Up-and-Down and the Percentile-Finding Problem

Details Up-and-Down and the Percentile-Finding Problem Up-and-Down and the Percentile-Finding Problem

2008-08-21

arxiv.org/abs/0808.3004v1

Statistics

Methodology

This is the full text of my Ph.D. dissertation as accepted at the University of Washington in Fall 2007. Further improvements

Scientific paper

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.

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