Random Walks for Quantile Estimation
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.
KeywordsRandom Walk American Statistical Association Quantile Estimation Bernoulli Random Variable Recurrent Markov Chain
Unable to display preview. Download preview PDF.
- Chiang, J. (1990). Sequential designs for the linear logistic model. Unpublished doctoral dissertation. Pennsylvania State University.Google Scholar
- Flournoy, N. (1990). Adaptive designs in clinical trials. Proceedings of the Biopharmaceutical Section of the American Statistical Association at the 1989 Joint Statistical Meetings. The American Statistical Association, Alexandria, VA.Google Scholar
- Flournoy, N. (1993). A clinical experiment in bone marrow transplantation: estimating the percentage point of a quantal response curve. Case Studies in Bayesian Statistics. Gatsonis, G., Hodges, J. S., Kass, R. E., Singpurwalla, N. D., editors. Springer-Verlag, New York, NY.Google Scholar
- Wetherill, G. B. and Glazebrook, K. D. (1986). Sequential estimation of points on quantal response curves. Chapter 10 in: Sequential Methods in Statistics, A Monograph on Statistics and Applied Probability. Third edition. Chapman and Hall, New York.Google Scholar