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 国家自然科学基金杰出青年基金项目

摘要

项目研究自适应设计与相关应用概率模型的概率极限理论，在含协变量的自适应设计方面，构建了平衡协变量的自适应设计的一般框架并建立了其理论、证明著名的Pocock & Simon（1975）设计如以前很多随机模拟显示的那样确实能达到协变量边际平衡但不能达到组内平衡，解决了这一设计理论问题，以此为依据建立了协变量自适应设计的统计推断理论； 在前期研究的基础上给出了适合广义线性模型的协变量调节的反应变量自适应设计及其理论；在反应变量自适应设计方面，构建了随机加强罐子模型中随机分配之收敛性、极限的分布、条件中心极限定理、高斯逼近等系列理论，解决May & Flournoy (Annals Statistics, 2009)中的关于极限分布一个公开问题，构建统计功效最佳、渐近选择偏差最小的双重最优自适应设计；利用随机迭代算子这应用概率模型建立了几类自适应设计，包括罐子模型、Hu & Zhang(2004)提出的推广双重自适应抛偏硬币设计，的联系，给出了这类模型在各种情形下的中心极限定理，以此为工具解决Hu & Rosenberger（2006）专著中提到了两个关于广义罐子模型的理论问题；由于非线性模型在金融风险不确定性研究中有广泛的应用背景，项目在彭实戈(2009)提出的非线性期望框架下，建立了矩不等式、指数不等式等概率工具，证明了独立同分布随机变量的泛函中心极限定理、重对数律、自正则化极限定理等系列定理，为建立了非线性期望的极限理论体系打下了基础。项目完成论文20多篇，一些发表在《Annals of Applied Probability》、《Journal of the American Statistical Association》、《Advances in Applied Probability》等概率统计重要刊物上。

The project aims at studying the adaptive randomization in clinical trials and the limit theorems of related applied probability models. For the covariate-adaptive randomization, the general framework of randomization procedures for balancing covariates are proposed and the limit theorems are established, and it is proved that the famous design proposed by Pocock & Simon（1975）is actually able to achieve the marginal balance of covariates as various simulation studies showed,  but it is not able to achieve the within-strata balance. Basing on these theoretical properties, the theory on statistical inference is given for testing the difference between treatments basing on the data from the covariate-adaptive randomization.  Basing on previous study on covariate-adjusted response-adaptive designs, the project further proposes a general design model for a poplar case that responses and covariates follow a generalized linear model and establish its related statistical theory. For the response-adaptive randomization, the project fist has studied the limit theory of the designs basing on two-color and general multi-color reinforced randomized urn (RRU) model, including, the convergence, the rate of convergence, the distribution of the limit, the conditional central limit theorem and the Gaussian approximation.  An open problem on the limit distribution stated in May & Flournoy (Annals Statistics, 2009) is solved. The project then proposes a general framework of efficient response-adaptive designs on which the randomization procedure can attain both the lower bound of the asymptotic variability and the lower bound of the asymptotic selection bias so that the design is the doubly best in the sense of power and the randomness. The project also finds a general recursive stochastic algorithm which links several response-adaptive designs including urn models and the generalized doubly adaptive biased coin design proposed by Hu and Zhang (2004). The central limit theorem on the recursive stochastic algorithm is established for various cases of the eigenvalues of its generating matrix, and as an application two open problems stated in the book by Hu and Rosenberger (2006) about adaptive designs which bases on the generalized urn model are solved. Because the non-linear expectation probability has wide application in modeling the finance uncertain, the project finally considers several kinds of limit theorems under the framework of the non-linear expectation proposed by Peng (2008). Several probability tools as moment inequalities on maximum partial sums and exponential inequalities are established. The functional central limit theorem, the law of the iterated logarithm and the self-normalized limit theorem are proved for independent and identically distributed random variables under the non-linear expectation. The results provide a foundation for establishing the system of the limit theorems under the non-linear expectation. The project produces more than 20 papers, some of which have been published in important probability and statistics journals as  Annals of Applied Probability、Journal of the American Statistical Association、Advances in Applied Probability, etc.

完成论文目录

[1] ZHANG LI-Xin (2016), Convergence to a self-normalized G-Brownian motion. Probability, Uncertainty and Quantitative Risk （accepted）

[2]ZHANG Li-Xin(2016) ,Central limit theorems of a recursive stochastic algorithm with applications to adaptive designs, Annals of Applied Probability,  Vol. 26(6), 3630–3658

[3] ZHANG Li-Xin(2016), Exponential inequalities under the sub-linear expectations with applications to laws of the iterated logarithm, Science in China- Mathematics, Vol. 59(12):2503-2526.

