Nonparametric estimation of the division rate of an age dependent branching process
Hoffmann, Marc; Olivier, Adélaïde (2016), Nonparametric estimation of the division rate of an age dependent branching process, Stochastic Processes and their Applications, 126, 5, p. 1433-1471. 10.1016/j.spa.2015.11.009
TypeArticle accepté pour publication ou publié
External document linkhttps://arxiv.org/abs/1412.5936v2
Journal nameStochastic Processes and their Applications
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Abstract (EN)We study the nonparametric estimation of the branching rate B(x)B(x) of a supercritical Bellman–Harris population: a particle with age xx has a random lifetime governed by B(x)B(x); at its death time, it gives rise to k≥2k≥2 children with lifetimes governed by the same division rate and so on. We observe in continuous time the process over [0,T][0,T]. Asymptotics are taken as T→∞T→∞; the data are stochastically dependent and one has to face simultaneously censoring, bias selection and non-ancillarity of the number of observations. In this setting, under appropriate ergodicity properties, we construct a kernel-based estimator of B(x)B(x) that achieves the rate of convergence View the MathML sourceexp(−λBβ2β+1T), where λBλB is the Malthus parameter and β>0β>0 is the smoothness of the function B(x)B(x) in a vicinity of xx. We prove that this rate is optimal in a minimax sense and we relate it explicitly to classical nonparametric models such as density estimation observed on an appropriate (parameter dependent) scale. We also shed some light on the fact that estimation with kernel estimators based on data alive at time TT only is not sufficient to obtain optimal rates of convergence, a phenomenon which is specific to nonparametric estimation and that has been observed in other related growth-fragmentation models.
Subjects / KeywordsGrowth-fragmentation; Cell division; Nonparametric estimation; Bias selection; Minimax rates of convergence; Bellman–Harris processes
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