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Unformatted text preview: STAT5044: Regression and ANOVA, Fall 2009 Exam 4 on Dec 14 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 i + β 3 X 2 2 i + σ g ( i ) ε i , i = 1 ,..., n where E ( ε i ) = 0, Var ( ε i ) = 1, Cov ( ε i , ε j ) = 0, and g ( i ) = i 2 • (a) When normality assumption is violated, how do you test the null hypothesis H : β 1 = β 3 = 0 ? Explain step by step. We can bootstrap approach. Let Y * = X * β + σε , where Y * = W 1 / 2 Y and X * = W 1 / 2 [ 1 , X 1 , X 2 , X 2 2 ] Step1: take a sample of size n from ( x * i , y * i ) with replacement Step2: from the above Bootstrap sample, calculate β b = ( X * X * ) 1 X * Y * Step3: Repeat B times Step4: Take ( α / 4 ) * 100 percentile and ( 1 α / 4 ) * 100 Step5: Check whether percentile based CI contains zero or not. If it contains zero, we fail to reject H • (b) Since we observe that there is no statistical significant evidence to reject H : β 1 = β 3 = 0, we fit the model (M1) Y i = β + β 2 X 2 i + σ g ( i ) ε i , i = 1 ,..., n What are the estimates of β 2 and σ 2 ? Are they independent to each other? Define X * = W 1 / 2 [ 1 , X 2 ] ˆ β 2 = ( X * X * ) 1 X * Y * σ 2 = Y * ( I * H ) Y * They are indepdent because ( X * X * ) 1 X * ( I * H ) = 0. • (c) In (b), however, the true model (M2) is Y i = β 2 X 2 i + σ g ( i ) ε i , i = 1 ,..., n . What are the mean and variance of slope estimator of your model (M1) and compare them with those of true model (M2). Which one is larger than the other? The slope estimator will be unbiased but M1 will overestimate the variance of the slope estimator. • (d) What is the ( 1 α ) × 100% prediction interval for a new observation Y when X = X new under the Model (M1) X * new ˆ β ± t α / 2 , n 2 s ˆ σ 2 ( 1 + 1 n + x * new ¯ x * s x * x * ) 2 Problem# 2....
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 Fall '11
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 Normal Distribution, Regression Analysis, Probability theory, Binomial distribution, Yi, Cumulative distribution function

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