Maximum Likelihood

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# Maximum Likelihood - PowerPoint PPT Presentation

Maximum Likelihood. We have studied the OLS estimator. It only applies under certain assumptions In particular, e ~ N(0, s 2 ) But what if the sampling distribution is not Normal? We can use an alternative estimator: MLE. See “Generalized Linear Models” in S-Plus. OLS vs. MLE.

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Maximum Likelihood
• We have studied the OLS estimator.
• It only applies under certain assumptions
• In particular, e ~ N(0,s2)
• But what if the sampling distribution is not Normal?
• We can use an alternative estimator: MLE. See “Generalized Linear Models” in S-Plus.
OLS vs. MLE
• If assumptions of OLS hold, OLS and MLE give exactly same estimates!
• So, using MLE instead of OLS is OK!
• MLE called “Generalized Linear Models” in S-Plus.
• More general than “Linear Regression”
• Allows you to specify dist’n of error.
Example: Ozone Attainment
• “Out of Attainment” if ozone exceeds standard on a given day.
• Model distribution of number of days out of attainment in a given county over 20 years.
• Use a Poisson Distribution
• Estimate the parameter using Maximum Likelihood.
MLE
• Principle: choose parameter(s) that make observing the given data the most probable (or “likely”).
• How do we measure “likelihood”?
• If we know sampling distribution, know how “probable” or “likely” any given data are.
• So we can measure likelihood.
• We must know the distribution.
Log-Likelihood
• Maximizing log-likelihood is equivalent to maximizing likelihood.
Solution
• We can model number of exceedences as Poisson distribution.
• 1 parameter.
• Estimated with maximum likelihood
• Estimated parameter (q) is 2.45