Introduction to Statistical Linear Models — DU B.A. (Prog.) with Statistics as Major Discipline Previous Year Question Papers

5. Consider the simple line.ar regression model Y = Po + PIX + € with usual assumptions. (i) Show that the sum of the residuals weighted by the corresponding value of the regressor variable is always equal to zero. (ii) Find a 100(1 -or)% confidence interval for P]. 6. (a) Discuss the problem of testing for lack of fit in simple linear regression model. (b) For the multiple regression model, derive the test statistics for testing the significance of individual regression coefficients. (6,9) 7. Describe the technique of analysis of variance. Derive the analysis of variance of two way classified with one observation per cell under fixed effects mod.el. (15) 8. Write short notes on: (i) No intercept regression model (ii). Coefficients of determination (iii) Orthogonal polynomials Statistical Linear Models : B.A./B.Sc. 0) under NHP- 2. Attempt Six questions in all, selecting Three from each Section. 1. (a) Describe the general linear model and discuss briefly the different studies derived from it. (b) Consider the models Y[ = Pi + P2 + €i9 Y2 = Pi + P3 + €23 and Y3 = P] + P2 + €3 With usual assumptions. Prove that ^[Pi estimable if and only if ^] = ^2 (a) For a given model Ynxi = Xnxp Ppxi + €nxi Wit.h E(€) = 0, V(€) = o2I and R(X) = p < n. Obtain an unbiased estimator of o2. (b) Consider the models Yi = Pi + P2 + 03 + €i7 e4 With usual assumptions. (i) Write the above general linear model in matrix notation. (ii) Obtain BLUE of the unknown parameters along with its variance-covariance matrix. 3. Let Y = (Y], Y2 ,..., Yn)' be a vector o.f n independent standard normal variates. Let Q] = Y'A]Y and Q2 = Y'A2Y be distributed as %2 with n[ and n2 d.f. respectively. Prove that the necessary and sufficient condition for Q] and Q2 to be independently distributed isAiA2=0. . (15) 4. (a) Let Y ~ N(LL, I), where LL' = (2-13) and Z] = 1. Find expected value of Y'AY. Does Y'AY have a chi-square distribution? (b) Four objects A, 8, C, D are involved in a weighing experiment. Put together they weighed Y] grams; when A and C are put in the left pan of the balance and 8 and D are put in the right pan, a weight of Y2 grams was necessary in the right pan for the balance. With A and 8 in the left pan and C and D in the right pan, Y3 grams were needed in the right pan. Finally A and D in the left pan and 8 and C in the right pan, Y4 grams were needed in the right pan. If the observations Y], Y2, Y3, Y4 are all Sinbject to uncorrelated errors with common variance o2, obtain the BLUE of individual weights and the total weight of the four objects, and variance of the estimate of total weight of four objects. (8,7)