Consider the the uniform distribution X ~ U(0,0). We showed in class that the MLE of is ML max(X1, ..., Xn). (a) Show that the MM estimator of is MM = 2X, where X is the sample mean. (b) It can be shown that the theoretical means and variances for the MM and MLE estimators are E[MM] = 00 Var[MM]= = 0% 3n n E[ÔMLE] -80 n+1 Var[OMLE] = 0% n(n+2) Are the MM and ML estimators unbiased and/or consistent? What are their asymptotic distributions? (c) Set the true 0 to 1. Repeat the following steps for n = 20, 100, 1000: i. Use a random number generator to draw 10,000 samples of length n from U(0,0) and compute the MLE and MM estimators of 0 for each of the 10,000 samples. ii. Plot the histograms of the 10,000 MLE and MM estimators, respectively. iii. Compute the mean, bias and standard error of the MLE and MM estimators in the simulated data. (d) For each n, compare the properties of the MLE and MM estimators in these simulations to their theoretical distributions. (e) Which properties of the estimators in the simulated data are expected, i.e. close to their theoretical prop- erties, and which ones are not?