An alternative to the model of Example 7.2.17 is the following, where we observe (Yi, Xi), i = 1,2, ...,n, where Yi ~ Poisson(mβτi) and (X1,..., Xn) ~ multinomial(m;Ï„), where Ï„ = (Ï„1, Ï„2,..., Ï„n) with ˆ‘ni=1 Ï„1= 1. So here, for example, we assume that the population counts are multinomial allocations rather than Poisson counts. (Treat m = ˆ‘xi as known.)
a. Show that the joint density of Y = (Y1,..., Yn) and X = (X1,..., Xn) is

b. If the complete data are observed, show that the MLEs are given by

c. Suppose that x1 is missing. Use the fact that X1 ~ binomial(m, t1) to calculate the expected complete-data log likelihood. Show that the EM sequence is given by

d. Use this model to find the MLEs for the data in Exercise 7.28, first assuming that you have all the data and then assuming that x1 = 3540 is missing.

ri yi Ti TL yi T -1,2,..n r1) and (r+1) nii j 1,2.,n