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Checking for normality in linear mixed models

science china mathematics 2012年第4期55卷 787-804页

作者： WU Ping 1,,ZHU LiXing 2,3 & FANG Yun 4 1 School of Finance and Statistics,East china normal university,shanghai 200241,china 2 School of Statistics and mathematics,Yunnan university of Finance and Economics,Yunnan 650221,china 3 The Department of mathematics,Hong Kong Baptist university,Hong Kong 999077,china 4 mathematics and science college,shanghai normal university,shanghai 200234,china School of Finance and Statistics East China Normal University Shanghai China School of Statistics and mathematics Yunnan University of Finance and Economics Yunnan China The Department of mathematics Hong Kong Baptist University Hong Kong China mathematics and science college Shanghai Normal University Shanghai China

Linear mixed models are popularly used to fit continuous longitudinal data, and the random effects are commonly assumed to have normal distribution. However, this assumption needs to be tested so that further analysis can be proceeded well. In this paper, we consider the Baringhaus-Henze-Epps-Pulley （BHEP） tests, which are based on an empirical characteristic function. Differing from their case, we consider the normality checking for the random effects which are unobservable and the test should be based on their predictors. The test is consistent against global alternatives, and is sensitive to the local alternatives converging to the null at a certain rate arbitrarily close to 1/V~ where n is sample size. ^-hlrthermore, to overcome the problem that the limiting null distribution of the test is not tractable, we suggest a new method： use a conditional Monte Carlo test （CMCT） to approximate the null distribution, and then to simulate p-values. The test is compared with existing methods, the power is examined, and several examples are applied to illustrate the usefulness of our test in the analysis of longitudinal data.

关键词： linear mixed models estimated best linear unbiased predictors BHEP tests conditional MonteCarlo test

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DISSIPATIVITY AND EXPONENTIAL STABILITY OF θ-METHODS FOR SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS WITH A BOUNDED LAG

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Journal of Computational mathematics 2003年第6期21卷 715-726页

作者： Hong-jiong Tian (Department of mathematics, shanghai normal university, shanghai 200234, china) Department of mathematics Shanghai Normal University Shanghai 200234 China

This paper deals with analytic and numerical dissipativity and exponential stability of singularly perturbed delay differential equations with any bounded state-independent lag. Sufficient conditions will be presented to ensure that any solution of the singularly perturbed delay differential equations (DDEs) with a bounded lag is dissipative and exponentially stable uniformly for sufficiently small ε > 0. We will study the numerical solution defined by the linear θ-method and one-leg method and show that they are dissipative and exponentially stable uniformly for sufficiently small ε > 0 if and only if θ = 1.

关键词： Singular perturbation θ-methods Dissipativity Exponential stability.

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CURVILINEAR PATHS AND TRUST REGION METHODS WITH NONMONOTONIC BACK TRACKING TECHNIQUE FOR UNCONSTRAINED OPTIMIZATION

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Journal of Computational mathematics 2001年第3期19卷 241-258页

作者： De-tong Zhu (Department of mathematics, shanghai normal university, shanghai 200234, china) 上海师范大学上海 200234

Focuses on a study which examined the modification of type approximate trust region methods via two curvilinear paths for unconstrained optimization. Properties of the curvilinear paths; Description of a method which combines line search technique with an approximate trust region algorithm; Information on the convergence analysis; Details on the numerical experiments.

关键词： curvilinear paths trust region methods nonmonotonic technique unconstrained optimization

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A MONOTONE COMPACT IMPLICIT SCHEME FOR NONLINEAR REACTION-DIFFUSION EQUATIONS

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Journal of Computational mathematics 2008年第2期26卷 123-148页

作者： Yuanming Wang Department of mathematics,East china normal university,shanghai 200241,china Division of Computational science,E-Institute of shanghai Universities,shanghai normal Benyu Guo Department of mathematics,shanghai normal university,shanghai 200234,china Division of Computational science,E-Institute of shanghai Universities,shanghai,china Department of mathematics East China Normal University Shanghai 200241 China Division of Computational science E-Institute of Shanghai Universities Shanghai Normal University Shanghai 200234 China Department of mathematics Shanghai Normal University Shanghai 200234 China Division of Computational science E-Institute of Shanghai Universities Shanghai China

A monotone compact implicit finite difference scheme with fourth-order accuracy in space and second-order in time is proposed for solving nonlinear reaction-diffusion equations. An accelerated monotone iterative method for the resulting discrete problem is presented. The sequence of iteration converges monotonically to the unique solution of the discrete problem, and the convergence rate is either quadratic or nearly quadratic, depending on the property of the nonlinear reaction. The numerical results illustrate the high accuracy of the proposed scheme and the rapid convergence rate *** iteration.

关键词： Nonlinear reaction-diffusion equation Monotone compact implicit scheme High accuracy Monotone iteration Rapid convergence rate.

