lec30 - Lecture 30 30.1 Joint distribution of the estimates...

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30.1 Joint distribution of the estimates. In our last lecture we found the maximum likelihood estimates of the unknown pa- rameters in simple linear regression model and we found the joint distribution of ˆ β 0 and ˆ β 1 . Our next goal is to describe the distribution of ˆ σ 2 . We will show the following: 1. ˆ σ 2 is independent of ˆ β 0 and ˆ β 1 . 2. n ˆ σ 2 2 has χ 2 n - 2 distribution with n - 2 degrees of freedom. Let us consider two vectors a 1 = ( a 11 , . . . , a 1 n ) = ± 1 n , . . . , 1 n ² and a 2 = ( a 21 , . . . , a 2 n ) where a 2 i = X i - ¯ X q n ( X 2 - ¯ X 2 ) . It is easy to check that both vectors have length 1 and they are orthogonal to each other since their scalar product is a 1 · a 2 = n X i =1 a 1 i a 2 i = 1 n n X i =1 X i - ¯ X q n ( X 2 - ¯ X 2 ) = 0 . Let us choose vectors a 3 , . . . , a n so that a 1 , . . . , a n is orthonormal basis and, as a result, the matrix A = a 11 ··· a n 1 a 12 ··· a n 2 . . . . . . . . . a

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lec30 - Lecture 30 30.1 Joint distribution of the estimates...

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