Random Walks for Quantile Estimation

Abstract

Quantile estimation is an important problem in many areas of application, such as toxicology, item response analysis, and material stress analysis. In these experiments, a treatment or stimulus is given or a stress is applied at a finite number of levels or dosages, and the number of responses at each level is observed. This paper focuses on sequentially assigning treatment levels to subjects in a manner that describes a random walk, with transition probabilities that depend on the prior response as well as the prior treatment. Criteria are given for random walk rules such that resulting stationary treatment distributions will center around an unknown, but prespecified quantile. It is shown how, when a parametric form for the response function is assumed, the stationary treatment distribution may be further characterized. Using the logistic response function as an example, a mechanism for generating new discrete probability distribution functions is revealed. In this example, three different estimates of the unknown quantile arise naturally.