This paper presents a new model of asymmetric bifurcating autoregressive process with random coefficients. We couple this model with a Galton-Watson tree to take into account possibly missing observations. We propose least-squares estimators for the various parameters of the model and prove their consistency, with a convergence rate, and asymptotic normality. We use both the bifurcating Markov chain and martingale approaches and derive new results in both these frameworks.
Mots clés : autoregressive process, branching process, missing data, least squares estimation, limit theorems, bifurcating Markov chain, martingale
@article{PS_2014__18__365_0,
author = {Saporta, Beno{\^\i}te de and G\'egout-Petit, Anne and Marsalle, Laurence},
title = {Random coefficients bifurcating autoregressive processes},
journal = {ESAIM: Probability and Statistics},
pages = {365--399},
publisher = {EDP-Sciences},
volume = {18},
year = {2014},
doi = {10.1051/ps/2013042},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps/2013042/}
}
TY - JOUR AU - Saporta, Benoîte de AU - Gégout-Petit, Anne AU - Marsalle, Laurence TI - Random coefficients bifurcating autoregressive processes JO - ESAIM: Probability and Statistics PY - 2014 SP - 365 EP - 399 VL - 18 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps/2013042/ DO - 10.1051/ps/2013042 LA - en ID - PS_2014__18__365_0 ER -
%0 Journal Article %A Saporta, Benoîte de %A Gégout-Petit, Anne %A Marsalle, Laurence %T Random coefficients bifurcating autoregressive processes %J ESAIM: Probability and Statistics %D 2014 %P 365-399 %V 18 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps/2013042/ %R 10.1051/ps/2013042 %G en %F PS_2014__18__365_0
Saporta, Benoîte de; Gégout-Petit, Anne; Marsalle, Laurence. Random coefficients bifurcating autoregressive processes. ESAIM: Probability and Statistics, Tome 18 (2014), pp. 365-399. doi : 10.1051/ps/2013042. http://www.numdam.org/articles/10.1051/ps/2013042/
[1] , Proliferating parasites in dividing cells: Kimmel's branching model revisited. Ann. Appl. Probab. 18 (2008) 967-996. | MR | Zbl
[2] and , Non-Gaussian bifurcating models and quasi-likelihood estimation. J. Appl. Probab. A 41 (2004) 55-64. | MR | Zbl
[3] , and , Asymptotic analysis for bifurcating autoregressive processes via a martingale approach. Electron. J. Probab. 14 (2009) 2492-2526. | MR | Zbl
[4] , Asymptotic results for bifurcating random coefficient autoregressive processes (2012). Preprint ArXiv: 1204.2926.
[5] , The stochastic equation Yn + 1 = AnYn + Bn with stationary coefficients. Adv. Appl. Probab. 18 (1986) 211-220. | MR | Zbl
[6] and , Inference for the random coefficients bifurcating autoregressive model for cell lineage studies. J. Statist. Plann. Inference 81 (1999) 253-262. | MR | Zbl
[7] and , The bifurcating autoregressive model in cell lineage studies. Biometrics 42 (1986) 769-783. | Zbl
[8] , Tail of the stationary solution of the stochastic equation Yn + 1 = anYn + bn with Markovian coefficients. Stochastic Process. Appl. 115 (2005) 1954-1978. | MR | Zbl
[9] , and , Parameters estimation for asymmetric bifurcating autoregressive processes with missing data. Electron. J. Stat. 5 (2011) 1313-1353. | MR | Zbl
[10] , and , Asymmetry tests for bifurcating autoregressive processes with missing data. Stat. Probab. Lett. 82 (2012) 1439-1444. | MR | Zbl
[11] and , Detection of cellular aging in a Galton-Watson process. Stoch. Process. Appl. 120 (2010) 2495-2519. | MR | Zbl
[12] , Random iterative models, Applications of Mathematics, vol. 34. Springer-Verlag, Berlin (1997). | MR | Zbl
[13] , Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging. Ann. Appl. Probab. 17 (2007) 1538-1569. | MR | Zbl
[14] , , , , and , Statistical study of cellular aging, in CEMRACS 2004, mathematics and applications to biology and medicine, vol. 14, ESAIM: Proc. EDP Sci., Les Ulis (2005) 100-114 (electronic). | MR | Zbl
[15] and , Martingale limit theory and its application. Probability and Mathematical Statistics. Academic Press Inc., New York (1980). | MR | Zbl
[16] , Time series analysis. Princeton University Press, Princeton, NJ (1994). | MR | Zbl
[17] , The theory of branching processes. Die Grundlehren der Mathematischen Wissenschaften, Bd. 119. Springer-Verlag, Berlin (1963). | MR | Zbl
[18] , Robust inference for variance components models for single trees of cell lineage data. Ann. Statist. 24 (1996) 1145-1160. | MR | Zbl
[19] and , Extensions of the bifurcating autoregressive model for cell lineage studies. J. Appl. Probab. 36 (1999) 1225-1233. | MR | Zbl
[20] and , Inference for the extended bifurcating autoregressive model for cell lineage studies. Aust. N. Z. J. Stat. 42 (2000) 423-432. | MR | Zbl
[21] and , Variance components models for dependent cell populations. J. AMS 89 (1994) 19-29. | Zbl
[22] and , Branching Markov processes and related asymptotics. J. Multivariate Anal. 100 (2009) 1155-1167. | MR | Zbl
[23] and , Asymptotic optimal inference for multivariate branching-Markov processes via martingale estimating functions and mixed normality. J. Multivariate Anal. 102 (2011) 1018-1031. | MR | Zbl
[24] , and Random coefficient autoregressive models: an introduction. In vol. 11, Lect. Notes Statist. Springer-Verlag, New York (1982). | MR | Zbl
[25] , , and , Aging and death in an organism that reproduces by morphologically symmetric division. PLoS Biol. 3 (2005) e45.
[26] , Adaptive prediction by least squares predictors in stochastic regression models with applications to time series. Ann. Statist. 15 (1987) 1667-1682. | MR | Zbl
[27] and , Least-squares estimation for bifurcating autoregressive processes. Statist. Probab. Lett. 74 (2005) 77-88. | MR | Zbl
[28] and , Maximum likelihood estimation for a first-order bifurcating autoregressive process with exponential errors. J. Time Ser. Anal. 26 (2005) 825-842. | MR | Zbl
Cité par Sources :
