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Random coefficients bifurcating autoregressiveprocesses

Published online by Cambridge University Press:  03 October 2014

Benoîte de Saporta ,
Anne Gégout-Petit  and
Laurence Marsalle
Benoîte de Saporta
Affiliation:
Univ. Bordeaux, Gretha, UMR 5113, IMB, UMR 5251, 33400 Talence, France CNRS, Gretha, UMR 5113, IMB, UMR 5251, 33400 Talence, France INRIA Bordeaux Sud Ouest, team CQFD, 33400 Talence, France
Anne Gégout-Petit
Affiliation:
Univ. Bordeaux, IMB, UMR 5251, 33400 Talence, France CNRS, IMB, UMR 5251, 33400 Talence, France INRIA Bordeaux Sud Ouest, team CQFD, 33400 Talence, France
Laurence Marsalle
Affiliation:
Univ. Lille 1, Laboratoire Paul Painlevé, UMR 8524, 59655 Villeneuve d’Ascq, France CNRS, Laboratoire Paul Painlevé, UMR 8524, 59655 Villeneuve d’Ascq, France. [email protected]
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Abstract

This paper presents a new model of asymmetric bifurcating autoregressive process withrandom coefficients. We couple this model with a Galton−Watson tree to take into account possiblymissing observations. We propose least-squares estimators for the various parameters ofthe 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 resultsin both these frameworks.

Keywords

Autoregressive processbranching processmissing dataleast squares estimationlimit theoremsbifurcating Markov chainmartingale
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 365 - 399
Copyright
© EDP Sciences, SMAI 2014

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References

Bansaye, V., Proliferating parasites in dividing cells: Kimmel’s branching model revisited. Ann. Appl. Probab. 18 (2008) 967–996. Google Scholar
Basawa, I.V. and Zhou, J., Non-Gaussian bifurcating models and quasi-likelihood estimation. J. Appl. Probab. A 41 (2004) 55–64. Google Scholar
Bercu, B., de Saporta, B. and Gégout-Petit, A., Asymptotic analysis for bifurcating autoregressive processes via a martingale approach. Electron. J. Probab. 14 (2009) 2492–2526. Google Scholar
V. Blandin, Asymptotic results for bifurcating random coefficient autoregressive processes (2012). Preprint ArXiv: 1204.2926.
Brandt, A., The stochastic equation Y n + 1 = A nY n + B n with stationary coefficients. Adv. Appl. Probab. 18 (1986) 211–220. Google Scholar
Bui, Q.M. and Huggins, R.M., Inference for the random coefficients bifurcating autoregressive model for cell lineage studies. J. Statist. Plann. Inference 81 (1999) 253–262. Google Scholar
Cowan, R. and Staudte, R.G., The bifurcating autoregressive model in cell lineage studies. Biometrics 42 (1986) 769–783. Google Scholar
de Saporta, B., Tail of the stationary solution of the stochastic equation Y n + 1 = a nY n + b n with Markovian coefficients. Stochastic Process. Appl. 115 (2005) 1954–1978. Google Scholar
de Saporta, B., Gégout-Petit, A. and Marsalle, L., Parameters estimation for asymmetric bifurcating autoregressive processes with missing data. Electron. J. Stat. 5 (2011) 1313–1353. Google Scholar
de Saporta, B., Gégout Petit, A. and Marsalle, L., Asymmetry tests for bifurcating autoregressive processes with missing data. Stat. Probab. Lett. 82 (2012) 1439–1444. Google Scholar
Delmas, J.-F. and Marsalle, L., Detection of cellular aging in a Galton-Watson process. Stoch. Process. Appl. 120 (2010) 2495–2519. Google Scholar
M. Duflo, Random iterative models, Applications of Mathematics, vol. 34. Springer-Verlag, Berlin (1997).
Guyon, J., Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging. Ann. Appl. Probab. 17 (2007) 1538–1569. Google Scholar
J. Guyon, A. Bize, G. Paul, E. Stewart, J.-F. Delmas and F. Taddéi, 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).
P. Hall and C.C. Heyde, Martingale limit theory and its application. Probability and Mathematical Statistics. Academic Press Inc., New York (1980).
J.D. Hamilton, Time series analysis. Princeton University Press, Princeton, NJ (1994).
T.E. Harris, The theory of branching processes. Die Grundlehren der Mathematischen Wissenschaften, Bd. 119. Springer-Verlag, Berlin (1963).
Huggins, R.M., Robust inference for variance components models for single trees of cell lineage data. Ann. Statist. 24 (1996) 1145–1160. Google Scholar
Huggins, R.M. and Basawa, I.V., Extensions of the bifurcating autoregressive model for cell lineage studies. J. Appl. Probab. 36 (1999) 1225–1233. Google Scholar
Huggins, R.M. and Basawa, I.V., Inference for the extended bifurcating autoregressive model for cell lineage studies. Aust. N. Z. J. Stat. 42 (2000) 423–432. Google Scholar
Huggins, R.M. and Staudte, R.G., Variance components models for dependent cell populations. J. AMS 89 (1994) 19–29. Google Scholar
Hwang, S.Y. and Basawa, I.V., Branching Markov processes and related asymptotics. J. Multivariate Anal. 100 (2009) 1155–1167. Google Scholar
Hwang, S.Y. and Basawa, I.V., Asymptotic optimal inference for multivariate branching-Markov processes via martingale estimating functions and mixed normality. J. Multivariate Anal. 102 (2011) 1018–1031. Google Scholar
Nicholls, D. F., and Quinn, B. G. Random coefficient autoregressive models: an introduction. In vol. 11, Lect. Notes Statist. Springer-Verlag, New York (1982).
Stewart, E., Madden, R., Paul, G. and Taddei, F., Aging and death in an organism that reproduces by morphologically symmetric division. PLoS Biol. 3 (2005) e45. Google Scholar
Wei, C.Z., Adaptive prediction by least squares predictors in stochastic regression models with applications to time series. Ann. Statist. 15 (1987) 1667–1682. Google Scholar
Zhou, J. and Basawa, I.V., Least-squares estimation for bifurcating autoregressive processes. Statist. Probab. Lett. 74 (2005) 77–88. Google Scholar
Zhou, J. and Basawa, I.V., Maximum likelihood estimation for a first-order bifurcating autoregressive process with exponential errors. J. Time Ser. Anal. 26 (2005) 825–842. Google Scholar