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Unformatted text preview: STAT5044: Regression and ANOVA, Fall 2010 Exam 2 on Nov 17 Your Name: Please make sure to specify all of your notations in each problem GOOD LUCK! 1 Problem# 1. Consider the following model y i = β + β 1 x 1 i + β 2 x 2 1 i + β 3 x 2 i + ε i , i = 1 ,..., n where E ( ε i ) = 0, Var ( ε i ) = σ 2 (unknown), Cov ( ε i , ε j ) = 0. • (1.a) Under the assumption of ε i ∼ N ( , σ 2 ) , we would like to test whether H : β 2 = vs H a : β 2 6 = 0. What are the test statistics and its distribution? What is a pvalue of this testing? Test stattistic is F = RSS ( X 1 , X 2 ) RSS ( X 1 , X 2 1 , X 2 ) / (( n 3 ) ( n 4 )) RSS ( X 1 , X 2 1 , X 2 ) / ( n 4 ) = Y T ( H 3 H 2 ) Y Y T ( I H 3 ) Y / ( n 4 ) = Y T ( H 3 H 2 ) Y / σ 2 Y T ( I H 3 ) Y / ( n 4 ) σ 2 ∼ F 1 , n 4 Y T ( H [ 3 H 2 ) Y / σ 2 ∼ χ 1 because H [ 3 H 2 σ 2 * σ 2 is idempotent matrix. Y T ( I H 3 ) Y / ( n 4 ) σ 2 ∼ χ n 4 because I H 3 σ 2 * σ 2 is idempotent matrix and H 3 H 2 and I H 3 are independent. Decision rule: reject H if F > F 1 , n 4 , α • (1.b) Suppose that we found that normal assumption is violated. we still want to test whether H : β 2 = 0 or not. What is the test statistics and decision rule. Explain your testing procedure in detail. We can use Bootstrap. The procedure is the following. Step1: sample ( Y b , X 1 b , X 2 , b ) with replacement and calculate ˆ β 2 , b using ( X b X b ) 1 X b Y b , where X b = [ 1 , X 1 b , X 2 1 b , X 2 b ] . Step2: repeat step 1, B times Step3: Sort all β 2 , b , b = 1 ,..., B and obtain 95% CI using 2.5percentile and 97.5 per centile. Step4: If the 95% CI contains zero, conlculde H and otherwise reject H • (1.c) Using Boxcox transformation, we found that normality assumption holds using y 1 / 3 i . Then we would like to obtain prediction interval of new observation y i , new for a given new x 1 i , new The model after BoxCox transformation is y * i = y 1 / 3 i 1 1 / 3 = β + β 1 x 1 i + β 2 x 2 1 i + β 3 x 2 i + ε i , i = 1 ,..., n 2 A point estimator for ˆ y * i , new given X i , new = [ 1 , x 1 inew , x 2 1 inew , x 2 i ] is ˆ y new = X 1 new ˆ β , where ˆ β = ( ˆ β , ˆ β 1 , ˆ β 2 , ˆ β 3 ) . The prediction interval for y * inew is X 1 new ˆ β ± t n 4 q ˆ σ 2 + ˆ var ( ˆ y new ) = [ a , b ] ....
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This note was uploaded on 01/02/2012 for the course STAT 5044` taught by Professor Staff during the Fall '11 term at Virginia Tech.
 Fall '11
 Staff

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