stat675-test1-2014-15.doc - Addis Ababa University Department of Statistics Stat 675 Linear Models test 1 Date Time allowed 3 hours Instructions 1

stat675-test1-2014-15.doc - Addis Ababa University...

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Addis Ababa University Department of Statistics Stat. 675: Linear Models test 1 Date: December 29, 2014 Time allowed: 3 hours Instructions 1. Attempt all questions 2. Show all the necessary steps 3. Write legibly 3 1. When are the system of linear equations Ax =y (where A is an m n real matrix, y is an m 1 vector and x is an n 1 real vector) said to be consistent? Does the formal definition of consistent equations demand that linear relationships must exist among the rows of A? Explain. Under what condition are the equations Ax =y always consistent? 2. What does bilinear form mean? 3. a. If A n n is positive definite and symmetric, show that n n n x A x R A dx dx e 2 1 2 1 2 1 2 ... ...   b. For any vector g and a positive definite matrix W, show that g W g n n x g x W x e W dx dx e 2 1 2 1 2 1 2 1 2 ... ... 1 4. When x n 1 ~N( ,V), show that x 'Ax and x 'Bx are distributed independently iff
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Unformatted text preview: AVB = 0 or BVA=0 given that the r th cumulant of x 'Ax is 2 r-1 (r-1)![tr(AV) r +r 'A(VA) r-1 ] 5. For x n 1 ~N( ,V), and k<n, show that the marginal density function of y 1 where y' 1 =(x 1 ,x 2 ,...,x k ) is multivariate normal. 6. State and prove loynes lemma. 7. For the full rank model y =X + , show that the total sum of squares can be expressed as a sum of quadratic forms and give the distribution of the ratio of each to 2 . 8. For the model of full rank, y =X + for which all assumptions are met, derive an expression for the vector of estimators, H ˆ under the general linear hypothesis H: K =m. 9. What is the difference between outliers and influential data points? 10. What are the indicators of multicollinearity? Page 1 of 1...
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