LINEAR MODELS — DU B.Sc (hons) Statistics Previous Year Question Papers

Linear Models B.Sc. (Hons.) Statistics under NEP-UGCF 3. Attempt six questions in all. Consider the model E (Yr) = 2$ + $2, E (Y, = 0r - gz, E (Yr) = 0r - pr with usual assumptions. Obtain the BLUE of pr+ 2pz and its variance. State all three necessary and sufticienl conditions for which a linear parametric function is estimable. Suppose yi (i = 1,2, ..., n) is a random sample from a standard normal distribution. Show that Xilr yiand XILr@; - !)z are independently distributed. Complete the following ANOVA Table: Sources of variation Between groups Within groups l2 l) 25 Consider the model Ytxt = Xnxp\pxt * €nxt with E(g) = Q,lz(e) = o2I alnd, p(X) = r <p <-tL. State and prove the Gauss-Markov theorem when X is a non-full rank matrix. Also, obtain an unbiased estimator of the error vari*." o'. Suppose y : (1tr,y.r, ... ,y)'is avector of n independent standard normal variates. Prove that a necessary and sufficient condition for y'Ay Nrdy'By to be distributed as independent t' variates with nt and nz degree of freedom respeclively is that AB=0. Give the fixed effect mathematical model for the two-way classification with one observation per cell, stating clearly the assumptions involved. Also obtain: (D The estimates of the parameters in the model. (iD The expectation of the m (b) In an experiment to determine whether four different makes of automobile average the same gasoline mileage, a random sample of two cars of each make was taken from each of four cities. Each car was then test run on 5 gallons of gasoline of the same brand. The following ANOVA table was obtained for the number of miles obtained: Source of DF SS MS F-ratio Complete all the missing entries in the table above. (10, 4) 5. (a) Consider the simple linear regression model, Y = Bo + Bix + e with E(E) = 0, V(E) = o, E's are uncorrelated. (i) Verify the bias and Variance Properties of Bo and Br (i) Show that E(MSR) = 02+ B}Sxx (i) Show that Cov (Bo. B1) = -x. ? (b) Suppose the postulated model is E(Y) = Bo + B1x, but the true model is E(Y) = Bo + BIxy + Bzxz. Show that both Bo and By are biased by an amount that depends the values of x's. (9,5) 6. (a) Explain the test for the significance of individual regression coefficient in multiple linear regression model. (b) Show that, for any linear model (c) For a general linear model, find the 100(1 - a)% prediction interval for the future observation on y corresponding to particular values of the regressor variables. 7. (a) Explain the role of orthogonal polynomials in fitting polynomial models in one variable. (b) Let Xi, Yi, 4j, i = 1,2,3 be 9 independent observations with common variance o?. Find the BLUEs of 1,02,01 + 02 (ii) Obtain the V (01), V (02), cov (01, 02). 8. Write notes on any Two of the following: Coefficient of determination Stepwise regression method. Lack of fit in simple linear regression model. (7,7)