Statistical Computing · End Term · May 2023 · AN

1. Let Xj, Xo,...,X, ~ iid F, where F is a 2component Geometric(p) mixture distribution with pmf L(x | piyp2, 71,7) = mifilz | pr) + m2fa(z | pr), where 0 < py,py < 1 and 7, 7% € (0,1), +7 = 1. Suppose we want to obtain the maximum likelihood estimate of # = (11,7, p1, po) using the EM algorithm. Note: Use the version of Geometric that models the number of trials needed to get the first success. (i) Recall that the E-step has the function: Q(0 | (4) = Eflog f(x,z | 8) | X =2,0 =O) Simplify this expression for the given likelihood. 6 marks| (ii) M-step: n > YVine,(k) (a) Show that pe(k+1) = it. (3 marks yy TiYie,(k) i=l n D Yelk) (b) Show that 71) == 3 marks| (c) Show that pe4+1), %e(de+1) is the maxima of q(p,7 | pa), 79). 2 marks| i Ly where e(k) = Sele | Peay) (k) > Jil | Pi,09)%5.09 # (iti) Write the steps of the EM algorithm for the 2-component Geometric(p) mixture distribution. {2 marks] 2. Consider a random variable X with the following cumulative distribution function: (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. [7 marks}