Up-and-Down and the Percentile-Finding Problem

Up-and-Down (U&D) is a popular sequential design for estimating threshold percentiles in binary experiments. However, U&D application practices have stagnated, and significant gaps in understanding its properties persist. The first part of my work aims to fill gaps in U&D theory. New results concerning stationary distribution properties are proven. A second focus of this study is nonparametric U&D estimation. An improvement to isotonic regression called “centered isotonic regression” (CIR), and a new averaging estimator called “auto-detect” are introduced and their properties studied. Bayesian percentile-finding designs, most notably the continual reassessment method (CRM) developed for Phase I clinical trials, are also studied. In general, CRM convergence depends upon random run-time conditions – meaning that convergence is not always assured. Small-sample behavior is studied as well. It is shown that CRM is quite sensitive to outlier sub-sequences of thresholds, resulting in highly variable small-sample behavior between runs under identical conditions. Nonparametric CRM variants exhibit a similar sensitivity. Ideas to combine the advantages of U&D and Bayesian designs are examined. A new approach is developed, using a hybrid framework, that evaluates the evidence for overriding the U&D allocation with a Bayesian one.

💡 Research Summary

The paper addresses the classic problem of locating a target percentile (threshold) in binary response experiments, focusing on two major families of sequential designs: the traditional Up‑and‑Down (U&D) method and Bayesian approaches, most notably the Continual Reassessment Method (CRM). The author first identifies a stagnation in the practical use of U&D and a lack of rigorous theoretical understanding of its long‑run behavior. To fill this gap, the study derives new results on the stationary distribution of the U&D Markov chain. By modeling the stimulus level as a discrete‑time Markov process with step size Δ, the author proves that the stationary distribution is symmetric around the target percentile p and that its variance scales as Δ²·p(1‑p)/