MATH 423 midterm -2007

MATH 423 midterm -2007 - Student Name: Student Number:...

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Student Name: Student Number: Faculty of Science MIDTERM EXAMINATION Mathematics 423 Regression and Analysis of Variance Sample Midterm You must do all THREE questions. Calculators are allowed. One 8.5” × 11” sheet of notes is allowed. Language dictionaries are allowed. There are 8 pages to this exam and the total number of marks for the exam is 100. 1
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Question 1: Suppose that in the standard linear regression model that (1) y i = β 0 + β 1 x i + ǫ i where ǫ i Normal(0 , γ 2 i σ 2 ) (i.e. are normally distributed with mean 0 and variance γ 2 i σ 2 ) where the γ i values are all known constants, but σ 2 is unknown. (a) List the usual assumptions of the Simple Linear Regression model. (b) Can we use the usual Simple Linear Regression model to estimate the parameters β 0 and β 1 ? Why or why not? Now consider the following regression model: (2) γ - 1 i y i = γ - 1 i β 0 + γ - 1 i β 1 x i + γ - 1 i ǫ i (c) Show that model 2 satisFes the assumptions of the standard statistical model. Hint: Re- write model (2) as z i = u i β 0 + v i β 1 + δ i and relate z i , u i , v i and δ i to the variables in the original model. (d) ±ind the least squares estimates of β 0 and β 1 . (e) Show that performing least squares on the new model in part (b) is equivalent minimizing n i =1 ( y i - β 0 + β 1 x i ) 2 ( γ i ) - 2 . 2
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Consider the usual multiple linear regression model, Y = X β + ǫ. (a) Show (via matrix algebra) that the sample residuals are orthogonal to every column of X . Hint: Write the residuals in terms of Y , X , and ˆ β . (b) Show that the sum of the sample residuals is equal to zero. Hint: Use part (a). Sometimes, if the dimension of X is large, it is computationally easier to work with the Q-R decomposition of X , where X = Q R where Q is an orthogonal matrix ( Q t Q = I ) and R is upper-triangular ( r ij = 0 for i > j ). This is because, with a Q-R decomposition, we do not ever have to calculate the matrix X t X . (c) Show that
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This note was uploaded on 01/15/2010 for the course MATH 423 taught by Professor Steele during the Spring '06 term at McGill.

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MATH 423 midterm -2007 - Student Name: Student Number:...

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