A Phase-Type Distribution for the Sum of Two Concatenated Markov
Processes. Application to the Analysis Survival in Bladder Cancer.
Abstract
Stochastic processes are very useful and have a very important role in
modeling the evolution of processes that take different states over
time, a situation frequently found in fields like Medical Research and
Engineering. In a previous paper and within this framework, we developed
the sum of two independent phase-type (PH) distributed variables, each
of them being associated with a Markovian process of one absorbing
state. In that analysis, we computed the distribution function, and its
associated survival function, of the sum of both variables also
PH–distributed. In this work, in one more step, we have developed a
first approximation of that distribution function to avoid the
calculation of an inverse matrix due to the possibility of bad
conditioning of the matrix involved in the expression of the
distribution function in the previous paper. Next, in a second step, we
improve this result, giving a second more accurate approximation. Two
numerical applications, one with simulated data and the other one with
bladder cancer data, are used to illustrate the two proposed approaches
to the distribution function. We compare and argue the accuracy and
precision of every one of them using their error bounds and the
application to real data of bladder cancer.