Binary response models are used in estimating the probability, P(x), of a successful response when an experimental unit is exposed to a stimulus level, x. Frequently, the aim of the investigator is to locate a specific level of stimulus for which P(x) takes on a known value, p. The stimulus level sought, x$\sb{\rm 100p}$, is referred to as the p$\rm \sp{th}$ quantile of P(x). When p lies outside the interval (.25,.75), x$\rm \sb{100p}$ is considered to be an extreme quantile. The focus of this paper is the estimation of the extreme x$\rm \sb{100p}$.

All estimation techniques for extreme quantiles seek, with varying degrees of success to limit estimate bias, lack of precision, and the importance of parametric assumption. The extreme quantile estimate described here utilizes, through a transformation of the responses, median estimation techniques where such problems are not so acute. The feasibility of the approach is demonstrated in a Monte Carlo study. It is shown that the design aspect of the approach has properties important to c-optimality, and it is argued that most data sets should be fit well by the model employed. This paper also explores the relationship between data structure and the existence of maximum likelihood estimates in finite samples.