The number N of species of insect caught in a trap during one night in a certain region is modelled by a distribution of the form 0" P(N = n)- n In(1-0) for n=1,2,3,..., where the unknown parameter p must lie between 0 and 1. Forty independent observations NN,..., N.o are made. (i) Show that the mean of this distribution is E(N)=-0[(1 - 0) In(1-0)]'. (4 marks) (ii) Find an equation that determines the maximum likelihood estimator, ê, of e. [Do not attempt to solve this equation.] (5 marks) (iii) The second derivative of the log-likelihood is given by + 40[1 + ln(1-7)] [(1 - 0) In(1 - 0)] Derive the Fisher information and hence find an approximate 95% confidence interval for 0, assuming that the maximum likelihood estimator is asymptotically efficient. Evaluate this confidence interval for the case where Ô = 0.75 (10 marks) Cont./... (iv) Suppose now that N, = 100. Describe an iterative method for finding the maximum likelihood estimate. Demonstrate three iteration steps of this method, using a starting value of p=0.70.
