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BATE curve in assessment of clinical utility of predictive biomarkers

Science china Mathematics 2012年第8期55卷 1529-1552页

作者： ZHOU XiaoHua 1,2,3,& MA YunBei 4,5 1 Northwest HSR&D Center of Excellence,VA Puget Sound Health Care System,Seattle,WA 98198,USA 2 Department of Biostatistics,university of Washington,Seattle,WA 98198,USA 3 Beijing International Center for Mathematical Research,Peking university,Beijing 100871,china 4 Department of Operations Research and Financial Engineering,Princeton university,Princeton,NJ 08540,USA 5 school of statisitcs,southwest university of finance and economics,chengdu 611130,china 1. Northwest HSR&D Center of Excellence VA Puget Sound Health Care System Seattle WA 98198 USA2. Department of Biostatistics University of Washington Seattle WA 98198 USA3. Beijing International Center for Mathematical Research Peking University Beijing 100871 China4. Department of Operations Research and Financial Engineering Princeton University Princeton NJ 08540 USA5. School of Statisitcs Southwest University of Finance and Economics Chengdu 611130 China

In this paper, for time-to-event data, we propose a new statistical framework for casual inference in evaluating clinical utility of predictive biomarkers and in selecting an optimal treatment for a particular patient. This new casual framework is based on a new concept, called Biomarker Adjusted Treatment Effect （BATE） curve. The BATE curve can be used for assessing clinical utility of a predictive biomarker, for designing a subsequent confirmation trial, and for guiding clinical practice. We then propose semi-p^rametric methods for estimating the BATE curves of biomarkers and establish asymptotic results of the proposed estimators for the BATE curves. We also conduct extensive simulation studies to evaluate finite-sample properties of the proposed estimation methods. Finally, we illustrate the application of the proposed method in a real-world data set.

关键词： predictive biomarker cutoff points interaction BATE curve time-to-event outcome

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Beurling's theorem and invariant subspaces for the shift on Hardy spaces

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Science china Mathematics 2008年第1期51卷 131-142页

作者： QIU ZhiJian school of Economic Mathematics, southwestern university of finance and economics, chengdu 610074, china 1. school of Economic Mathematics Southwestern University of Finance and Economics Chengdu 610074 China

Let G be a bounded open subset in the complex plane and let H 2(G) denote the Hardy space on G. We call a bounded simply connected domain W perfectly connected if the boundary value function of the inverse of the Riemann map from W onto the unit disk D is almost 1–1 with respect to the Lebesgue measure on ?D and if the Riemann map belongs to the weak-star closure of the polynomials in H ∞(W). Our main theorem states: in order that for each M ∈ Lat (M z ), there exist u ∈ H ∞(G) such that M = ∨{uνH 2(G)}, it is necessary and sufficient that the following hold:each component of G is a perfectly connected domainthe harmonic measures of the components of G are mutually singularthe set of polynomials is weak-star dense in H ∞(G).Moreover, if G satisfies these conditions, then every M ∈ Lat (M z ) is of the form uH 2(G), where u ∈ H ∞(G) and the restriction of u to each of the components of G is either an inner function or zero.

关键词： Hardy space invariant subspace shift operator 47B20 30H05 30E10 46E15

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Carleson Measures on Planar Sets

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Acta Mathematica Sinica,English Series 2009年第11期25卷 1881-1892页

作者： Zhijian QIU school of Mathematics,southwestern university of finance and economics,chengdu 610074,P. R. china school of Mathematics Southwestern University of Finance and Economics Chengdu 610074 P. R. China

In this paper, we investigate what are Carleson measures on open subsets in the complex plane. A circular domain is a connected open subset whose boundary consists of finitely many disjoint circles. We call a domain G multi-nicely connected if there exists a circular domain W and a conformal map ψ from W onto G such that ψ is almost univalent with respect the arclength on δW. We characterize all Carleson measures for those open subsets so that each of their components is multinicely connected and harmonic measures of the components are mutually singular. Our results suggest the extension of Carleson measures probably is up to this class of open subsets

关键词： Carleson measure circular domain harmonic measure multi-nicely connected domain

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Local asymptotic behavior of the survival probability of the equilibrium renewal model with heavy tails

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Science china Mathematics 2005年第3期48卷 300-306页

作者： JIANG Tao & CHEN Yiqing school of finance. Nanjing university of finance and economics, Nanjing 210003, china school of economics and Management, Guangdong university of Technology, Guangzhou 510090, china 1. school of finance Nanjing University of Finance and Economics 210003 Nanjing China 2. school of economics and Management Guangdong University of Technology 510090 Guangzhou China

Recently, Tang established a local asymptotic relation for the ruin probability in the Cramer-Lundberg risk model. In this short note we extend the corresponding result to the equilibrium renewal risk model.

关键词： geometric sums, heavy-tailed distribution, ladder height, the equilibrium renewal model, the ruin probability.

