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 ² x i | μ, σ 2 ³ = ² 2 πσ 2 ³ - N 2 exp ´ - 1 2 σ 2 N µ 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 2 2 ³ - N 2 exp ´ - 1 2 σ 2 N µ i =1 ( x i - μ ) 2 (2) 4. Taking the Frst-order conditions with respect to μ Frst 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 ² - N 2 exp ³ - 1 2 σ 2 N ´ i =1 ( x 1 - μ ) 2 µ × - 1 2 σ 2 N ´ i =1 [ - 2( x i - μ )] · =0 1 σ 2 N ´ i =1 x i - · ˆ μ = 1 N N ´ i =1 x i (3) 5. In order to solve for the ±rst-order conditions with respect to the variance, we treat σ 2 as a single variable ∂σ 2 = - N 2 (2 π ) - N 2 ± σ 2 ² - N 2 - 1 exp ³ - 1 2 σ 2 N ´ i =1 ( x i - μ ) 2 µ + ± 2 2 ² - N 2 ± σ 2 ² - 2 2 N ´ i =1 ( x i - μ ) 2 exp ³ - 1 2 σ 2 N ´ i =1 ( x i - μ ) 2 µ = - N ± σ 2 ² - 1 2 + ± σ 2 ² - 2 2 N ´ i =1 ( x i - μ ) 2 = - 2 N ´ i =1 ( x i - μ ) 2 ˆ σ 2 = 1 N N ´ i =1 ( x i - μ ) 2 (4) 6. The derivation of the maximum likelihood estimates can be simpli±ed 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 ² - 1 2 σ 2 N ³ i =1 ( x i - μ ) 2 ln ( L ) ∂μ = - 1 2 σ 2 N ³ i =1 [ - 2( x i - μ )] = 0 N ³ i =1 x i - =0 ˆ μ = 1 N N ³ i =1 x i ln ( L ) ∂σ 2 = - N 2 1 σ 2 + 1 2 ± σ 2 ² 2 N ³ i =1 ⇒- 2 + N ³ i =1 ( x i - μ ) 2 ˆ σ 2 = 1 N N ³ 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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Lecture07-2010 - Maximum Likelihood and Examples: Lecture...

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