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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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The Cobordism Classification of Involution on DOLD Manifold P(1,2l)

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Acta Mathematica Sinica,English Series 1997年第1期13卷 9-20页

作者： wu Zhende Liu Zongze (department of mathematics,Hebei normal university,Shijazhuang 050016,china)Bai Ruipu (department of mathematics,Hebei university,Baoding 071002,china) department of mathematics Hebei Normal University 050016 Shijazhuang China

Supposing the smooth involution of the DOLD manifold P(2,2l)satisfies the following condition:the fiberation π:P(1,2l)×T_×(-1)S~∞→RP(∞)is totally nonhomologous to zero(cf.[1, p373]),this paper determines the cla... 详细信息

关键词： Stiefel-Whitney class Involution on smooth manifold Bordism class wu formulas

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THE NEIGHBORHOOD INTERSECTIONS OF ESSENTIAL SETS AND HAMILTONICITY OFGRAPHS

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Systems Science and Mathematical Sciences 1998年第3期11卷 230-237页

作者： wu Zhengsheng(department of mathematics, Naming normal university, Naming 210097, china)XU Xinping(department of mathematics, Jiangsu Education College, Naming 210013, china)ZHOU Xinghe(department of mathematics, Naming normal university, Nabbing 210097,

Let G be a graph. An independent set Y in G is called an essential independent set (or essential set for simplicity) if there is {yi , y2} Y such that dist(y1 , y2) - 2. For integer t > 0, let It(G) = {Y| Y is an independent set of G, |Y| = t}, It(G) = {Y|Y is an essential set of G, |Y| = t}. For ∈E It(G), let si(y) = |{v|v ∈V(G), |N(v) n Y| = i}|(i = 0, 1,…, t). Let X, Y g V(G). Define dist(X, Y) = dist(u, v), n(Y) = |{v|v ∈V(G), dist({v}, Y) ≤ 2}|. A non-negative rational sequence (a1，a2,…, ak+1) (k ≥2) is called an LTW-sequence, if it satisfies 1) a1 ≤ 1; 2) for arbitrary i1, i2,…，ih. ∈{2,3,……, k + 1}, The main new results of this paper are as follows: Let (a1， a2，… ak+1) be all LTW-sequence, and k ≥ 2. If G is a k-connected graph, and then G has a Hamilton cycle; if G is a (k + 1)-connected graph and for each then G is Hamilton-connected. The existing results are generalized by these since Ik+1(G) is replaced by I(G). We introduce a new technique of T-insertion in this paper， by using the T-vertex inserting lemmas we give a unified proof for a graph to be hamiltonian or Hamilton-connected.

关键词：篖TW-sequence essential sets T-vertex insertion hamiltonicity

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ASYMPTOTIC normalITY OF MAXIMUM QUASI-LIKELIHOOD ESTIMATORS IN GENERALIZED LINEAR MODELS WITH FIXED DESIGN

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Journal of Systems Science & Complexity 2008年第3期21卷 463-473页

作者： Qibing GAO Yaohua wu Chunhua ZHU Zhanfeng WANG School of mathematics and Computer Science Nanjing Normal University Nanjing 210097 China department of Statistics and Finance University of Science and Technology of China Hefei 230026 China School of mathematics Science Anhui University Hefei 230039 China

In generalized linear models with fixed design, under the assumption λ↑_n→∞ and other regularity conditions, the asymptotic normality of maximum quasi-likelihood estimator ^↑βn, which is the root of the quasi-likelihood equation with natural link function ∑i=1^n Xi（yi -μ（Xi′β）） = 0, is obtained, where λ↑_n denotes the minimum eigenvalue of ∑i=1^nXiXi′, Xi are bounded p × q regressors, and yi are q × 1 responses.

关键词： Asymptotic normality fixed design generalized linear models maximum quasi-likelihood estimator

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ON NICE SEQUENCES CONCERNING HAMILTONICITY OF GRAPHS

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Systems Science and Mathematical Sciences 1993年第4期6卷 373-375页

作者： LIU Yiping wu Zhengsheng department of mathematics, Nanjing normal university, Nanjing 210024, china

In the paper, the （k-1）-traceable-nice （（k-1）-T-nice） and k-homogeneously-traceable-nice （k-HT-nice） sequence are defined similarly to the definition of k-Hamilton-nice （k-H-nice） and （k+1）-Hamilton-connected-nice （（k+1）-HC-nice） sequence. Therelationships among these four nice sequences are discussed. The main results are asfollows: Letη=(a1, a2,…, ak+1 be a non-negative rational sequence, k≥2. （1） If η is（k+1）-HC-nice and ak+1=2, then η is k-HT-nice, （2） If η is k-HT-nice and ak+1=2,then η is （k-1）-T-nice, （3） If η is k-H-nice, then η is k-HT-nice. Meanwhile, four unsolvedproblems on these topics are proposed.

关键词： k-H-nice sequence (k-1)-T-nice sequence k-HT-nice sequence (k+1)-HC-nice sequence

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Behavior of gravity waves with limited amplitude in the vicinity of critical layer

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Chinese Science Bulletin 2002年第18期47卷 1547-1550页

作者： wu Shaoping & Yl Fanwuhan Institute of Physics and mathematics, Chinese Academy of Sciences, wuhan 430071, china department of Radio Science and Technology, wuhan university, wuhan 430072, china department of Physics, Huazhong normal Univ wuhan Institute of Physics and mathematics Chinese Academy of Sciences Wuhan China department of Physics Huazhong Normal University Wuhan China department of Radio Science and Technology Wuhan University Wuhan China

By using the FICE scheme, a numerical simulation of three-dimensional nonlinear propagation of gravity wave packet in a wind-stratified atmosphere is presented. The whole nonlinear propagation process of the gravity wave packet is shown; the propagation behavior of gravity waves in the vicinity of critical layer is analyzed. The results show that gravity waves encounter the critical layer when propagating in the fair winds whose velocities increase with height, and the height of critical layer propagating nonlinearly is lower than that expected by the linear gravity waves theory; the amplitudes of gravity waves increase with height as a whole before gravity waves encounter the critical layer, but the increasing extent is smaller than the result given by the linear theory of gravity waves, while the amplitudes of gravity waves reduce when gravity waves meet the critical layer; the energy of wave decreases with height, especially at the critical layer; the vertical wavelength reduces with the height

关键词： gravity waves critical layer FICE scheme simulation.

