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Unformatted text preview: Empirical Maximum Likelihood: Lecture VIII Charles B. Moss September 10, 2010 I. Empirical Maximum Likelihood and Stochastic Process A. To demonstrate the estimation of the likelihood functions using maximum likelihood, we formulate the estimation problem for the gamma distribution for the same dataset including a trend line in the mean. B. The basic gamma distribution function is L = N i =1 1 Γ( α ) β α x α- 1 exp- x β ⇒ ln ( L ) = N i =1- ln (Γ ( α ))- α ln ( β ) + ( α- 1) x i- x i β (1) C. Next, we add the possibility of a trend line max α ,α 1 ,β ln ( L ) = T t =1 [- ln (Γ ( α + α 1 t ))- ( α + α 1 t ) ln ( β ) + ( α + α 1 t- 1) x t- x t β (2) 1. Given the implicit nonlinearity involved, we will solve for the optimum using the nonlinear optimization techniques. 2. Most students have been introduced to the first-order condi- tions for optimality. 1 AEB 6182 Agricultural Risk Analysis and Decision Making Professor Charles B. Moss Lecture VIII Fall 2010 3. For our purposes, we will redevelop these conditions within3....
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- Fall '08
- Derivative, Taylor series expansion, Professor Charles B. Moss, Agricultural Risk Analysis, Empirical Maximum Likelihood