random variables used in experimentation finite confidence intervals for the zero of $R(\cdot)$ may be found, such confidence intervals providing an upper and a lower bound on the zero of $R(\cdot)$ with prespecified level of confidence.