Web-based Supplementary Materials for Likelihood-Based Inference for Discretely Observed Birth-Shift-Death and Multi-Type Branching Processes

Here we derive and solve the Kolmogorov backward equations of the two-type branching process necessary for evaluating the generating functions whose coefficients yield transition probabilities. See [Bailey, 1990] for an exposition on this solution technique. Our two-type branching process is represent by a vector (X1(t), X2(t)) that denotes the numbers of particles of two types at time t. Recall the quantities a1(k, l), the rates of producing k type 1 particles and l type 2 particles, starting with one type 1 particle, and a2(k, l), analogously defined but beginning with one type 2 particle. Then we may introduce respective pseudo-generating functions ui(s1, s2) = ∑ k ∑ l ai(k, l)s k 1s l 2 for i = 1, 2, and the probability generating functions can be expressed