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Approximate solutions to infinite dimensional LQ problems over infinite time horizon

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Science china Mathematics 2006年第7期49卷 865-876页

作者： PAN Liping, ZHANG Xu & CHEN Qihong school of Mathematical Sciences, Fudan university, Shanghai 200433, china school of Mathematics, Sichuan university, chengdu 610064, china Department of Applied Mathematics, Shanghai university of finance & economics, Shanghai 200433, china school of Mathematical Sciences Fudan University Shanghai 200433 China school of Mathematics Sichuan University Chengdu 610064 China Department of Applied Mathematics Shanghai University of Finance & Economics Shanghai 200433 China

This paper is addressed to develop an approximate method to solve a class of infinite dimensional LQ optimal regulator problems over infinite time horizon. Our algorithm is based on a construction of approximate solutions which solve some finite dimensional LQ optimal regulator problems over finite time horizon, and it is shown that these approximate solutions converge strongly to the desired solution in the double limit sense.

关键词： approximate solution LQ problem infinite dimensional infinite time hori-

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Moving collocation methods for time fractional differential equations and simulation of blowup

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Science china Mathematics 2011年第3期54卷 611-622页

作者： MA JingTang1 & JIANG YingJun2 1school of Economic Mathematics, southwestern university of finance and economics, chengdu 611130, china 2Department of Mathematics and Scientific Computing, Changsha university of Science and Technology, Changsha 410076, china 1. school of Economic Mathematics Southwestern University of Finance and Economics Chengdu 611130 China2. Department of Mathematics and Scientific Computing Changsha University of Science and Technology Changsha 410076 China

A moving collocation method is proposed and implemented to solve time fractional differential equations. The method is derived by writing the fractional differential equation into a form of time difference equation. The method is stable and has a third-order convergence in space and first-order convergence in time for either linear or nonlinear equations. In addition, the method is used to simulate the blowup in the nonlinear equations.

关键词： moving collocation methods time fractional differential equations blowup

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A local probability exponential inequality for the large deviation of an empirical process indexed by an unbounded class of functions and its application

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Science china Mathematics 2004年第6期47卷 821-830页

作者： ZHANG DixinDepartment of finance, Business school, Nanjing university, Nanjing 210093, china 1. Department of finance Business School Nanjing University 210093 Nanjing China

A local probability exponential inequality for the tail of large deviation of an empirical process over an unbounded class of functions is proposed and studied. A new method of truncating the original probability space and a new symmetrization method are given. Using these methods, the local probability exponential inequalities for the tails of large deviations of empirical processes with non-i.i.d. independent samples over unbounded class of functions are established. Some applications of the inequalities are discussed. As an additional result of this paper, under the conditions of Kolmogorov theorem, the strong convergence results of Kolmogorov on sums of non-i.i.d. independent random variables are extended to the cases of empirical processes indexed by unbounded classes of functions, the local probability exponential inequalities and the laws of the logarithm for the empirical processes are obtained.

关键词： empirical process large deviation unbounded class of functions local probability exponential inequality

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Almost sure central limit theorems for random functions

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Science china Mathematics 2006年第12期49卷 1788-1799页

作者： LU Chuanrong, QIU Jin & XU Jianjun school of Mathematics and Statistics, Zhejiang university of finance and economics, Hangzhou 310018, china Department of Mathematics, Zhejiang university, Hangzhou 310028, china 1. school of Mathematics and Statistics Zhejiang University of Finance and Economics Hangzhou 310018 China 2. Department of Mathematics Zhejiang University Hangzhou 310028 China

Let {Xn,-∞< n <∞} be a sequence of independent identically distributed random variables with EX1 = 0, EX12 = 1 and let Sn =∑k=1∞Xk, and Tn = Tn(X1,…,Xn) be a random function such that Tn = ASn + Rn, where supn E|Rn| <∞and Rn = o(n^(1/2)) a.s., or Rn = O(n1/2-2γ) a.s., 0 <γ< 1/8. In this paper, we prove the almost sure central limit theorem (ASCLT) and the function-typed almost sure central limit theorem (FASCLT) for the random function Tn. As a consequence, it can be shown that ASCLT and FASCLT also hold for U-statistics, Von-Mises statistics, linear processes, moving average processes, error variance estimates in linear models, power sums, product-limit estimators of a continuous distribution, product-limit estimators of a quantile function, etc.

关键词： statistics,random function,almost sure central limit theorem,logarithm average.

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Asymptotics of discounted aggregate claims for renewal risk model with risky investment

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Applied Mathematics(A Journal of Chinese Universities) 2010年第2期25卷 209-216页

作者： JIANG Tao school of finance, Zhejiang Gongshang university, Hangzhou 310018, china school of finance Zhejiang Gongshang University Hangzhou China

Under the assumption that the claim size is subexponentially distributed and the insurance surplus is totally invested in risky asset, a simple asymptotic relation of tail probability of discounted aggregate claims for renewal risk model within finite horizon is obtained. The result extends the corresponding conclusions of related references.

关键词： Discounted aggregate claims ruin probability within finite horizon renewal risk model risky investment subexponential class.

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Ruin probability of the renewal model with risky investment and large claims

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Science china Mathematics 2009年第7期52卷 1539-1545页

作者： WEI Li school of finance,Renmin university of china,Beijing 100872,china 1. school of finance Renmin University of China Beijing 100872 China

The ruin probability of the renewal risk model with investment strategy for a capital market index is investigated in this *** claim sizes with common distribution of extended regular variation,we study the asymptotic behaviour of the ruin *** a corollary,we establish a simple asymptotic formula for the ruin probability for the case of Pareto-like claims.

关键词： asymptotics extended regular variation renewal risk model risky investment strategy ruin probability 60G70 60K30 60K37

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