Statistical Computing · End Term · May 2023 · FN

1. Let Xy, Xo,..., Xv iid F, where F is a 2-component Exponential(\) mixture distribution with density f(x | At, A2, 1,72) = mfa(z | 1) + rafolz | 2), where Ay, Ay > 0 and 7,7 € (0,1),7 + 7 = 1. Suppose we want to obtain the maximum likelihood estimate of § = (77,7, 44, Ag) using the EM algorithm. (i) Recall that the E-step has the function: Q(0 | Ag) = Ellog f(x,z| 6) | X = 2,0 =O] Simplify this expression for the given likelihood. 6 marks (ii) M-step: n py Yie(h) (a) Show that A. 41) == : (3 marks » Tick) n y Fisch) (b) Show that m,a41) == 3 marks (c) Show that .(441), 7(k+1) 18 the maxima of q(A, 7 | An), 7(4))- 2 marks Se(ti | rea) Ret) E » Filta | Ax). Fs (iti) Write the steps of the EM algorithm for the 2-component Exponential(.) mixture distribution. [2 marks] Where Yin(t) = 2. Consider a random variable X with the following cumulative distribution function: 2 F(z) =" = O<c<l (a) Write down the steps of the Inverse transform method to sample from the given distribution. [3 marks] (b) Write down the steps for the Accept-reject to sample from this distribution. [7marks}