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

(ii) Obtain the V(Ô1), V(Ô2) and cov (01, 02) 3. Attempt six questions in all. and Adjusted R-Squared Statistic (Adj-R?). (b) Write the statement of Gauss Markov Theorem for full rank case. (c) Define a statistical linear model. Differentiate between fixed effects, random effects, and mixed effects linear models. • (d) Consider the following model with the usual assumptions: E (y,) = 2B,+ Bz, E(v2) = B, - Bz, EU3) = B, + aBy. Determine a so that the BLUES of By, Bz are uncorrelated. (e) Explain residual analysis and discuss its uses. -0.28xz. What is the distribution of 0.29x2 - 0.86xy + 0.64y2+0.072? + 0.28xz -0.42yz? Are these two quadratic forms independent? (b) Suppose we postulate the model E(Y) = Bot B,t but the model E(Y) = Bot B,×+B, x2+Bx3 is actually the true response function, unknown to us. If we use observations of Y at x= -2, -1, 0, 1, 2 to estimate B, and B, in the postulated model, what biases will be introduced? (7,7) 7. (a) Define multiple regression model and polynomial regression model. (b) Let X;, Y;, Z,, i=1,2,3 be 9 independent observations with common variance o?. If E(X) = O,, E(Y,) = (i) Find the BLUEs of 01,02 and 0, +02 (iii) At 0.05 level of significance, can one conclude that the classes of factor A have different effects? Why? (Use F0.0512,231=3.42, Fo.0517,141=2.76, 5. Let Y= (YI, Y2,.... Yn)' be a vector of n independent standard normal variates. Let Q,= =y'A Yand Q2= X'AzY be distributed as x? with ni and n2 degrees of freedom respectively. Prove that the necessary and sufficient condition for l, and Q2 to be independently distributed is A,A, = O. 6. (a) Let x, y, and z denote three independent standard normal variates. Stating appropriate theorem, find the distribution of homogenous equation of second degree, 0.71x2 + 0.86xy + 0.36y2+0.93z2 +0.42yz (f) An advertising company is investigating which type of advertisement (ad) increases the sale most. It chooses 16 cities, 4 selling agents and 4 types of ads. In each city an ad is assigned randomly to an agent for a period of four weeks and the figures on sales are used to find the effects of the ads. Give the name of the experimental design used in this case and also write the ANOVA table of the design. Consider the simple linear regression model (SLRM) y=Bo+BIXtE given that E(a) = O, V(E) = o, e's are uncorrelated. (i) Obtain the least squares estimates of Bo and B, and show that they are unbiased. (ii) Obtain the variance of Bo and Br: (iii) Find the confidence intervals of Bo,B1 and o2. (14) 3. For the one-way classified fixed effect model, yi = M + a, + Eiji i = 1,2,...,k; j= 1,2, ...,n, where the symbols have their usual meaning. (i) Obtain the estimates of the parameters 4 and aí's. (i) Derive the expectations of the various sum of squares. (iii) Write the ANOVA Table. (14) 4. (a) For the multiple linear regression model derive the test statistic for testing the significance of individual regression coefficients.