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DERIVATIONS ON DIFFERENTIAL OPERATOR ALGEBRA AND WEYL ALGEBRA

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Chinese Annals of mathematics,Series B 1996年第2期17卷 199-212页

作者： CHEN CAOYU Department of mathematics, shanghai normal university, shanghai 200234, china. 上海师范大学数学系上海

Let L be an n-dimensional nilpotent Lie algebra with a basis {x1,…,xn}, and every xiacts as a locally nilpotent derivation on algebra A. This paper shows that there exists a setof derivations {y1,…,yn} on U(L) such that (A#U(L))#k[yi,…,yn] is isomorphic to theWeyl algebra An(A). The author also uses the derivations to obtain a necessary and sufficientcondition for a finite dimensional Lie algebra to be nilpotent.

关键词： Crossed porduct Smash product Derivation Nilpotent Lie algebra Weyl algebra

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A LINE SEARCH FILTER SECANT METHOD FOR NONLINEAR EQUALITY CONSTRAINED OPTIMIZATION

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Journal of Systems science & Complexity 2010年第2期23卷 343-361页

作者： Zhujun WANG Detong ZHU Department of mathematics Shanghai Normal University Shanghai 200234 China. Business college Shanghai Normal University Shanghai 200234 China.

This paper formulates and analyzes a line search method for general nonlinear equalityconstrained optimization based on filter methods for step acceptance and secant methods for *** feature of the new algorithm is that the secant algorithm is used to produce a searchdirection,a backtracking line search procedure is used to generate step size,some filtered rules areused to determine step acceptance,second order correction technique is used to reduce infeasibility andovercome the Maratos *** convergence properties of this method are analyzed:under mildassumptions it is showed that every limit point of the sequence of iterates generated by the algorithmis feasible,and that there exists at least one limit point that is a stationary point for the ***,it is also established that the Maratos effect can be overcome in our new approach by addingsecond order correction steps so that fast local superlinear convergence to a second order sufficient localsolution is ***,the results of numerical experiments are reported to show the effectivenessof the line search filter secant method.

关键词： Filter method nonlinear programming secant algorithm second order correction.

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THE LARGE TIME CONVERGENCE OF SPECTRAL METHOD FOR GENERALIZED KURAMOTO-SIVASHINSKY EQUATION (Ⅱ)

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Journal of Computational mathematics 1998年第3期16卷 203-212页

作者： Xin-ming Xiang (shanghai normal university, shanghai 200234, china) 上海师范大学上海

In this paper, we study fully discrete spectral method and long time behavior of solution of generalized Kuramoto-Sivashinsky equation with periodic initial condition. We prove that the large time error estimation for fully discrete solution of spectral method. We prove the existence of approximate attractors AN, ANk respectively and d(AN, A) → 0, d(ANk, A) → 0.

关键词： Kuramoto- Sivashinsky equation,large time convergence Approximate attractor Upper semicontinuity of attractors.

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The　θ－Methods　in　Numerical　Solution　of　Systems　of　Differential　Equations　with　Two　Delay　Terms

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Journal of Systems Engineering and Electronics 1994年第3期5卷 32-40页

作者： Tian Hongjiong & Kuang Jiaoxun (Department of mathematics, shanghai normal university, shanghai 200234, china) Department of mathematics Shanghai Normal University Shanghai 200234 China

This paper deals with the numerical solution of initial value problems for systems of differential equations with two delay terms. We investigate the stability of adaptations of the θ-methods in the numerical solution of test equations u'(t) = a 11 u(t) + a12v(t) + b11 u(t - τ1) + b12v(t-τ2,v'(t) = a21 u(t) + a22 v(t) + b21 u(t -τ1,) + b22 v(t -τ2), t>0,with initial conditionsu(t)=u0(t),v(t) =v0(t), t≤*** aij, bij∈C, τj >0, i,j = 1,2,, and u0(t), v0(t)are continuous and complex valued. Sufficient conditions for the asymptotic stability of test equation are derived. Furthermore, with respect to an appropriate definition of stability for the numerical method, it is proved that the linear θ-method is stable if and only if 1/2≤θ≤1 and the one-leg θ-method is stable if and only if θ= 1.

关键词： Delay differential equations Numerical solution θ-methods Asymptotic stability Schur polynomial.

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JACOBI SPECTRAL APPROXIMATIONS TO DIFFERENTIALEQUATIONS ON THE HALF LINE

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Journal of Computational mathematics 2000年第1期18卷 95-112页

作者： Ben-yu Guo(Department of mathematics, shanghai normal university, shanghai 200234, china)(COM2MAC, Pohang university of science and Technology, Pohang, Korea) 上海师范大学上海 200234

Presents a study which investigated some Jacobi approximations which are used for numerical solutions of differential equations on the half line. Application of the approximations to problems on unbounded domains; Alg... 详细信息

关键词： Jacobi spectral approximations differential equations on the half line stability and convergence

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THE GPL-STABILITY OF RUNGE-KUTTA METHODS FORDELAY DIFFERENTIAL SYSTEMS

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Journal of Computational mathematics 2000年第1期18卷 75-82页

作者： Biao Yang Lin Qiu Jiao-xun Kuang(Department of mathematics, shanghai normal university, shanghai 200234, china.) 上海师范大学上海 200234

Presents a study which dealt with stability of the implicit Runge-Kutta methods for the numerical solutions of the systems of delay differential equations. Stability behavior of the methods; Exponential solutions of t... 详细信息

关键词： delay differential equation implicit Runge-Kutta methods GPL-stability

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