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Dr. Hackney STA Solutions pg 104

Dr. Hackney STA Solutions pg 104 - Second Edition 7-7 We...

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Second Edition 7-7 We know that ¯ x and ˆ σ 2 X = i ( x i - ¯ x ) 2 /n maximizes A ; the question is whether given σ Y , μ Y , and ρ , does ¯ x , ˆ σ 2 X maximize B ? Let us first fix σ 2 X and look for ˆ μ X , that maximizes B . We have log B ∂μ X - 2 i ( y i - μ Y ) - ρσ Y σ X ( x i - μ X ) ρσ Y σ X set = 0 i ( y i - μ Y ) = ρσ Y σ X Σ( x i - ˆ μ X ) . Similarly do the same procedure for L ( θ | y ) L ( θ, y | x ) This implies i ( x i - μ X ) = ρσ X σ Y i ( y i - ˆ μ Y ). The solutions ˆ μ X and ˆ μ Y therefore must satisfy both equations. If i ( y i - ˆ μ Y ) = 0 or i ( x i - ˆ μ X ) = 0, we will get ρ = 1 , so we need i ( y i - ˆ μ Y ) = 0 and i ( x i - ˆ μ X ) = 0. This implies ˆ μ X = ¯ x and ˆ μ Y = ¯ y . ( 2 log B ∂μ 2 X < 0. Therefore it is maximum). To get ˆ σ 2 X take log B ∂σ 2 X i ρσ Y σ 2 X ( x i - ˆ μ X ) ( y i - μ Y ) - ρσ Y σ X ( x i - μ X ) set = 0 . i ( x i - ˆ μ X )( y i - ˆ μ Y ) = ρσ Y ˆ σ X ( x i - ˆ μ X ) 2 .
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