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Slides08-2010

# Slides08-2010 - Outline Empirical Maximum Likelihood...

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Outline Empirical Maximum Likelihood Empirical Maximum Likelihood: Lecture VIII Charles B. Moss September 10, 2010 Charles B. Moss Empirical Maximum Likelihood

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Outline Empirical Maximum Likelihood Empirical Maximum Likelihood Newton-Raphson Example Charles B. Moss Empirical Maximum Likelihood
Outline Empirical Maximum Likelihood Newton-Raphson Example Empirical Maximum Likelihood 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. 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) Charles B. Moss Empirical Maximum Likelihood

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Outline Empirical Maximum Likelihood Newton-Raphson Example Next, we add the possibility of a trend line max α 0 1 ln ( L ) = T 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) Given the implicit nonlinearity involved, we will solve for the optimum using the nonlinear optimization techniques.
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