σ 2 ln L n 2 σ 2 1 2 σ 2 2 n X i 1 x i μ 2 Solve for μ and σ 2 the simultaneous

Σ 2 ln l n 2 σ 2 1 2 σ 2 2 n x i 1 x i μ 2 solve

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( σ 2 ) ln L = - n 2 σ 2 + 1 2( σ 2 ) 2 n X i =1 ( x i - μ ) 2 Solve for μ and σ 2 , the simultaneous equations: ∂μ ln L = 0 , ( σ 2 ) ln L = 0 We also verify that L is concave at the solutions of these equations (Hessian matrix).

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Conclusion: The MLEs are ˆ μ = ¯ X , ˆ σ 2 = 1 n n X i =1 ( X i - ¯ X ) 2 ˆ μ is unbiased: E μ ) = μ . ˆ σ 2 is not unbiased: E σ 2 ) = n - 1 n σ 2 6 = σ 2 . For this reason, we use n n - 1 ˆ σ 2 S 2 .
Calculus cannot always be used to find MLEs. Return to the “cosmic ray composition” example. f ( x ; θ ) = exp( - ( x - θ )) if x θ 0 , if x < θ The likelihood function is L ( θ ) = f ( x 1 ; θ ) · · · f ( x n ; θ ) = exp[ - n i =1 ( x i - θ )] , if all x i θ, 0 , if any x i < θ
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• Maximum likelihood, Fisher Information

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