Statistical Computing · End Term · May 2025 · FN

(i). What role the moment argument plays in the function imp_exp? Explain your answer. [2 Marks} (ii). Identify h(x), f(z) and g(z) from the code and write down the importance sampling estimator 6). (4 Marks} (iii). What is an estimate of Var(),)? [2 Marks} 4. Consider an objective function f(z) that we want to maximize. The Minorize-Maximize algorithm uses a minorizing function f(6 | 6(4)) at the current iterate 64). Prove that the algorithm has the ascent property. [5 Marks} 5. Following are the BIC values obtained for different values of C for the clustering problem, where C denote the number of clusters: C=2:BIC=-113.1 C=3:BIC=-113.5 C=4:BIC=-110.0 C=5:BIC=-111.2 C=6:BIC=-1119 What value of C will the algorithm select: based on the BIC values obtained? [2 marks} Note: BIC(d | x) = 2logi(6 | x) — log(n)K 6. Let X1, Xo,...,Xn a F, where F has unknown mean @ and finite variance. Suppose we choose the estimator § = X to estimate 0. Write down the steps of bootstrapping to obtain a 100(1 —a)% confidence interval around #. Write all the steps clearly. [5 Marks}