General Point Process and Intensity Function

Let S1,S2,...,Sn be the arrival times of a general point process with conditional intensity function λ(t |Ht) over the observation interval [0,T].

(a) Write down an expression for the likelihood (joint distribution) of the arrival times and the fact that there were a total of n arrivals in the interval as a function of the conditional intensity function.

(b) Let Z1,Z2,...,Zn be a collection of random variables such that

Zi = ∫ λ(t |Ht)dt (Integral from Si-1 to Si).

Compute the marginal distribution of each Zi and show that they are mutually independent.

(c) If the point process is a homogeneous Poisson process, that is λ(t |Ht) = λ, find an estimator of λ that maximizes the likelihood function