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(m Icirc sup2 Iuml i) and (X1, , Xn) multinomial(m Iuml ), where Iuml ( Iuml 1, Iuml 2, , Iuml n) with ni 1 Iuml 1 1 So here, for example, we assume that the population counts are m...

a The joint density is the product of the individual densities b The log likelihood ...

An alternative to the model of Example 7.2.17 is the following, where we observe (Yi, Xi), i

Question:

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(mÎ²Ï„i) 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

An alternative to the model of Example 7.2.17 is the

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.