Answer Happy

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.)