Algorithm - Eberhardt - Newton-Raphson - Fisher - Statistics Assignment Help

Assessment Task:

Algorithm,  Eberhardt, Newton-Raphson, Fisher, Statistics Assignment Help

The following questions relate to the following field experiment. Edwards and Eberhardt (1967) conducted a live-trapping study on a confined population of known size. In their study, wild cottontail rabbits were penned in a 4-acre rabbit-proof enclosure. Live trapping was conducted for 18 consecutive nights. Recorded capture frequencies were as follows: 43 rabbits were caught at exactly 1 night, 16 rabbits at 2 nights, 8 at 3 nights, 6 at four nights, 2 at 6 nights and 1 rabbit was caught at 7 nights. In
other words, we have observations from X1, · · · Xm count variables of m = 76 di erent, caught rabbits. So, in this sample, 43 of the x1, · · · , xm were ones, 16 twos, etc. Every count can take a maximum of T = 18 captures. The number of of zero counts, i.e. the number of rabbits that were not caught, is unobserved. We denote this additional sample as Xm+1, · · · , Xn. Assuming independence of trapping

1. Determine the log-likelihood function for the observed sample as well as its first and second derivatives. 
2. Give the exact forms of the Newton-Raphson algorithm, the Fisher-scoring algorithm and the Gauss-Newton algorithm. It is enough to provide forms how the new iterate i+1 is built from
i. For each algorithm, provide the estimator of the variance of the MLE. 
3. Provide the details of the EM algorithm, in particular give details
(i) on the Q(j+1, j) function
(ii) on the E-step and M-step
(iii) why the EM algorithm is particularly attractive here. [20 marks]
4. Important for the convergence of an algorithm can be the choice of the starting values. Show that a moment estimator is available here in closed form, based on the fact that E(X) = T and E(X2) = T + (T)2. For details see lecture 11. [10 marks]
5. Write R-code that implements the four algorithms. Provide the iterations for the MLE, the associated log-likelihood of the observed, truncated likelihood as well as an estimate of the variance of the MLE at termination for each of the algorithms (except the EM). Comment on the convergence behaviour of the algorithms such as speed of convergence (number of iterations until convergence) or dependence on starting value. Which one would you prefer and why?
[30 marks]
6. What is your estimate of the size of the cottontail rabbit population for this defined community?How would you achieve an estimate of the size if you would use the Newton-Raphson algorithm instead of the EM algorithm?

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