- Problem 5 2 In Exercise 6 6 2 Provide An Expression For O 1 Based On K K 1 2 What Is The Limit Of O K 1 (24.05 KiB) Viewed 36 times
6.6.1. Rao (page 368, 1973) considers a problem in the estimation of linkages in genetics. McLachlan and Krishnan (1997) also discuss this problem and we present their model. For our purposes, it can be described as a multinomial model with the four categories C1,C2,C3, and C4. For a sample of size n, let X = (X1, X2, X3, X4)' denote the observed frequencies of the four categories. Hence, n = Li-1X;. The Χ probability model is = , 4 = C1 +20 C2 C3 C4 1-20 1-10 | 20 4 4 where the parameter 0 satisfies 0) < 0 < 1. In this exercise, we obtain the mle of 0. (a) Show that likelihood function is given by X 1 X2+x3 24 L(@x) n! 1 x}!x2!*3!*4! 2 = mabababa [3+" 14-4** *** 7) 1 0 0 (6.6.22)
(b) Show that the log of the likelihood function can be expressed as a constant (not involving parameters) plus the term Xı log[2 +0] + [x2 + x3] log[1 – 0] + x4 log 0. (c) Obtain the partial derivative with respect to 0 of the last expression, set the result to 0, and solve for the mle. (This will result in a quadratic equation which has one positive and one negative root.)
6.6.2. In this exercise, we set up an EM algorithm to determine the mle for the situation described in Exercise 6.6.1. Split category Cį into the two subcategories C11 and C12 with probabilities 1/2 and 0/4, respectively. Let Z11 and Z12 denote the respective "frequencies.” Then X1 211 + Z12. Of course, we cannot observe 211 and 212. Let Z = (Z11, Z12)'. = (a) Obtain the complete likelihood Lº(@|x, z). (b) Using the last result and (6.6.22), show that the conditional pmf k(z|0, x) is binomial with parameters Xı and probability of success 0/(2+0). (c) Obtain the E step of the EM algorithm given an initial estimate ô(0) of 0. That is, obtain Q[0@0), x) = Egolog L*(0|X, Z) 60), x). (l ] = Recall that this expectation is taken using the conditional pmf k(zlão), x). Keep in mind the next step; i.e., we need only terms that involve 0. (d) For the M step of the EM algorithm, solve the equation Q(0]@0), x)/30 = 0. Show that the solution is B1), x1@0) + 2x4 + xx@co) nő0) + 2(x2 + x3 + x4) (6.6.23)