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1、回归分析复习题 1.(15)设1P和2P为两个投影阵，则（1）为投影阵；1PPP2212210PPP P（2）1MPP为投影阵。12212PPP PP2.在用逐步回归的方法来选择自变量的过程中，为什么剔除自变量的显著性水平不能小于引入自变量的显著性水平。若引入自变量的显著性水平和剔除自变量的显著性水平相等，且第一步和第二步能引入两个自变量，试证明第三步不可能剔除任何自变量。3.(a)Let 1,nZZ be independent normally distributed random variables with(),iiE Z Define the non-central chi-squa
2、re distribution in terms of the distribution of a random variable,which is some function of the()iVar ZC1.iZs.(You dont need to derive a density function,just state that the non-central chi-square distribution is defined as the distribution of).What is the degrees of freedom?What is the non-centrali
3、ty parameter?C(b)Let(,)YNI，where is an Y1n random vector.Prove that if A is an symmetric idempotent matrix of rank，then the quadratic form has a non-central chi-square distribution.Give the degrees of freedom and the non-centrality parameter(which should be given as a function of n ndA and).(c)Consi
4、der the general linear model of full rank YX assuming()0Eand 2()CovI and X is np of rank p.Derive where，2E(S)2()SYXX()/(Yn)p is the LSE of。Assuming is normally distributed,prove that Y2()np S2/is distributed chi-square.What is the degrees of freedom?What is the non-centrality parameter.(d)Consider t
5、he general linear model of full rank YX with()0E and，where 2()CovV1(,p)XJ XX，()r Xp1n Y22,，.Derive the distributions of 11(PX X X(1),VI(/)nXnY IJn Y/(1)SSTand(1,1),nnnnJJ Q,YY SSE/(1)SSE11),XSSTJ()nIP YJJ,XSSRSSTSSE ,and prove that and are independent.SSRSSE4.(a)State the Gauss-Markov theorem in the
6、 context of the ordinary linear model,where()E YX and 2()Cov YI.(b)Again in the context of the ordinary linear model,compare to as estimators of a YXa P Ya X(recall that )in terms of their bias and variances.Which estimator has the smaller variance?1()XPX X XX(c)How does your result in part(b)change
7、 if 2()Cov YV1 and，Show that is an idempotent matrix，and11()XPX X VXX V11()1()XPX X VXX VX VXX VY is the value of that minimizes 1()(SYXV()XY.5.(a)Derive the generalized likelihood ratio test for 0H:d when,where 2(,)YNI is estimatable.(b)Suppose the model for the pre-intervention observations is 1n1
8、11YX1 and the model for the post-intervention data is 2n222YX2,where 1Xand2X are observation matrices of the same k explanatory variables,both with rank k.Show how the two sets of data can be combined into a single regression model of the formYX so that the hypothesis 012H:can be expressed in terms
9、of this single regression model as0H:0.Carefully define all notation you use,including and,Y X.And develop the regression F test for testing this hypothesis.6.Consider the standard linear model with()E YX where Yis a random vector with elements;nis a r-vector of parameters;and Xis an n r matrix not
10、necessarily of full rank.(a)Let()a be a linear function of.Give the definition that()is estimable.Derive(with justifications)a necessary and sufficient condition for so that a(is estimable in terms of the matrix X and how it relates to.You need to show both necessity and sufficiency.a(b)Consider a f
11、ixed effects one-way ANOVA model without restriction ijiijY with 2(0,)ijNfor 1,2;1,iijn.In each of the following questions,if your answer is yes,then provide an estimator;if your answer is no,then prove it.i.Is 1 estimable?ii.Is 12 estimable?iii.Is 12 estimable?7.Suppose a 2-way cross-classification
12、 with interactions has 2 levels of each treatment.The linear model can be written as:,1,2;1,2;1,2,.ijkijijijkijYijkn (a)Show that,in usual notations,i.21.2.1.2.(|)()/Rn nyyn ii.2.1.111.212.11 121.21222.(|,)()/(/)Ryn yn yn nnn nn and iii.211.1 12.21.22.11122122(|,)()/(1/1/1/1/).Ryyyynnnn (b)Illustrat
13、e the above results by calculating the analysis of variance table for the following data:1 1 2 4 2 1 2 9 10 2 factorfactorObservations，1 63 2 2 10 12，8.对有一个协变量的单向分类模型,1,;1,ijkiijijYziaj,n 其中2(0,)ijN，所有ij相互独立，(a)求对照,lmlm 的BLUE.(b)求回归系数的BLUE.(c)导出假设0H:0 和假设01H:a的检验统计量。(d)列出相应的协方差分析表。9.为n阶方阵。证明为投影阵的充分必
14、要条件是存在AAM为nR的线性子空间，使得nR在M上的投影算子。MPA 10.设(0,),nXNIUAX VBX WCX令(,)cov(,)0U VU W，这里皆为矩阵，且秩为r，若cov，证明UV,A B CrnW与独立。11.设X为维随机向量。证明：nX服从维正态分布n),(uNn)的充要条件是对任意非零向量，有服从正态分布nRaXa,(aauaN。（提示：必要性可用多元正态分布的特征函数来证明）12.举例：a)随机变量X和Y不相关，但不相互独立 b)随机变量X与Y独立，X与Z独立，但X与(,)f Y Z不独立 13.设(,),nnXN u IC为n阶对称矩阵。证明TX CX服从非中心柯方
15、分布的充要条件是C为投影阵。14.证明在正态线性模型中，参数),0(2nnINXY的最小二乘估计为的极大似然估计。15.证明在线性模型中，参数nIDEXY2)(,0)(的最小二乘估计YX为XX)(1的最小方差线性无偏估计（其中X为列满秩矩阵）。16.(20)证明在线性模型2()0,()YXED中，参数的广义最小二乘估计*11()XXXY为的最小方差线性无偏估计，即对的任一线性无偏估计Ay，有*cov，其中(Ay)cov()0X为列满秩矩阵。17.设,211011 1(,),1,iiyNxin2220212(,),1,jjyNxjm,且相互独立。试写出相应的线性模型，并导出检验假设12,1,1,
16、ijyin yjm21110:H的似然比统计量和否定域。18.设,211011 1(,),1,iiyNxin2220212(,),1,jjyNxjm,且相互独立。试写出相应的线性模型，并导出检验假设12,1,1,ijyin yjm01020,1121:H的似然比统计量和否定域。19.天平称重时，随机误差为服从。实重为),0(2N321,的物体按下列方法称重 其中 1 表示物体放在天平左边，-1 表示物体放在天平右边，0 表示物体没有放在天平上。为所需加砝码的重量，若，表示砝码放在天平右边，y00yy则表示砝码加在天平左边。试估计321,。若把三物体都放在左边再称一次，求所需加砝码的水平为 95%的置信区间（可认为y,00）。182.3)3(025.0t1x 2x 3x y-1 1 1-0.5 1-1 1 9.1 1 1-1 10.3 1 0 0 9.5 0 1 0 5 0 0 1 4.5