Lecture08-2010 - Empirical Maximum Likelihood: Lecture VIII...

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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 Y i =1 1 Γ( α ) β α x α 1 exp x β ! ln ( L )= N X 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 0 1 ln ( L )= T X t =1 [ ln (Γ ( α 0 + α 1 t )) ( α 0 + α 1 t )ln( β )+ ( α 0 + 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 Frst-order condi- tions for optimality. 1
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This note was uploaded on 07/15/2011 for the course AEB 6182 taught by Professor Weldon during the Fall '08 term at University of Florida.

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Lecture08-2010 - Empirical Maximum Likelihood: Lecture VIII...

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