An administrator at a certain prestigious university in the Northeast U.S. wants to estimate the time-to-degree of successful graduate students in the history department. Unfortunately, for the past 15 years the entering cohorts have been rather small, so there is not a great deal of data on program completers. Looking at the data, he observes that the minimum number of years in the sample is 4 and the maximum is 11. He decides to model the distribution as a “started exponential”; that is, T = 4+ Y, where Y has pdf = θ exp(-θy), with y >0 and θ > 0. Since the gamma distribution is conjugate to the exponential distribution, he decides to use that as a prior. He then manages to obtain some data from a few history departments at comparable universities. Unfortunately, he is only able to obtain mean times to degree. A reasonable aggregate value for the mean time to degree is about 6.5 years and he decides to use that as the mean of his prior.

1. He considers two different choices for the parameters of the Gamma prior: (a,b) = (5,2) and (a.b) = (7.5,3). Given the problem setting, which do you think is the better choice? Explain your answer.

2. The data from his institution consists of 45 observations with a mean of 6.1 years. What is the posterior distribution?

3. Compare the prior and posterior distributions. Comment.

4. What are some potential problems in using the “started exponential” as a model in this setting?