Given   determine the best estimator  under quadratic loss. Show that this estimator is a generalized Bayes estimator and discuss its admissibility.

(Hobert and Robert (1999)) Consider  with 

a. Give necessary and sufficient conditions on (a, b) for the posterior distribution to be defined and the prior distribution to be improper.

b. Give the transition kernel (8.2.7) in this case. In the special case b = −1/2 and a = k/2, show that the transition is a non-central chi-squared distribution.

c. Show that the transition kernel (8.7.1) is

where p = (b + 1)/(b + 2).

d. Show that this distribution corresponds to the usual negative binomial distribution when a is a positive integer.

e. In the general case, the distribution with mass function

f. Deduce that (xt) associated with the kernel K∗ is a branching process,

Where  And 

g. Conclude that the chain  is recurrent when b = 0 and 0 <>< 1,=”” and=”” transient=”” otherwise.=””>