1. Maximum Likelihood We have studied the OLS estimator. It only applies under certain assumptions In particular,  ~ N(0, 2 ) 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 dist’n 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 () is 2.45