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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
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
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
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
Log-Likelihood

  • Maximizing log-likelihood is equivalent to maximizing likelihood.


Solution
Solution

  • We can model number of exceedences as Poisson distribution.

    • 1 parameter.

    • Estimated with maximum likelihood

  • Estimated parameter (q) is 2.45


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