problem 1. (Parcto MLE and MOME in astronomy) In astronomy, the distribution of asteroid sizes is of great interest (ever watched "Armageddon"?). Denoting the size of asteroid i X;, a common model is: fx (:) = else Here, A is the minimum size of an asteroid; let's call it 1 meter for now. Suppose we observe a sample of n asteroid sizes X1, ..., Xn. 1. Holding A fixed, what is the likelihood function for o? 2. What is the MLE of o? 3. Now, assume o is fixed. What is the likelihood function for #? What is its MLE? 4. Now assume that 8 = 3 meters. Find a MOM estimator for o (The mean of a pareto is of for a > 1). It is ok to assume o is at least 1. 5. Draw observations from a Pareto with o = 2 and 8 = 3. Compute the MLE and MOM estimates for o for various sample sizes. Compare the accuracy of the estimates in a plot, as a function of sample size