stat5044Exam2fall10v2sol

stat5044Exam2fall10v2sol - STAT5044: Regression and ANOVA,...

Info iconThis preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 = + 1 x 1 i + 2 x 2 1 i + 3 x 2 i + i , i = 1 ,..., n where E ( i ) = 0, Var ( i ) = 2 (unknown), Cov ( i , j ) = 0. (1.a) Under the assumption of i N ( , 2 ) , we would like to test whether H : 2 = vs H a : 2 6 = 0. What are the test statistics and its distribution? What is a p-value 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 / 2 Y T ( I- H 3 ) Y / ( n- 4 ) 2 F 1 , n- 4 Y T ( H [ 3- H 2 ) Y / 2 1 because H [ 3- H 2 2 * 2 is idempotent matrix. Y T ( I- H 3 ) Y / ( n- 4 ) 2 n- 4 because I- H 3 2 * 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 , (1.b) Suppose that we found that normal assumption is violated. we still want to test whether H : 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 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 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 Box-cox 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 Box-Cox transformation is y * i = y 1 / 3 i- 1 1 / 3 = + 1 x 1 i + 2 x 2 1 i + 3 x 2 i + i , i = 1 ,..., n 2 A point estimator for y * i , new given X i , new = [ 1 , x 1 inew , x 2 1 inew , x 2 i ] is y new = X 1 new , where = ( , 1 , 2 , 3 ) . The prediction interval for y * inew is X 1 new t n- 4 q 2 + var ( y new ) = [ a , b ] ....
View Full Document

Page1 / 11

stat5044Exam2fall10v2sol - STAT5044: Regression and ANOVA,...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online