Asymptotic Analysis for Bifurcating AutoRegressive Processes via a Martingale Approach

Bercu, Bernard; Saporta, Benoı̂te de; Gégout-Petit, Anne

doi:10.1214/ejp.v14-717

Cited by 34 publications

(55 citation statements)

References 15 publications

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“…The next result is an adaptation of Lemma A.2 in Bercu et al (2009) to the GW tree framework. It gives a correspondence between sums on one generation and sums on the whole tree.…”

Section: B1 Preliminary Results: From Single-trees To Multiple Treesmentioning

confidence: 91%

“…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.…”

Section: Introductionmentioning

confidence: 99%

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Statistical study of asymmetry in cell lineage data

Saporta

Gégout-Petit

Marsalle

2014

Computational Statistics & Data Analysis

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A rigorous methodology is proposed to study cell division data consisting in several observed genealogical trees of possibly different shapes. The procedure takes into account missing observations, data from different trees, as well as the dependence structure within genealogical trees. Its main new feature is the joint use of all available information from several data sets instead of single data set estimation, to avoid the drawbacks of low accuracy for estimators or low power for tests on small single-trees. The data is modeled by an asymmetric bifurcating autoregressive process and possibly missing observations are taken into account by modeling the genealogies with a two-type Galton-Watson process. Least-squares estimators of the unknown parameters of the processes are given and symmetry tests are derived. Results are applied on real data of Escherichia coli division and an empirical study of the convergence rates of the estimators and power of the tests is conducted on simulated data.

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“…The next result is an adaptation of Lemma A.2 in Bercu et al (2009) to the GW tree framework. It gives a correspondence between sums on one generation and sums on the whole tree.…”

Section: B1 Preliminary Results: From Single-trees To Multiple Treesmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 99%

Statistical study of asymmetry in cell lineage data

Saporta

Gégout-Petit

Marsalle

2014

Computational Statistics & Data Analysis

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“…Basawa and Huggins [2,3] and Basawa and Zhou [4,5]. More recently, de Saporta et al [8,19] introduces asymmetry and take into account missing data while Blandin [14], Bercu and Blandin [7], and de Saporta et al [20] study an extension with random coefficients. Bitseki-Penda and Djellout [10] prove deviation inequalities and moderate deviations for estimators of parameters in linear BAR processes.…”

Section: )mentioning

confidence: 99%

“…This enables us to extend the recent results of Doumic et al in several directions: adaptive estimation, extension of the smoothness classes and the loss functions considered, and also a proof of a minimax lower bound. In Section 4.2, we show how bifurcating autoregressive models (BAR) as developped for instance in de Saporta et al [8] and Bitseki Penda and Olivier [13] are embedded into our generic framework of estimation. A numerical illustration highlights the feasibility of our procedure in practice and is presented in Section 4.3.…”

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confidence: 99%

Adaptive estimation for bifurcating Markov chains

Penda¹,

Hoffmann²,

Olivier³

2017

Bernoulli

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Abstract. In a first part, we prove Bernstein-type deviation inequalities for bifurcating Markov chains (BMC) under a geometric ergodicity assumption, completing former results of Guyon and Bitseki Penda, Djellout and Guillin. These preliminary results are the key ingredient to implement nonparametric wavelet thresholding estimation procedures: in a second part, we construct nonparametric estimators of the transition density of a BMC, of its mean transition density and of the corresponding invariant density, and show smoothness adaptation over various multivariate Besov classes under L p -loss error, for 1 ≤ p < ∞. We prove that our estimators are (nearly) optimal in a minimax sense. As an application, we obtain new results for the estimation of the splitting size-dependent rate of growth-fragmentation models and we extend the statistical study of bifurcating autoregressive processes.Mathematics Subject Classification (2010): 62G05, 62M05, 60J80, 60J20, 92D25,

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“…This allowed him to conclude a statistical evidence in aging in E. Coli. Let us also mention [6], where Bercu & Al. using the martingale approach give asymptotic analysis of the least squares estimator of the unknown parameters of a general asymmetric pth-order BAR processes.…”

Section: Introductionmentioning

confidence: 99%

Deviation inequalities for bifurcating Markov chains on Galton−Watson tree

Penda¹

2015

ESAIM: PS

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Abstract. We provide deviation inequalities for properly normalized sums of bifurcating Markov chains on Galton-Watson tree. These processes are extension of bifurcating Markov chains (which was introduced by Guyon to detect cellular aging from cell lineage) in case the index set is a binary Galton-Watson process. As application, we derive deviation inequalities for the least-squares estimator of autoregressive parameters of bifurcating autoregressive processes with missing data. These processes allow, in case of cell division, to take into account the cell's death. The results are obtained under an uniform geometric ergodicity assumption of an embedded Markov chain.

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Asymptotic Analysis for Bifurcating AutoRegressive Processes via a Martingale Approach

Cited by 34 publications

References 15 publications

Statistical study of asymmetry in cell lineage data

Statistical study of asymmetry in cell lineage data

Adaptive estimation for bifurcating Markov chains

Deviation inequalities for bifurcating Markov chains on Galton−Watson tree

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