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Unformatted text preview: 1 The University of Western Ontario Department of Statistical and Actuarial Sciences Statistical Sciences 3859a Assignment 1 Solutions 1. Suppose ( x 1 , y 1 ) , ( x 2 , y 2 ) , . . . , ( x n , y n ) constitute a sample of independent observations. Consider the model y i = β 1 x i + β 2 i + ε i where the ε i are i.i.d. N(0, σ 2 ) random variables, for i = 1 , 2 , . . . , n . (a) Derive the leastsquares estimators for β 1 and β 2 . Under what condition on the predictor ( x i ) are these estimators not welldefined? β 1 = ∑ i 2 ∑ x i y i ∑ ix i ∑ iy i ∑ i 2 ∑ x 2 i ( ∑ ix i ) 2 β 2 = ∑ x 2 i ∑ iy i ∑ ix i ∑ x i y i ∑ i 2 ∑ x 2 i ( ∑ ix i ) 2 If x i ∝ i , then the estimator is not welldefined. (b) For the case where the coefficient estimators are welldefined, write down an unbi ased estimator for σ 2 . Set y i = β 1 x i + β 2 i and e i = y i y i . Then an unbiased estimator for σ 2 is given by σ 2 = ∑ e 2 i n 2 2. Use R to complete the following problem, but do not use builtin functions such as lm() and predict() . A toy car has been released from ramps having nine different angles. Distance travelled (in m) has been measured in each case. angle 1.3 4.0 2.7 2.2 3.6 4.9 0.9 1.1 3.1 distance 0.43 0.84 0.58 0.58 0.70 1.00 0.27 0.29 0.63 (a) Plot the data. What is the predictor and what is the response? Is a linear model reasonable? (On physical grounds?, On statistical grounds?) > a13 > attach(a13) angle distance 1 1.3 0.43 4 4.0 0.84 7 2.7 0.58 10 2.2 0.58 13 3.6 0.70 16 4.9 1.00 19 0.9 0.27 22 1.1 0.29 25 3.1 0.63 > plot(distance ~ angle, pch=16) The predictor is angle, and the response is distance. (We want to see how distance changes with angle.) On the basis of the plot below, we see on statistical grounds that a linear model is reasonable. On physical grounds, such a model is sensible. We expect (for small angles) that the distance travelled would increase with angle, with the relation being at least approximately linear. 2 ● ●...
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This note was uploaded on 11/25/2010 for the course AS 3859 taught by Professor Braun during the Fall '10 term at UWO.
 Fall '10
 Braun

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