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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.
slide4
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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