The random variables are independent and identically distributed with the geometric distribution
where is a parameter in the range of to be estimated. The mean of the above geometric distribution is .
Write down formulae for the maximum likelihood estimator for and for Fisher’s information;
Write down what you know about the distribution of the maximum likelihood estimator for this example when is large.
In a particular experiment, , .
Compute an approximate 95% confidence interval for based on the asymptotic distribution of the maximum likelihood estimator;
Compute the deviance and sketch it over the range . Use your sketch to describe how to use the deviance to obtain an approximate 95% confidence interval for ;
If you were asked to produce an approximate 95% confidence interval for the mean of the distribution , what would be your recommended approach? Justify your answer.