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Lecture07-2010 - Maximum Likelihood and Examples Lecture...

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Maximum Likelihood and Examples: Lecture VII Charles B. Moss September 2, 2010 I. Maximum Likelihood A. An alternative objective approach to estimating the parameters of a distribution function is by maximum likelihood. 1. The argument behind maximum likelihood is to choose those parameters that maximize the likelihood or relative probabil- ity of drawing a particular sample. 2. The likelihood function (or the probability of a particular sam- ple) can then be written as L = N i =1 f x i | μ, σ 2 = 2 πσ 2 N 2 exp 1 2 σ 2 N i =1 ( x i μ ) 2 (1) 3. Maximizing this function with respect to the parameters μ and σ 2 implies max μ,σ 2 L = 2 πσ 2 N 2 exp 1 2 σ 2 N i =1 ( x i μ ) 2 (2) 4. Taking the first-order conditions with respect to μ first 1
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AEB 6182 Agricultural Risk Analysis and Decision Making Professor Charles B. Moss Lecture VII Fall 2010 ∂L ∂μ = 2 πσ 2 N 2 exp 1 2 σ 2 N i =1 ( x 1 μ ) 2 × 1 2 σ 2 N i =1 [ 2 ( x i μ )] = 0 1 σ 2 N i =1 x i = 0 ˆ μ = 1 N N i =1 x i (3) 5. In order to solve for the first-order conditions with respect to the variance, we treat σ 2 as a single variable ∂L ∂σ 2 = N 2 (2 π ) N 2 σ 2 N 2 1 exp 1 2 σ 2 N i =1 ( x i μ ) 2 + 2 πσ 2 N 2 σ 2 2 2 N i =1 ( x i μ ) 2 exp 1 2 σ 2 N i =1 ( x i μ ) 2 = 0 = N σ 2 1 2 + σ 2 2 2 N i =1 ( x i μ ) 2 = 0 = 2 N i =1 ( x i μ ) 2 = 0 ˆ σ 2 = 1 N N i =1 ( x i μ ) 2 (4) 6. The derivation of the maximum likelihood estimates can be simplified by maximizing the logarithm of the likelihood func- tion. 2
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AEB 6182 Agricultural Risk Analysis and Decision Making Professor Charles B. Moss Lecture VII Fall 2010 ln ( L ) = N 2 ln σ 2 1 2 σ 2 N i =1 ( x i μ ) 2 ln ( L ) ∂μ = 1 2 σ 2 N i =1 [ 2 ( x i μ )] = 0 N i =1 x i = 0 ˆ μ = 1 N N i =1 x i ln ( L ) ∂σ 2 = N 2 1 σ 2 + 1 2 σ 2 2 N i =1 = 0 ⇒ − 2 + N i =1 ( x i μ ) 2 = 0 ˆ σ 2 = 1 N N i =1 ( x i μ ) 2 (5) B. The method of moments estimator and maximum likelihood esti- mator of the parameters of the normal distribution are the same.
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