Consider the model (14.10) and suppose that there is a single observation

=y, from each Level 2 unit i, and that

b, = b is the same for every unit

(i.e., the regression is a constant).

(a) Derive the negative binomial marginal distribution of y, I ( I , h. Show that the probability parameter of this distribution is p = l / ( b + 1).

(b) For the remainder of this problem, amme that (I = 4 i s known. The following data were obtained from 25 units:

4,2,8,ll,1,3,2,4,7,5,13,12,1,8,15,8,11,10,10,10, 2, 8, 7, 2, 14 Obtain the maximum likelihood estimate of p and hence of b.

(c) What is the posterior distribution of p I y under a uniform, that is, Beta( 1.1 ), prior distribution for p?

(d) What is the MLEB posterior distribution of (posterior mean, mode, and variance), given that y, = 8?

(e) Use a random number generator to obtain 10 draws from the posterior distribution of p (and hence of b). Approximate the full Bayes’
inference (mean and variance) by integrating over the distribution of b using your 10 draws. How good is the MLEB inference as an approximation to full Bayes inference?