Lecture20a-2010 - Empirical Maximum Likelihood and Method...

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Empirical Maximum Likelihood and Method of Moment Estimation: Lecture XXa Charles B. Moss October 21, 2010 I. Gamma Distribution Function f ( x | α, β )= x α 1 exp x β ! Γ( α ) β α (1) A. The likelihood function for a sample of random variables dis- tributed gamma can be expressed as L ( x 1 ,x 2 , ··· x n | α, β )= 1 (Γ ( α ) β α ) n n Y i =1 x α 1 i exp " x i β # . (2) B. Taking the logarithm of the likelihood function ln ( L ( x 1 ,x 2 , ··· x n | α, β )) = n ln (Γ ( α )) ln ( β ) +( α 1) n X i =1 ln ( x i ) 1 β n X i =1 x i (3) C. Note that if we de±ne two statistics T 1 = n X i =1 ln ( x i ) T 2 = n X i =1 x i (4) 1
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AEB 6571 Econometric Methods I Professor Charles B. Moss Lecture XXa Fall 2010 The sample likelihood function becomes ln ( L ( x 1 ,x 2 , ··· x n | α, β )) = n ln (Γ ( α )) ln ( β ) +( α 1) T 1 1 β T 2 (5) D. Code to estimate the parameters of the gamma in R is then
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This note was uploaded on 07/15/2011 for the course AEB 6180 taught by Professor Staff during the Spring '10 term at University of Florida.

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Lecture20a-2010 - Empirical Maximum Likelihood and Method...

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