Exercises for Tue Apr 10_亚博娱乐官网_亚博pt手机客户端登录

nils
2018-04-06T13:50:00.361Z
type:structured-message
www.uio.no

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 s...