Maximum Likelihood

1
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.

2
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 distn of error.

3
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.

4
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.

5
Graph of Likelihood
6
Log-Likelihood

• Maximizing log-likelihood is equivalent to
  maximizing likelihood.

7
Solution

• We can model number of exceedences as Poisson
  distribution.
• 1 parameter.
• Estimated with maximum likelihood
• Estimated parameter (q) is 2.45