Parameters estimation for asymmetric bifurcating autoregressive processes with missing data

Abstract

We estimate the unknown parameters of an asymmetric bifurcating autoregressive process (BAR) when some of the data are missing. In this aim, we model the observed data by a two-type Galton-Watson process consistent with the binary Bee structure of the data. Under independence between the process leading to the missing data and the BAR process and suitable assumptions on the driven noise, we establish the strong consistency of our estimators on the set of non-extinction of the Galton-Watson process, via a martingale approach. We also prove a quadratic strong law and the asymptotic normality.AMS 2000 subject classifications: Primary 62F12, 62M09; secondary 60J80, 92D25, 60G42.

Statistical study of asymmetry in cell lineage data

“…0.0369). These data were investigated in Guyon et al, 2005;Guyon, 2007;de Saporta et al, 2011;de Saporta et al, 2012). Stewart et al (2005) proposed a statistical study of the averaged genealogy and pair-wise comparison of sister cells.…”

“…The author investigates the method on the 94 data sets but it is not clear how he manages missing data. More recently, de Saporta et al (2011) proposed a single-tree analysis with a rigorous method to deal with the missing data and carried out their analysis on the largest data set, concluding to asymmetry on this single set. Further single-tree studies of the 51 data sets issued from the 94 colonies containing at least 8 generations were conducted in de Saporta et al (2012).…”

“…We first define our parametric model and introduce our notations. The BAR and GW processes have the same dynamics as in de Saporta et al (2011), the main difference is that our inference is here based on several i.i.d. realizations of the processes, instead of a single one.…”

“…An asymmetric BAR model was therefore proposed by Guyon (2007), where the two sets of parameters corresponding to sister cells are allowed to be different. Inference for this model was only investigated for single-trees growing to infinity, see Guyon (2007), Bercu et al (2009) for the fully observed model or Delmas and Marsalle (2010), de Saporta et al (2011), de Saporta et al (2012 for missing data models.…”

“…Indeed, we propose an inference based on a finite fixed number of replicated trees when the total number of observed cells tends to infinity. We use the missing data asymmetric BAR model introduced by de Saporta et al (2011). In this approach, the observed genealogies are modeled with a two-type Galton-Watson (GW) process.…”