1. Consider the tobit model with an unknown censoring point. All model details and assumptions are as in the body of the chapter (see (9.1) and (9.2)), except that (9.3) is replaced by yi D yŁi if yŁi > c yi D0 ifyŁi  c and c is an unknown parameter known to lie in the interval .0; 1/.

(a) Assuming a U.0; 1/ prior for

c, show how the Gibbs sampler with data augmentation described in Section 9.3 can be modified to allow for posterior inference about c.

(b) Write a program which uses your result from part

(a) to carry out Bayesian inference in the tobit model with unknown censoring point. Use your program on an artificial data set of the sort described in Section 9.3.1.

Note: p.cjyŁ; þ; h/ might not have a convenient form, but c is a scalar and will be defined over a restricted interval so even very crude methods might work well. For instance, an independence chain Metropolis–

Hastings algorithm which uses a U.0; 1/ candidate generating density would be very simple to program. Investigate whether such an algorithm is efficient. Can you come up with a more efficient algorithm?