19  Maximum Likelihood Estimation: Calculus

Example 19.1 We wish to estimate the parameter \(\mu\) for a Poisson(\(\mu\)) distribution based on a single observed value \(X\).

1. Suppose that \(x=3\). Carefully write the likelihood function.
   
   
   
   
   
2. Suppose that \(x=3\). Carefully write the log-likelihood function.
   
   
   
   
   
3. Use calculus to find the MLE of \(\mu\) given \(x=3\). Compare to what we found previously using graphical methods.
   
   
   
   
   
4. Now consider a general \(x\). Carefully write the likelihood function.
   
   
   
   
   
5. Now consider a general \(x\). Carefully write the log-likelihood function.
   
   
   
   
   
6. Use calculus to find the MLE of \(\mu\) given \(x\). Compare to what we found previously using graphical methods.
   
   
   
   
   

Example 19.2 We wish to estimate the parameter \(\mu\) for a Poisson(\(\mu\)) distribution based on a random sample of \(n\) values \(X_1\ldots, X_n\).

1. Suppose that \(n=3\) and the sample is (3, 0, 2). Carefully write the likelihood function.
   
   
   
   
   
2. Suppose that \(n=3\) and the sample is (3, 0, 2). Carefully write the log-likelihood function.
   
   
   
   
   
3. Use calculus to find the MLE of \(\mu\) given \(n=3\) and the sample (3, 0, 2). Compare to what we found previously using graphical methods.
   
   
   
   
   
4. Now consider a general sample of size \(n\). Carefully write the likelihood function.
   
   
   
   
   
5. Now consider a general sample of size \(n\). Carefully write the log-likelihood function.
   
   
   
   
   
6. Use calculus to find the MLE of \(\mu\) given a sample of size \(n\). Compare to what we found previously using graphical methods.