Isotonic regression under sequential designs.

Given an independent sequence of random tuples $(X\sb{k},Y\sb{k})$ identically distributed as (X,Y), the problem of estimating the regression function $\mu(x)={\bf E}\{Y\vert X=x\},\ \alpha\le x\le b,$ for monotonic $\mu$ is discussed under the context of sequential designs, where after observing $X\sb{k},$ the experimenter may choose to observe or not to observe $Y\sb{k}.$ Characterization of decision rules that yield consistency for the isotonic regression is given. A non-Bayesian, nonparametric treatment of the bandit problem with covariates is presented, asymptotically optimal decision rules are discussed and illustrated through a simulation study. Attempts are made to improve the results by the introduction of the isotonic smoothing spline as an extension to the isotonic estimate. The existence, uniqueness, and characterization of the isotonic smoothing spline, which is a piecewise linear function, is presented together with computational considerations. Its comparative treatment under sequential designs is also demonstrated. A Bayesian, parametric treatment of the bandit problem with covariates and delayed responses is presented, where under a simple Dirichlet model, the optimal decision rules are tractable with the method of dynamic programming.Ph.D.Pure SciencesStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/128954/2/9303830.pd