Mathematical statistics

Hint: Use the definition in part (a) and (b) for multinomial distribution

Suppose the random vector Y = (Y1, Y2, Y3, Y4) follows a multinomial distribution with cell probabilities (1/2 - 0/2, 0/4, 0/4, 1/2). Here 0 6 0, 1] is an unknown parameter. (a) If Y is observed, show that the MLE of d is D Yz + Y3 Yit Yz +Y3 (b) Suppose instead of Y, the observed data is y1 = 38, y2 = 34, and y3 + 94 = 125. Thus, we get to observe Yl, Y2, and the sum Y's + Y4, but not Y's and Y4 separately. Set up an EM algorithm which upon convergence gives the MLE of 0. Specifically, derive the E- and M-steps of the algorithm and show that at the end of an iteration of EM, Gold - the current value of 0 - will be updated as Anew 34 + 93,old where y3,old 125(Gold/4) 72 + 93,old 1/2 + Gold/4Definition 4.6.2 Let n and m be positive integers and let p1, ...,P. be num- bers satisfying 0 S p.