Generalized two-color urn processes and approximations.

Abstract

This dissertation is an investigation into the mechanics of generalized two-color urn processes and associated inference problems. Generalized urn processes were introduced by Hill, Lane and Sudderth [1980], who proved convergence theorems for the process <italic>X_n</italic>. We use their results as a starting point and obtain central limit theorems for linear urn processes converging almost surely to downcrossings. Our result can be looked upon as a generalization of the well known De-Moivre Laplace central limit theorem for Bernoulli random variables. The limit theorems are used to make inference about the slope of an urn function at a downcrossing and an illustrative example is discussed. We also establish a connection between generalized urn processes and adaptive biased coin designs. We also provide local and global approximation methods based on Bayesian methodology, for estimating urn-functions. The methods are illustrated and compared using simulated data and their limitations are discussed.