Quasi- and pseudo-maximum likelihood estimators for discretely observed continuous-time Markov branching processes

Abstract

This article deals with quasi- and pseudo-likelihood estimation in a class of continuous-time multi-type Markov branching processes observed at discrete points in time. “Conventional” and conditional estimation are discussed for both approaches. We compare their properties and identify situations where they lead to asymptotically equivalent estimators. Both approaches possess robustness properties, and coincide with maximum likelihood estimation in some cases. Quasi-likelihood functions involving only linear combinations of the data may be unable to estimate all model parameters. Remedial measures exist, including the resort either to non-linear functions of the data or to conditioning the moments on appropriate sigma-algebras. The method of pseudo-likelihood may also resolve this issue. We investigate the properties of these approaches in three examples: the pure birth process, the linear birth-and-death process, and a two-type process that generalizes the previous two examples. Simulations studies are conducted to evaluate performance in finite samples.

Parameter estimation for discretely observed linear birth‐and‐death processes

“…x(x−1) to derive that 3) and observing that N/( N n=1 Z n−1 ) → 0 a.s. as N → ∞ on Z N → ∞ [15], we obtain that the first and the third summand on the right-hand side of (7.3) tend to zero in probability conditionally on {Z N → ∞} as N → ∞. For the middle term, we use consistency ofm Z 0 ,N and (3.5) to derive that…”

“…Several authors have proposed approaches to address this instability, and more generally to estimate the parameters of discretely-observed general birth-anddeath processes, for which an analytic expression for the population size distribution may be unavailable. Chen and Hyrien [3] propose quasi-and pseudo-likelihood estimators, but their empirical analysis suggests that these are inferior to the maximum likelihood estimator (MLE). Crawford and Suchard [5] and Crawford et al [4] use continued fractions to obtain expressions for the Laplace transform of the population size distribution.…”

Parameter estimation for multivariate population processes: a saddlepoint approach

“…There is a considerable body of work on inverse problems for continuous-time population processes. In the first place we refer to for example [1][2][3][4][5] for parameter estimation procedures for univariate birth-death processes without modulation. In these papers the case is considered where the population is observed at discrete times only, hence the individual births and deaths are not observed directly.…”

A branching process model of heterogeneous DNA damages caused by radiotherapy in in vitro cell cultures

“…Unfortunately, the associated computational cost is heavy. Several stochastic modeling paradigms have been proposed to describe heterogeneity in tumors such as Markov chains [23,28,29,36], branching processes [25,35,10,16,37,41,13,31] and even stochastic di↵erential equations [11,7,1], but they were all focused on the steady-state responses of cell populations.…”