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Modified Chaplygin gas as an interacting holographic dark energy model

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Science china(Physics,Mechanics & Astronomy) 2010年第4期53卷 598-606页

作者： wu YaBo, YANG WeiQiang, WANG Cong & WANG Lei department of Physics, Liaoning normal university, Dalian 116029, china department of Physics Liaoning Normal University Dalian China

The modified Chaplygin gas (MCG) as an interacting model of holographic dark energy in which dark energy and dark matter are coupled together is investigated in this paper. Concretely, by studying the evolutions of related cosmological quantities such as density parameter Ω, equation of state w, deceleration parameter q and transition redshift zT, we find the evolution of the universe is from deceleration to acceleration, their present values are consistent with the latest observations, and the equation of state of holographic dark energy can cross the phantom divide w = -1. Furthermore, we put emphasis upon the geometrical diagnostics for our model, i.e., the statefinder and Om diagnostics. By illustrating the evolutionary trajectories in r - s, r - q, w -w and Om planes, we find that the holographic constant c and the coupling constant b play very important roles in the holographic dark energy (HDE) model. In addition, we also plot the LCDM horizontal lines in Om diagrams, and show the discrimination between the HDE and LCDM models.

关键词： modified Chaplygin gas interacting holographic dark energy geometrical diagnostics

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Commuting Involutions with Fixed Point Set of Variable Codimension

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Acta Mathematica Sinica,English Series 2001年第3期17卷 425-430页

作者： Yan Ying WANG Zhen De wu Zhao Jing MENG department of mathematics,Hebei normal university,Shijiazhuang 050016.P.R.china E-mail:Wang Yangying@*** department of mathematics Hebei Normal University Shijiazhuang 050016 P. R. China

Special generators of the unoriented cobordism ring MO* are constructed to determine some groups Jn,kl1,l2,…,lm of cobordism classes in MOn containing a representative Mn admitting a （Z2）k-action with the fixed poi... 详细信息

关键词： Indecomposable class Fixed point set Projective space bundle (Z2)k-action

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Molten solvent extraction behavior of trivalent La, Sm, Dy, and Yb with tri-n-butyl phosphate into molten paraffin wax at 60 ℃

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Rare Metals 2002年第1期21卷 7-11页

作者： PENG Bo, GAO Jinzhang, FAN Haiyan, KANG Jingwan, and YANG wu department of Chemistry, Northwest normal university, Lanzhou 730070, china department of Chemistry Northwest Normal University Lanzhou 730070

The extraction behavior of La^3+, Sm^3+, Dy^3+, and Yb^3+ insodium acetate-acetic acid medium was studied with tri-N-butylphosphate (TBP) at 60 deg. C using paraffin wax as a diluent. Theextraction percentage is greater than 85/100 in the pH range of 6 to8. The result of slope analysis method indicates that thecompositions of the extracted species are different between the lightand heavy rare ear this. The formula of the extracted species isfound to be La (TBP) (OH) (Ac)_2 for La^3+ and Yb (TBP) (OH)_3 forYb^3+. The effects of extracting time, the concentration of TBP inthe organic phase and salts on the extraction efficiency were alsodiscussed.

关键词： solvent extraction rare earth TBP Paraffin wax

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A MODIFIED HOMOGENEOUS AND SELF-DUAL LINEAR PROGRAMMING ALGORITHM

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Systems Science and Mathematical Sciences 1995年第3期8卷 270-277页

作者： GUO Tiande(mathematics department,Qufu normal university, Qufu 273165,china)wu Shiquan(Institute of Applied mathematics,Academia Sinia, Beijing 100080, china) mathematics department Qufu Normal University Qufu 273165 China

ＡＭＯＤＩＦＩＥＤＨＯＭＯＧＥＮＥＯＵＳＡＮＤＳＥＬＦ－ＤＵＡＬＬＩＮＥＡＲＰＲＯＧＲＡＭＭＩＮＧＡＬＧＯＲＩＴＨＭ￥ＧＵＯＴｉａｎｄｅ（ＭａｔｈｅｍａｔｉｃｓＤｅｐａｒｔｍｅｎｔ，ＱｕｆｕＮｏｒｍａｌＵｎｉｖｅｒｓ... 详细信息

ＡＭＯＤＩＦＩＥＤＨＯＭＯＧＥＮＥＯＵＳＡＮＤＳＥＬＦ－ＤＵＡＬＬＩＮＥＡＲＰＲＯＧＲＡＭＭＩＮＧＡＬＧＯＲＩＴＨＭ￥ＧＵＯＴｉａｎｄｅ（ＭａｔｈｅｍａｔｉｃｓＤｅｐａｒｔｍｅｎｔ，ＱｕｆｕＮｏｒｍａｌＵｎｉｖｅｒｓｉｔｙ，Ｑｕｆｕ２７３１６５，Ｃｈ...

关键词： Linear programming interior point algorithms homogeneity self-dual.

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