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Unformatted text preview: STAT5044: Regression and ANOVA, Fall 2011 Exam 1 on Nov 08 Brief Solution Please make sure to specify all of your notations in each problem GOOD LUCK! 1 Problem# 1. M 1 : y i = + i M 2 : y i = + 1 ( x i x )+ i (1.a) Assume that the true model is M 1 . But your fitted model is M 2 . What are the mean and variance of your estimators? Compare them with those in true model. Are your estimators unbiased or biased? Are your estimators larger or smaller variance than those in true model? Parameter estimators: Fitted: 1 f = s xy s xx , f = y , where s xy = ( x i x ) y i True: t = y and 1 t = 0. Mean Fitted: E ( 1 f ) = E ( 1 t ) = 0 because E ( x i x ) = True: E ( t ) = E ( y ) = Hence they are unbiased estimators. Variance: Fitted: var ( f ) = 2 1 n 1 s xx Hence var ( 1 f ) = 2 s xx and var ( f ) = 2 n True: var ( t ) = var ( y ) = 2 n and var ( 1 t ) = Hence var ( f ) = var ( t ) and var ( 1 f ) > var ( 1 t ) (1.b) In (1.a), the error follows a normal distribution. I would like to calculate confi dence interval (CI) of intercept parameter based fitted model. Compare your CI with CI in true model. Which one is wider? ( 1 ) 100 for t is ( t t / 2 , n 1 q 2 t n ) = ( y t / 2 , n 1 q 2 t n ) ( 1 ) 100 for f is ( f t / 2 , n 2 q 2 f n ) = ( y t / 2 , n 2 q 2 f n ) Width for true CI is 2 * t / 2 , n 1 q ( y i y ) 2 ( n 1 ) * n and Width for fitted CI is 2 * t / 2 , n 1 q ( y i y i ) 2 ( n 2 ) * n The difference between the square of width for true CI and the square of width for fitted CI is t 2 / 2 , n 1 ( y i y ) 2 ( n 1 ) * n t 2 / 2 , n 2 ( y i y i ) 2 ( n 2 ) * n z 2 / 2 { ( y i y ) 2 n 2 ( y i y i ) 2 n 2 } > because ( y i y ) 2 > ( y i y i ) 2 , CI for t is wider. 2 (1.c) Assume that the true model is M 2 . But your fitted model is M 1 . What are the mean and variance of your estimators? Compare them with those in true model. Are your estimators unbiased or biased? Are your estimators larger or smaller variance than those in true model? parameter estimators: Fitted: f = f = y True: c = ( X c X c ) 1 X c Y = y s xy s xx , where c = c 1 c and X c = [ 1 , x x ] mean Fitted: E ( 1 f ) = E ( 1 t ) = 1 True: E ( f ) = E ( y ) = Hence they are unbiased estimators....
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 Fall '11
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