Exercises for Tue Feb 20

For Tue Feb 20, we attempt to finish what was not completed for the exercises last week. In addition:

Look at the Beta process construction via a "fine grid": Let A_m(t) be the sum of independent B_i \sim Beta(c a_0(i/m) (1/m), c (1 - a_0(i/m)(1/m)), over all i/m \le t. Work at the mean and variance of A_m(t), and take the limit as m goes to infinity. Then work with the Laplace transform \E\exp(-\theta A_m(t)), and find an expression for its limit.

Then *apply* the Beta process, with A the cumulative hazard for survival data. Let A be a Beta process with parameters (c, A_0), with A_0(t) = a_0 t, i.e. an exponential. Simulate paths from A(t). Play with a_0 and c.

Then take n data t_1, ..., t_n from some given distribution on (0,\infty). Update the A process, following the result that A given data is another Beta process, with parameters (c + Y, \hat A), with Y(s) = number of individuals at risk at time s, and \hat A(t) = \int_0^t (c dA_0 + dN) / (c + Y). Here N(t) = number of events over [0, t], which means that dN(s) is the number over events on [s, s+ds].