Deviation inequalities and moderate deviations for estimators of parameters in bifurcating autoregressive models

S. Valère Bitseki Penda; Hacène Djellout

Annales de l'I.H.P. Probabilités et statistiques (2014)

  • Volume: 50, Issue: 3, page 806-844
  • ISSN: 0246-0203

Abstract

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The purpose of this paper is to investigate the deviation inequalities and the moderate deviation principle of the least squares estimators of the unknown parameters of general p th-order asymmetric bifurcating autoregressive processes, under suitable assumptions on the driven noise of the process. Our investigation relies on the moderate deviation principle for martingales.

How to cite

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Bitseki Penda, S. Valère, and Djellout, Hacène. "Deviation inequalities and moderate deviations for estimators of parameters in bifurcating autoregressive models." Annales de l'I.H.P. Probabilités et statistiques 50.3 (2014): 806-844. <http://eudml.org/doc/271956>.

@article{BitsekiPenda2014,
abstract = {The purpose of this paper is to investigate the deviation inequalities and the moderate deviation principle of the least squares estimators of the unknown parameters of general $p$th-order asymmetric bifurcating autoregressive processes, under suitable assumptions on the driven noise of the process. Our investigation relies on the moderate deviation principle for martingales.},
author = {Bitseki Penda, S. Valère, Djellout, Hacène},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {deviation inequalities; moderate deviation principle; bifurcating autoregressive process; martingale; limit theorems; least squares estimation; binary tree structure; least-squares estimator; superexponential convergence},
language = {eng},
number = {3},
pages = {806-844},
publisher = {Gauthier-Villars},
title = {Deviation inequalities and moderate deviations for estimators of parameters in bifurcating autoregressive models},
url = {http://eudml.org/doc/271956},
volume = {50},
year = {2014},
}

TY - JOUR
AU - Bitseki Penda, S. Valère
AU - Djellout, Hacène
TI - Deviation inequalities and moderate deviations for estimators of parameters in bifurcating autoregressive models
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2014
PB - Gauthier-Villars
VL - 50
IS - 3
SP - 806
EP - 844
AB - The purpose of this paper is to investigate the deviation inequalities and the moderate deviation principle of the least squares estimators of the unknown parameters of general $p$th-order asymmetric bifurcating autoregressive processes, under suitable assumptions on the driven noise of the process. Our investigation relies on the moderate deviation principle for martingales.
LA - eng
KW - deviation inequalities; moderate deviation principle; bifurcating autoregressive process; martingale; limit theorems; least squares estimation; binary tree structure; least-squares estimator; superexponential convergence
UR - http://eudml.org/doc/271956
ER -

