In the next example, there is no exact formula for the standard error of the estimator.

Question: Geometric random sample

If \(\{x_1, x_2, \dots, x_n\}\) is a random sample from a geometric distribution with parameter \(\pi\), find a large-sample 90% confidence interval for the parameter \(\pi\).

(Solved in full version)

In the next example, the Newton-Raphson algorithm should be used to obtain the maximum likelihood estimate and its standard error.

Question: Log-series distribution

The following data set that is assumed to arise from a log-series distribution with probability function

\[ p(x) \;=\; \frac {-1} {\log(1-\theta)} \times \frac {\theta^x} x \quad\quad \text{for } x=1, 2, \dots \]
3 5 1 4 8 10 2 1 1 2
1 8 1 6 13 1 6 2 1 3
1 1 1 2 1 6 1 1 1 1

Find a large-sample 95% confidence interval for the parameter \(\theta\).

(Solved in full version)