STA6934: Monte Carlo Statistical Methods Spring 2010Prof. CasellaAssignment Chapter Due Date Problems1 1 Friday 1/29 1.1, 1.4, 1.7, 1.13, 1.22,1.23ab,1.24 (Details are in Casella and Berger),1.28Assignment Chapter Due Date Problems2 2-3 Friday 2/12 2.6cd, 2.17abc,2.19,2.25a - also- write an R program to generatethe noncentral χ2using both representations2.31,3.1, 3.3b, 3.14 (just use g1and g4), 3.21abNote: In Problem 3.14, g4should be the normal density.Assignment Chapter Due Date Problems3 5 and 6 Fri 2/26 5.1, 5.9ab, 5.10, 5.17, 5.24, 5.296.7, 6.12a, 6.21, 6.22Note: I have never assigned Problem 5.24 before, so I don’t know how hardit is. I suggest that you do not leave it to the end!Assignment Chapter Due Date Problems4 6 and 7 Friday 3/19 6.22, 6.35ab, 6.40a, 6.677.2, 7.9, 7.21abc, 7.27, 7.30a,7.39, 7.44Hints:1. 6.40a: Use the notation that we discussed in class, and show thatsupA|F1(A) − F2(A)| =Z|f1(x) − f2(x)|dx.First show that0 =ZA(f1(x) − f2(x))dx +ZAc(f1(x) − f2(x))dx,1and that this implies |RAf1(x)−f2(x)dx| = |RAcf1(x)−f2(x)dx|. Therest follows.2. 7.39: You might want to look at problems 6.47 and 6.50 - you can as-sume the the results in those problems are true. In particular, writingthe result of Problem 6.50 in m atrix form sh ows thatlimN →∞1Nvar NXi=1h(Xi)!= h′B[2Z − I − A]h,where A and Z are defined in Problem 6.47 and B is a diagonal matrixwhose entries are πi, the stationary distribution.Assignment Chapter Due Date Problems5 7 and 8 Friday 4/9 8.1(a), 8.4, 8.5, 9.2, 9.4, 9.8(a), 9.14(a),9.17(1)(2), 9.26(a)(b)Notes:1. 9.8 is a difficult problem. You do n ot h ave to use the ARS algorithmif you can find another way. You also do not have to compare to theanswer in Example 9.7.2. After you do 9.14(a), give the details of a Gibbs sampler to generate theθs. Write an R (or other) program to get a histogram of the posteriordistribution of the θs.3. For 9.26, fit the model to the data of Table 11.1 (see also Figure 11.1)for k=3 and k=4. You will have to choose values of the hyperparam-eters. The data are in a .txt file that you can download.Assignment Chapter Due Date Problems6 10 and 12 Friday 4/23 10.10cd,10.12,10.13, (consider only the case p = 2)10.14, 10.19ab, 10.28Additional Problems1. Problem 5.18: Solve the problem as stated and, in addition, write aGibbs sampler to estimate the parameters. Compare the answers fromEM and Gibbs. → Next Page22. This is based on the material in Chapter 12, but you have the toolsto solve it. For the model and data of Problem 10.28, derive threeestimators of the αi. Monitor convergence of the chain with the threeestimators, by plotting their cumulative