[4] ZANG, Qingpei and ZAHNG　Li-Xin (2016), Parameter estimation for generalized diffusion processes with reflected boundary, Science in China- Mathematics, Vol.59(6): 1163-1174.

[5] ZANG, Qingpei and ZAHNG　Li-Xin (2016), A general lower bound of parameter estimation for reflected Ornstein–Uhlenbeck processes. Journal of Applied Probability, Vol. 53(1), 22-32.  

[6] ZAHNG Li-Xin (2016), Rosenthal's inequalities for independent and negatively dependent random variables under sub-linear expectations with applications, Science in China- Mathematics, Vol.59 (4), 751–768.

[7] ZAHNG　Li-Xin (2016), Self-normalized moderate deviation and laws of the iterated logarithm under G-expectation, Communications in Mathematics and Statistics, Vol.4:229–263

[8] MA Wei, HU Feifang and Zhang Li-Xin (2015), Testing Hypotheses of Covariate-Adaptive Randomized Clinical Trials. Journal of the American Statistical Association. Vol. 110(510): 669-680.

[9] ZAHNG　Li-Xin (2015), Donsker’s Invariance principle under the sub-linear expectation with an application to Chung’s law of the iterated logarithm, Communications in Mathematics and Statistics, Vol. 3 (2): 187-214.

[10] ZAHNG　Li-Xin (2015), Response-adaptive randomization: an overview of designs and asymptotic theory, Modern Adaptive Randomized Clinical Trials--Statistical and Practical Aspects, Edited by O. Sverdlov, pp 221-250, Chapman & Hall/CRC Biostatistics Series, CRC Press, 2015.  

[11] HU Feifang., HU Yanqing, MA Wei, ZAHNG　Li-Xin and ZHU Hongjian (2015), Statistical inference of adaptive randomized clinical trials for personalized medicine. Clinical Investigation, Vol. 5(4), 415–425.

[12] CHAN Wai-Sum, CHEUNG Siu Hung, CHOW Wai Kit, and ZHANG Li-Xin (2015),  A robust test for threshold-type non-linearity in multivariate time-series analysis,  Journal of Forecasting.Vol.34( 6): 441-454

[13] ZHANG Yang and ZHANG LI-Xin (2015),  On the almost sure invariance principle for dependent Bernoulli random variables. Statistics & Probability Letters, Vol. 107: 264-271

[14] ZHANG LI-Xin and ZHANG Yang (2015), Asymptotics for a class of dependent random variables, Stattistics & Probability Letters, Vol. 105: 47-56.

[15] 陈颖瑜, 张立新(2015), 随机采样下随机微分方程Milstein方法的渐近误差,中国科学:数学. Vol.45(3):287-300

[16] ZHANG Li-Xin, HU Feifang, CHEUNG Siuhung and CHAN Waisum (2014),   Asymptotic properties of multi-color randomly reinforced Polya urns. Advances in Applied Probability, Vol 46: 585-602.

[17] ZHANG Li-Xin (2014).  A Gaussian process approximation for two-color randomly reinforced urns.  Electronic Journal of Probability, Vol. 19, no. 86, 1–19.

[18] CHEUNG Siuhung, ZHANG Li-Xin, HU Feifang and CHAN Waisum, Covariate-adjusted response-adaptive designs for generalized linear models,   Journal of Statistical Planning and Inference, Vol. 149: 152-161.

[19] ZHANG Zhongyang and ZHANG Li-Xin (2013), Scaling limits for one-dimensional long-range percolation: Using the corrector method, Statistics & Probability Letters，  Vol. 83(11): 2459-2466  

[20] ZHU Yunzhou, ZHANG Li-Xin and  ZHANG Yi (2013), Optimal reinsurance under the Haezendonck risk measure, Statistics & Probability Letters，Vol. 83(4): 1111-1116

[21] HU Feifang and ZHUANG Li-Xin, On the theory of covariate adaptive design, Manuscript.  

[22] ZHANG Li-Xin, HU Feifang, CHEUNG Siuhung and CHAN Waisum (2016), Efficient response-adaptive designs with minimum selection bias. Manuscript.  

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