tut3_09 - Show that for appropriate values of ν 1 and ν 2...

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1 The University of Western Ontario Department of Statistical and Actuarial Sciences SS 3859a Regression Analysis Tutorial 3 Questions: 1. Suppose n observations have been taken on a variable y and on k fixed explanatory variables x 1 , x 2 , . . . , x k . Consider the multiple regression model y ˜ = X β ˜ + ˜ , where ˜ = B z ˜ for an invertible n × n matrix B which has the property B 1 ˜ = 1 ˜ (1 ˜ is the vector of ones), and z ˜ is a vector of n independent standard normal random variables. An intercept is included in the model. The least-squares estimator for β ˜ is given by β ˜ = ( X T B - T B - 1 X ) - 1 X T B - T B - 1 y ˜ (a) Show that β ˜ is an unbiased estimator for β ˜ with variance-covariance matrix given by ( X T B - T B - 1 X ) - 1 . (b) It can be shown that the residual sum of squares is given by SS E = z ˜ T I - B - 1 X X T B - T B - 1 X - 1 X T B - T z ˜ , and under the hypothesis ( H 0 : β 1 = β 2 = · · · = β k = 0, it can be shown that the regression sum of squares is given by SS R = z ˜ T B - 1 X ( X T B - T B - 1 X ) - 1 X T - J /n B - T z ˜ . J is the n ×
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Unformatted text preview: Show that for appropriate values of ν 1 and ν 2 , SS R /ν 1 SS E /ν 2 has an F-distribution. Iden-tify ν 1 and ν 2 . 2. Suppose { ( x i , y i ) , i = 1 , 2 , . . . , n } are independent observations taken from data gov-erned by the model y = β 1 x + ε where ε is exponentially distributed with parameter 1. Suppose that the x values are all positive. (a) Find the maximum likelihood estimator for β 1 . (b) Estimate β 1 given the following data: x 1 2 3 4 5 y 21 41 60 80 100 (c) Does ε satisfy all of the usual assumptions for linear regression? (d) Bonus: Show that the maximum likelihood estimator for β 1 is biased, and provide an alternative estimator which is unbiased. Also, calculate the variance of the maximum likelihood estimator....
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