Asymptotic theorems of sequential estimation-adjusted urn models

The Generalized P\'{o}lya Urn (GPU) is a popular urn model which is widely used in many disciplines. In particular, it is extensively used in treatment allocation schemes in clinical trials. In this paper, we propose a sequential estimation-adjusted urn model (a nonhomogeneous GPU) which has a wide spectrum of applications. Because the proposed urn model depends on sequential estimations of unknown parameters, the derivation of asymptotic properties is mathematically intricate and the corresponding results are unavailable in the literature. We overcome these hurdles and establish the strong consistency and asymptotic normality for both the patient allocation and the estimators of unknown parameters, under some widely satisfied conditions. These properties are important for statistical inferences and they are also useful for the understanding of the urn limiting process. A superior feature of our proposed model is its capability to yield limiting treatment proportions according to any desired allocation target. The applicability of our model is illustrated with a number of examples.