References

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  1. [1] R. Adamczak and P. Milos. CLT for Ornstein–Uhlenbeck branching particle system. Preprint. Available at arXiv:1111.4559. Zbl1321.60035
  2. [2] I. V. Basawa and J. Zhou. Non-Gaussian bifurcating models and quasi-likelihood estimation. J. Appl. Probab.41 (2004) 55–64. Zbl1049.62115MR2057565
  3. [3] B. Bercu, B. de Saporta and A. Gégout-Petit. Asymtotic analysis for bifurcating autoregressive processes via martingale approach. Electron. J. Probab.14 (2009) 2492–2526. Zbl1190.60019MR2563249
  4. [4] B. Bercu and A. Touati. Exponential inequalities for self-normalized martingales with applications. Ann. Appl. Probab.18 (2008) 1848–1869. Zbl1152.60309MR2462551
  5. [5] V. Bitseki Penda, H. Djellout and A. Guillin. Deviation inequalities, moderate deviations and some limit theorems for bifurcating Markov chains with application. Ann. Appl. Probab.24 (2014) 235–291. Zbl1293.60036MR3161647
  6. [6] R. Cowan and R. G. Staudte. The bifurcating autoregressive model in cell lineage studies. Biometrics42 (1986) 769–783. Zbl0622.62105
  7. [7] V. H. de la Peña, T. L. Lai and Q.-M. Shao. Self-Normalized Processes. Limit Theory and Statistical Applications. Probability and Its Applications (New York). Springer-Verlag, Berlin, 2009. Zbl1165.62071MR2488094
  8. [8] B. de Saporta, A. Gégout-Petit and L. Marsalle. Parameters estimation for asymmetric bifurcating autoregressive processes with missing data. Electron. J. Stat.5 (2011) 1313–1353. Zbl1274.62192MR2842907
  9. [9] B. de Saporta, A. Gégout-Petit and L. Marsalle. Asymmetry tests for bifurcating auto-regressive processes with missing data. Statist. Probab. Lett.82 (2012) 1439–1444. Zbl1296.62161MR2929798
  10. [10] J. F. Delmas and L. Marsalle. Detection of cellular aging in a Galton–Watson process. Stochastic Process. Appl.120 (2010) 2495–2519. Zbl1206.60077MR2728175
  11. [11] A. Dembo. Moderate deviations for martingales with bounded jumps. Electron. Comm. Probab.1 (1996) 11–17. Zbl0854.60027MR1386290
  12. [12] A. Dembo and O. Zeitouni. Large Deviations Techniques and Applications, 2nd edition. Springer, New York, 1998. Zbl1177.60035MR1619036
  13. [13] H. Djellout. Moderate deviations for martingale differences and applications to φ -mixing sequences. Stoch. Stoch. Rep.73 (2002) 37–63. Zbl1005.60044MR1914978
  14. [14] H. Djellout, A. Guillin and L. Wu. Moderate deviations of empirical periodogram and non-linear functionals of moving average processes. Ann. Inst. Henri Poincaré Probab. Stat.42 (2006) 393–416. Zbl1100.60010MR2242954
  15. [15] H. Djellout and A. Guillin. Large and moderate deviations for moving average processes. Ann. Fac. Sci. Toulouse Math. (6) 10 (2001) 23–31. Zbl1002.60028MR1928987
  16. [16] N. Gozlan. Integral criteria for transportation-cost inequalities. Electron. Comm. Probab.11 (2006) 64–77. Zbl1112.60009MR2231734
  17. [17] N. Gozlan and C. Léonard. A large deviation approach to some transportation cost inequalities. Probab. Theory Related Fields139 (2007) 235–283. Zbl1126.60022MR2322697
  18. [18] J. Guyon. Limit theorems for bifurcating Markov chains. Application to the detection of cellular aging. Ann. Appl. Probab. 17 (2007) 1538–1569. Zbl1143.62049MR2358633
  19. [19] R. M. Huggins and I. V. Basawa. Extensions of the bifurcating autoregressive model for cell lineage studies. J. Appl. Probab.36 (1999) 1225–1233. Zbl0973.62100MR1746406
  20. [20] R. M. Huggins and I. V. Basawa. Inference for the extended bifurcating autoregressive model for cell lineage studies. Aust. N. Z. J. Stat.42 (2000) 423–432. Zbl1016.62098MR1802966
  21. [21] S. Y. Hwang, I. V. Basawa and I. K. Yeo. Local asymptotic normality for bifurcating autoregressive processes and related asymptotic inference. Stat. Methodol.6 (2009) 61–69. Zbl1220.62106MR2655539
  22. [22] M. Ledoux. The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs 89. American Mathematical Society, Providence, RI, 2001. Zbl0995.60002MR1849347
  23. [23] P. Massart. Concentration Inequalities and Model Selection. Lecture Notes in Mathematics 1896. Springer, Berlin, 2007. Zbl1170.60006MR2319879
  24. [24] A. Puhalskii. Large deviations of semimartingales: A maxingale problem approach. I. Limits as solutions to a maxingale problem. Stoch. Stoch. Rep. 61 (1997) 141–243. Zbl0890.60025MR1488137
  25. [25] J. Worms. Moderate deviations for stable Markov chains and regression models. Electron. J. Probab. 4 (1999) 28 pp. Zbl0980.62082MR1684149
  26. [26] J. Worms. Moderate deviations of some dependent variables. I. Martingales. Math. Methods Statist.10 (2001) 38–72. Zbl1007.60010MR1841808
  27. [27] J. Worms. Moderate deviations of some dependent variables. II. Some kernel estimators. Math. Methods Statist. 10 (2001) 161–193. Zbl1007.60011MR1851746
  28. [28] J. Zhou and I. V. Basawa. Least-squares estimation for bifurcating autoregressive processes. Statist. Probab. Lett.74 (2005) 77–88. Zbl1070.62075MR2189078
  29. [29] J. Zhou and I. V. Basawa. Maximum likelihood estimation for a first-order bifurcating autoregressive process with exponential errors. J. Time Series Anal.26 (2005) 825–842. Zbl1097.62091MR2203513

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