exercises for Tue Oct 8

1. On Tue Oct 8 I discussed aspects of Bayesian and classic frequentist bootstrapping, and illustrated this in the case of dtaa from an exponential distribution, with the task being to produce confidence intervals for each of \xi = mean, \sigma = standard deviation, \gamma = skewness. See com45a, the relevant Nils R script.

2. I also went through the basics of convergence of random processes on the space D[0, 1], of all right-continuous functions with left-hand limits, x:[0, 1] to R, with the Skorohod metric. Construction of various processes can be put up along the lines of Interesting Process = limit of simpler processes. Among these are Brownian Motion and the Gamma processes; see Exercises 20, 21, 22, 23, 24.

3. For Oct 8, do the following, an application of the Gamma process + Poisson process setup. (a) Consider a Gamma process Z(t) on [0, 1], with Z(t) being Gamma( b A(t), b), with A(t) the integral of a(s) = 10 + 0.75 \cos(2\pi s). Draw 25 samples of Z(t), for suitable values of the "prior certainty parameter" b. (b) Then create a Poisson process dataset Y(t), by simulating from an intensity process not equal to that of the prior for Z(t), but, rather, from Z_\true(t) = 10 + 0.05 \cos(2\pi s). (c) Construct the posterior process, Z(t) given the data Y. Simulate 25 samples from the posterior. Compute the Bayes estimator \hat Z(t), the posterior mean, along with a credibility band. (d) Attempt to generalise, in a couple of directions, one of which should be "more data".

3. After a bit more of Gamma processes, with Poisson or Bernoulli observations on top, we go to Beta processes and its applciations in survival data analysis.

4. Note: I've uploaded "version C" of the Nils Exercises and Lecture Notes (as of Oct 3, 2019, now 52 pages). Print it our for your convenience.