The asymptotic formula for the standard error is
\[ \se(\hat {\theta}) \;\;\approx\;\; \sqrt {- \frac 1 {\ell''(\hat {\theta})}} \]The second derivative of the log-likelihood is found in the last iteration of the Newton Raphson algorithm.
Standard error for log-series distribution
The iterations of the Newton-Raphson algorithm for finding the MLE of the log-series distribution's parameter, \(\theta\), to the data on the previous page from an initial guess, \(\theta_0 = 0.7\) were:
Iteration, i | \(\theta_i\) | \(\ell'(\theta_i)\) | \(\ell''(\theta_i)\) |
---|---|---|---|
0 | 0.7000 | 52.656 | -240.78 |
1 | 0.9187 | -43.613 | -1200.14 |
: | : | : | : |
5 | 0.8628 | -0.000 | -526.28 |
6 | 0.8628 | -0.000 | -526.28 |
7 | 0.8628 |
The second derivative of the log-likelihood converges to \(\ell''(\hat{\theta}) = -526.28\). The approximate standard error of the estimate is therefore
\[ \se(\hat {\theta}) \;\;\approx\;\; \sqrt {- \frac 1 {\ell''(\hat {\theta})}} \;\;=\;\; \sqrt {\frac 1 {526.28}} \;\;=\;\; 0.0436 \]