Robust estimation in controlled branching processes: Bayesian estimators via disparities

Abstract

This paper is concerned with Bayesian inferential methods for data from controlled branching processes that account for model robustness through the use of disparities. Under regularity conditions, we establish that estimators built on disparity-based posterior, such as expectation and maximum a posteriori estimates, are consistent and efficient under the posited model. Additionally, we show that the estimates are robust to model misspecification and presence of aberrant outliers. To this end, we develop several fundamental ideas relating minimum disparity estimators to Bayesian estimators built on the disparity-based posterior, for dependent treestructured data. We illustrate the methodology through a simulated example and apply our methods to a real data set from cell kinetics.

Approximate Bayesian computation in controlled branching processes: the role of summary statistics

“…We consider a CBP with both the offspring and control laws belonging to each one-dimensional parametric families and denote the offspring and control parameters by θ and γ, respectively. Regarding the offspring law, it is usual to consider a parametric framework (see [3], [11], [12], [15], and [19]) since from previous observations or experiments, some information that suggests a family of distributions for the offspring law might be available (see [16] for further details). For instance, prokaryotic cells usually reproduce by binary fission and hence, one can parametrise the offspring distribution by considering the parameter θ defined as the probability that a cell splits off, and consequently, 1 − θ is the probability that a cell dies with no offspring.…”

“…Actually, this data is not so difficult to be observed because of recent advances in technology. For instance, within the cell kinetics setting in [12] a controlled two-type branching process is proposed for modelling a real data set where not only the number of individuals and progenitors are known in all the generations, but also the entire family tree. These data set is considered in Section 4.…”

“…In addition to illustrate that the methodology can be extended without too much difficulty, the aim of this example is to illustrate the ABC methodology in a situation in which the true posterior densities can be calculated. The population considered is a real population of oligodendrocyte cells already studied in [12], [17], and [29]. In these populations, one distinguishes two type of cells: the oligodendrocyte precursor cells -referred as T 1 -and the terminally differentiated oligodendrocytes -referred as T 2 .…”

“…With the aim of describing the proliferation of cells in these populations a continuous time branching process with emigration and a controlled two-type process for the embedded discrete structure were considered (see [12] and [29] for further details). Briefly, focussing on the embedded discrete structure, the controlled two-type branching process, denoted as {Zn} n∈N0 , is defined as follows:…”

Branching processes in varying environment with generation-dependent immigration

“…For instance, in [9] a multitype continuous-time branching process with immigration is used to describe a retrial queue, where the birth and death of the individuals represent the arrival and departure of a customer, respectively, and the situations when a customer is immediately served after the arrival into the system are modelled through the immigration. In the context of cell kinetics, a two-type age-dependent branching process with emigration is used to model the emigration of oligodendrocyte cells cultured in vitro out of the field of observation in [30]; more recently, in [7], a controlled two-type branching process with binomial control is used to describe its discrete branching structure as a result of an embedding of the aforesaid model. Another example is the application of an age-dependent branching process with time-inhomogeneous immigration for modelling the progression of Leukemia in mice in [13]; indeed, this process models the number of Leukemia cells in the blood and the immigration process represents the influx of cells from other tissues of the body.…”