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

\[ \se(\hat {\theta}) \;\;\approx\;\; \sqrt {- \frac 1 {\ell''(\hat {\theta})}} \;\;=\;\; \sqrt {\frac 1 {526.28}} \;\;=\;\; 0.0436 \]