《实用回归分析》复习题.pdf

回归分析复习题 1 15 设 1 P和 2 P为两个投影阵 则 1 为投影阵 1 PPP 2 2 1221 0PPP P 2 1 MPP 为投影阵 12212 PPP PP 2 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 n ZZ be independent normally distributed random variables with ii E Z Define the non central chi square distribution in terms of the distribution of a random variable which is some function of the i Var Z C 1 i Z s You don t 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 centrality 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 n d A and c Consider the general linear model of full rank YX assuming 0E and 2 CovI and X is np of rank p Derive where 2 E S 2 SYX X Yn p is the LSE of Assuming is normally distributed prove that Y 2 np S 2 is distributed chi square What is the degrees of freedom What is the non centrality parameter d Consider the general linear model of full rank YX with 0E and where 2 CovV 1 p XJ XX r Xp1n Y 22 Derive the distributions of 1 1 PX X X 1 VI n Xn Y IJn Y 1 SSTand 1 1 nn nn JJ Q YY SSE 1 SSE 1 1 XSST J n IP Y JJ X SSRSSTSSE and prove that and are independent SSR SSE 4 a State the Gauss Markov theorem in the 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 Y X a P Y a X recall that in terms of their bias and variances Which estimator has the smaller variance 1 X PX X XX c How does your result in part b change if 2 Cov YV 1 and Show that is an idempotent matrix and 11 X PX X VXX V 11 1 X PX X VXX V X VXX V Y is the value of that minimizes 1 SYXV XY 5 a Derive the generalized likelihood ratio test for 0 H d when where 2 YNI is estimatable b Suppose the model for the pre intervention observations is 1 n 111 YX 1 and the model for the post intervention data is 2 n 222 YX 2 where 1 Xand 2 X 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 012 H can be expressed in terms of this single regression model as 0 H 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 n is a r vector of parameters and Xis an n r matrix not 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 fixed effects one way ANOVA model without restriction ijiij Y with 2 0 ij N for 1 2 1 i ijn 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 with interactions has 2 levels of each treatment The linear model can be written as 1 2 1 2 1 2 ijkijijijkij Yijkn a Show that in usual notations i 2 1 2 1 2 Rn nyyn ii 2 1 111 212 11 121 21222 Ryn yn yn nnn nn and iii 2 11 1 12 21 22 11122122 1 1 1 1 Ryyyynnnn b Illustrate 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 ijkiijij Yziaj n 其中 2 0 ij N 所有 ij 相互独立 a 求对照 lm lm 的BLUE b 求回归系数 的BLUE c 导出假设 0 H 0 和假设 01 H 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要条件是存在AAM为 n R的线性子空间 使得 n R在 M上的投影算子 M PA 10 设 0 n XNIUAX VBX WCX 令 cov 0U VU W 这里皆为矩阵 且秩 为r 若cov 证明UV A B Crn W 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 n R aXa 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 nn XN u IC为n阶对称矩阵 证明 T X CX服从非中心柯方分布的充要条件是 C为投影阵 14 证明在正态线性模型 中 参数 0 2 nn IN XY 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 n IDE XY 2 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 YX ED 中 参数 的广义最小二乘估 计 11 XXX Y 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 0 X为列满秩矩阵 17 设 2 11011 1 1 ii yNxin 2 220212 1 jj yNxjm 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 ij yin yjm 21110 H的似然比统计量和否定域 18 设 2 11011 1 1 ii yNxin 2 220212 1 jj yNxjm 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 ij yin yjm 01020 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 2 N