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Lecture 20-2007

# Lecture 20-2007 - Concentrated Likelihood Functions Normal...

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Concentrated Likelihood Functions, Normal Equations, and Properties of Maximum Likelihood: Lecture XX I. Concentrated Likelihood Functions A. In the last lecture I introduced the concept of maximum likelihood using a known variance normal distribution of unknown mean. Undoubtedly this framework is fictional. Even if we know that the distribution is normal, it would be a rare event to know the distribution of the variance. B. What I would like to do now is to demonstrate how maximum likelihood functions may be applied by demonstrating how the variance can be concentrated out of the standard normal estimation. 1. The more general form of the normal likelihood function can be written as: 2 2 2 1 2 1 , exp 2 2 n i i X LX 2. Ignoring the constants, the natural logarithm of the likelihood function can be written as: 2 2 2 1 1 ln ln 22 n i i n 3. This expression can be solved for the optimal choice of 2 by differentiating with respect to 2 : 2 2 1 2 2 2 1 2 2 1 ln 1 0 2 2 0 1 ˆ n i i n i i n MLE i i L n X nX X n 4.

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Lecture 20-2007 - Concentrated Likelihood Functions Normal...

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