Abstract
We propose a novel procedure to test whether the immigration process of a discretely observed age-dependent branching process with immigration is time-homogeneous. The construction of the test is motivated by the behavior of the coefficient of variation of the population size. When immigration is time-homogeneous, we find that this coefficient converges to a constant, whereas when immigration is time-inhomogeneous we find that it is time-dependent, at least transiently. Thus, we test the assumption that the immigration process is time-homogeneous by verifying that the sample coefficient of variation does not vary significantly over time. The test is simple to implement and does not require specification or fitting any branching process to the data. Simulations and an application to real data on the progression of leukemia are presented to illustrate the approach.
Highlights
Age-dependent branching processes define a class of continuous-time stochastic processes that is increasingly popular in the biological sciences
The goal of this paper is to propose a test to determine whether the immigration process of an age-dependent branching process with immigration is time-homogeneous based on observations of the population size at discrete time points
We find that the coefficient of variation converges over time to a strictly positive constant when the immigration process is timehomogeneous
Summary
Age-dependent branching processes define a class of continuous-time stochastic processes that is increasingly popular in the biological sciences They offer flexible, yet tractable, stochastic models that describe population dynamics at the individual level, and allow the lifespan to follow arbitrary distributions (e.g., [1, 2, 3, 4, 5, 6, 7]). Sevastyanov (1957) was the first to study branching processes with immigration [8] He investigated the Markov case, and extensions to age-dependent processes were subsequently considered by Jagers and other authors in the single-type [9, 10, 11, 12, 13, 14, 15, 16, 17, 18 ] and the multi-type case [19, 20 ]. This class of processes has been proposed to model the dynamics of cell populations in vivo under the assumption that immigration obeys a (possibly time-inhomogeneous) Poisson process [17, 18, 20, 21 ]
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