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Unformatted text preview: Lecture 31 31.1 Statistical inference in simple linear regres sion. Let us first summarize what we proved in the last two lectures. We considered a simple linear regression model Y = β + β 1 X + ε where ε has distribution N (0 , σ 2 ) and given the sample ( X 1 , Y 1 ) , . . . , ( X n , Y n ) we found the maximum likelihood estimates of the parameters of the model and showed that their joint distribution is described by ˆ β 1 ∼ N β 1 , σ 2 n ( X 2 ¯ X 2 ) , ˆ β ∼ N β , 1 n + ¯ X 2 n ( X 2 ¯ X 2 ) σ 2 Cov( ˆ β , ˆ β 1 ) = ¯ Xσ 2 n ( X 2 ¯ X 2 ) and ˆ σ 2 is independent of ˆ β and ˆ β 1 and n ˆ σ 2 σ 2 ∼ χ 2 n 2 . Suppose now that we want to find the confidence intervals for unknown parameters of the model β , β 1 and σ 2 . This is straightforward and very similar to the confidence intervals for parameters of normal distribution. For example, using that n ˆ σ 2 /σ 2 ∼ χ 2 n 2 , if we find the constants c 1 and c 2 such that χ 2 n 2...
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This note was uploaded on 10/11/2009 for the course STATISTICS 18.443 taught by Professor Dmitrypanchenko during the Spring '09 term at MIT.
 Spring '09
 DmitryPanchenko
 Statistics, Linear Regression

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