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Li-Xin Zhang

Publications and source records attributed to Li-Xin Zhang.

44 records · Page 3Linked to original sources
arXiv2008

The Gaussian approximation for multi-color generalized Friedman's urn model

The Friedman's urn model 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 prove that both the urn composition process and the allocation proportion process can be approximated by a multi-dimensional Gaussian process almost surely for a multi-color generalized Friedman's urn model with non-homogeneous generating matrices. The Gaussian process is a solution of a stochastic differential equation. This Gaussian approximation together with the properties of the Gaussian process is important for the understanding of the behavior of the urn process and is also useful for statistical inferences. As an application, we obtain the asymptotic properties including the asymptotic normality and the law of the iterated logarithm for a multi-color generalized Friedman's urn model as well as the randomized-play-the-winner rule as a special case.

math.PR↗
arXiv2008

A New Family of Covariate-Adjusted Response Adaptive Designs and their Asymptotic Properties

It is often important to incorporating covariate information in the design of clinical trials. In literature, there are many designs of using stratification and covariate-adaptive randomization to balance on certain known covariate. Recently Zhang, Hu, Cheung and Chan (2007) have proposed a family of covariate-adjusted response-adaptive (CARA) designs and studied their asymptotic properties. However, these CARA designs often have high variabilities. In this paper, we propose a new family of covariate-adjusted response-adaptive (CARA) designs. We show that the new designs have smaller variabilities and therefore more efficient.

stat.ME↗
arXiv2006

Adaptive Allocation Theory in Clinical Trials

Various adaptive randomization procedures (adaptive designs) have been proposed to clinical trials. This paper discusses several broad families of procedures, such as the play-the-winner rule and Markov chain model, randomized play-the-winner rule and urn models, drop-the-loser rule, doubly biased coin adaptive design. Asymptotic theories are presented with several pivotal proofs. The effect of delayed responses, the power and variability comparison of these designs are also discussed.

math.ST↗
arXiv2006

Precise rates in the law of the iterated logarithm

Let $X$, $X_1$, $X_2$, $...$ be i.i.d. random variables, and let $S_n=X_1+... + X_n$ be the partial sums and $M_n=\max_{k\le n}|S_k|$ be the maximum partial sums. We give the sufficient and necessary conditions for a kind of limit theorems to hold on the convergence rate of the tail probabilities of both $S_n$ and $M_n$. These results are related to the law of the iterated logarithm. The results of Gut and Spataru (2000) are special cases of ours.

math.PR↗
arXiv2006

On the rates of the other law of the logarithm

By using the strong approximation, this paper establishes several limit results on the convergent rate of a infinite series of probabilities on the other law of iterated logarithm.

math.PR↗
arXiv2006

Asymptotic Properties of Covariate-Adjusted Adaptive Designs

Response-adaptive designs have been extensively studied and used in clinical trials. However, there is a lack of a comprehensive study of response-adaptive designs that include covariates, despite their importance in clinical experiments. Because the allocation scheme and the estimation of parameters are affected by both the responses and the covariates, covariate-adjusted response-adaptive (CARA) designs are very complex to formulate. In this paper, we overcome the technical hurdles and lay out a framework for general CARA designs for the allocation of subjects to $K (\geq 2)$ treatments. The asymptotic properties are studied under certain widely satisfied conditions. The proposed CARA designs can be applied to generalized linear models. Two important special cases, the linear model and the logistic regression model, are considered in detail.

math.ST↗