Problem
To model HIV sickness and mortality, an insurance company uses a time-inhomogeneous sickness-death model, with states H (healthy), S (sick) and D (dead), and age-dependent transition rates σ(t), µ(t), and ν(t) for transitions H → S, H → D, and S → D, respectively, where t denotes the current age (transition rate S → H is assumed to be 0). Write down the generator matrix for the process above. Draw the transition.
A company asks you to estimate the transition rates σ(t), µ(t), and ν(t) from the past data. Write down the formulas for the maximum likelihood estimators for the parameters in questions, explaining all notation used.
Assuming σ(t) = 0.001t, and µ(t) = 0.005t per annum, estimate the transition probability pHH(20,30).