Minimum disparity estimation in controlled branching processes

Abstract

Minimum disparity estimation in controlled branching processes is dealt with by assuming that the offspring law belongs to a general parametric family.Under some regularity conditions it is proved that the minimum disparity estimators proposed -based on the nonparametric maximum likelihood estimator of the offspring law when the entire family tree is observed-are consistent and asymptotic normally distributed. Moreover, it is discussed the robustness of the estimators proposed. Through a simulated example, focussing on the minimum Hellinger and negative exponential disparity estimators, it is shown that both are robust against outliers, being the negative exponential one also robust against inliers.

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.…”

“…The novelty of the CBP lies in the presence of a mechanism establishing the number of individuals with reproductive capacity (progenitors) in each generation. Thus, the evolution of populations suffering from the existence of predators, populations of invasive species or different migratory movements can be modelled by using this branching process (see [11] for further details). The nature of this control mechanism can be either deterministic or random -described in this latter case by what are referred to as the control laws -and it gives rise to the models introduced by [26] and [30], respectively.…”

Robust estimation in controlled branching processes: Bayesian estimators via disparities

“…( c) In [18], conditions that guarantee the continuity of D(q, •) on Θ, for each q ∈ Γ, are established. In particular, it is sufficient to assume that such a disparity measure is defined by a bounded function G(•) and p k (•) is a continuous function on Θ, for each k ∈ N 0 .…”

“…Nevertheless, such a great flexibility of CBPs comes with a cost; namely, they require specification of multiple distributions such as the offspring distribution and the control distributions. It is well-known (see [18]) that specifying simultaneously the offspring distribution and control distributions for data analysis is challenging and is prone to misspecifications. Divergence-based methods have been used to provide methodologies for inference in these settings that are robust to presence of aberrant outliers and efficient when the posited model is correct, see [30] for the GWP and the more general approach given in [18] for the CBP.…”

“…It is well-known (see [18]) that specifying simultaneously the offspring distribution and control distributions for data analysis is challenging and is prone to misspecifications. Divergence-based methods have been used to provide methodologies for inference in these settings that are robust to presence of aberrant outliers and efficient when the posited model is correct, see [30] for the GWP and the more general approach given in [18] for the CBP. While these papers deal with the problem from the frequentist standpoint, divergence-based methods in the Bayesian framework in the context of branching processes with strong theoretical guarantees do not exist.…”