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)