For the setup of Exercise 6.6.2, show that the following estimator of θ is unbiased:
Exercise 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 C1 into the two subcategories
Exercise 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
where the parameter θ satisfies 0 ≤ θ ≤ 1. In this exercise, we obtain the mle of θ.
(a) Show that likelihood function is given by
(b) Show that the log of the likelihood function can be expressed as a constant (not involving parameters) plus the term
(c) Obtain the partial derivative with respect to θ of the last expression, set the result to 0, and solve for the mle. (This will result in a quadratic equation that has one positive and one negative root.)