Answer the following question manually and show your process:

A Gamma random variable has pdf 1 ,1 , a e 3W, 0 S 0,3 > 0. Suppose that we have a random sample of size n from a Gamma distribution (a) Find a twodimensional suicient statistic for a and 6. (b) Justify that the sufcient statistic also complete. (i.e. this a nice exponential family of distributions) (c) Find the method of moments estimators for a and ,6. (d) Suppose that or = 2 is known. Find the maximum likelihood estimator of F. (e) Suppose that both a and are unknown. In this case, there is no closed form solution for the pair (MLE, MLE). Show that you can get a numerical estimate of the MLE pair from a given dataset using simulation in R as follows: (1) Use set.seed(361561) and then simulate 100 random observations from a Gamma(a = 2, [3 = 5) using the R function rgamma(n = 100, shape = 2, scale = 5); (2) write an R function() for the gamma loglikelihood function (follow the normal example from lecture); (3) use the 'optim()' function in R to solve for the MLE (supply the initial values a = .5 and = .5 to the optim() function); (4) report the MLE estimate alongside the method of moments estimates for your simulated dataset for comparison. * Please provide the code and output to establish these methods and their results