Exercises for Tue Apr 10

1. For a Gamma process Z(t), with parameters (a t, 1), followed over the time interval [0, 1], find the distribution of its biggest jump J. Generalise to the biggest jump J(\tau) when Z(t) is followed over [0, tau], and also to the case where Z is Gamma with parameters (a M(t), 1). Hint: work out that the cdf \Gamma_\eps(t) for a Gamma(\eps, 1) can be expressed as 1 - \eps E_1(t) + small, where E_1(t) = \int_t^\infty (1/x) \exp(-x) dx is the so-called exponential integral. Study aspects of this distribution for J(\tau).

2. Suppose certain creatures have gamma processes (a M(t), 1) in their rucksacks, and that they live until the biggest jump reaches a threshold c. Find the distribution for their lifetimes T. Create regression models that are consistent with the Cox regression model (and others that are not).

3. Consider iid data x_1, .., x_n from an unknown distribution P on [0, 1], where the problem is estimating the mean \theta(P) = \int_0^1 x dP(x), with squared error loss. With P from a Dir (a P_0), find the Bayes estimator; then its risk function; then the minimum Bayes risk, say mbr(a, P_0). When is this at its maximal value? Find a minimax estimator for \theta. Check the Nils exercise on page 34.

4. Do points (a) and (b) for the skiing days at Bj?rnholt exercise, page 32-33 in Nils Collection.

I will otherwise continue with Bayesian nonparametric regression themes.