Adaptive estimation of the conditional intensity of marker-dependent counting processes

F. Comte; S. Gaïffas; A. Guilloux

Annales de l'I.H.P. Probabilités et statistiques (2011)

  • Volume: 47, Issue: 4, page 1171-1196
  • ISSN: 0246-0203

Abstract

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We propose in this work an original estimator of the conditional intensity of a marker-dependent counting process, that is, a counting process with covariates. We use model selection methods and provide a nonasymptotic bound for the risk of our estimator on a compact set. We show that our estimator reaches automatically a convergence rate over a functional class with a given (unknown) anisotropic regularity. Then, we prove a lower bound which establishes that this rate is optimal. Lastly, we provide a short illustration of the way the estimator works in the context of conditional hazard estimation.

How to cite

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Comte, F., Gaïffas, S., and Guilloux, A.. "Adaptive estimation of the conditional intensity of marker-dependent counting processes." Annales de l'I.H.P. Probabilités et statistiques 47.4 (2011): 1171-1196. <http://eudml.org/doc/243314>.

@article{Comte2011,
abstract = {We propose in this work an original estimator of the conditional intensity of a marker-dependent counting process, that is, a counting process with covariates. We use model selection methods and provide a nonasymptotic bound for the risk of our estimator on a compact set. We show that our estimator reaches automatically a convergence rate over a functional class with a given (unknown) anisotropic regularity. Then, we prove a lower bound which establishes that this rate is optimal. Lastly, we provide a short illustration of the way the estimator works in the context of conditional hazard estimation.},
author = {Comte, F., Gaïffas, S., Guilloux, A.},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {marker-dependent counting process; conditional intensity; model selection; adaptive estimation; minimax and nonparametric methods; censored data; conditional hazard function},
language = {eng},
number = {4},
pages = {1171-1196},
publisher = {Gauthier-Villars},
title = {Adaptive estimation of the conditional intensity of marker-dependent counting processes},
url = {http://eudml.org/doc/243314},
volume = {47},
year = {2011},
}

TY - JOUR
AU - Comte, F.
AU - Gaïffas, S.
AU - Guilloux, A.
TI - Adaptive estimation of the conditional intensity of marker-dependent counting processes
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2011
PB - Gauthier-Villars
VL - 47
IS - 4
SP - 1171
EP - 1196
AB - We propose in this work an original estimator of the conditional intensity of a marker-dependent counting process, that is, a counting process with covariates. We use model selection methods and provide a nonasymptotic bound for the risk of our estimator on a compact set. We show that our estimator reaches automatically a convergence rate over a functional class with a given (unknown) anisotropic regularity. Then, we prove a lower bound which establishes that this rate is optimal. Lastly, we provide a short illustration of the way the estimator works in the context of conditional hazard estimation.
LA - eng
KW - marker-dependent counting process; conditional intensity; model selection; adaptive estimation; minimax and nonparametric methods; censored data; conditional hazard function
UR - http://eudml.org/doc/243314
ER -

References

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  1. [1] P. K. Andersen, O., Borgan, R. D. Gill and N. Keiding. Statistical Models Based on Counting Processes. Springer, New York, 1993. Zbl0824.60003MR1198884
  2. [2] Y. Baraud. A Bernstein-type inequality for suprema of random processes with an application to statistics. Bernoulli (2010). To appear. Zbl05858609MR2759169
  3. [3] Y. Baraud and L. Birgé. Estimating the intensity of a random measure by histogram type estimators. Probab. Theory Related Fields 149 (2009) 239–284. Zbl1149.62019MR2449129
  4. [4] A. Barron, L. Birgé and P. Massart. Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 (1999) 301–413. Zbl0946.62036MR1679028
  5. [5] J. Beran. Nonparametric regression with randomly censored survival data. Technical report, Dept. Statist., Univ. California, Berkeley, 1981. 
  6. [6] L. Birgé and P. Massart. Minimum contrast estimators on sieves: Exponential bounds and rates of convergence. Bernoulli 4 (1998) 329–375. Zbl0954.62033MR1653272
  7. [7] E. Brunel, F. Comte and C. Lacour. Adaptive estimation of the conditional density in presence of censoring. Sankhyā A 69 (2007) 734–763. Zbl1193.62055MR2521231
  8. [8] G. Castellan and F. Letué. Estimation of the Cox regression function via model selection. Chapter of the PhD thesis of F. Letué, Univ. Paris XI-Orsay, 2000. 
  9. [9] A. Cohen, I. Daubechies and P. B. Vial. Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon. Anal. 1 (1993) 54–81. Zbl0795.42018MR1256527
  10. [10] F. Comte. Adaptive estimation of the spectrum of a stationary Gaussian sequence. Bernoulli 7 (2001) 267–298. Zbl0981.62075MR1828506
  11. [11] F. Comte, S. Gaïffas and A. Guilloux. Adaptive estimation of the conditional intensity of marker-dependent counting processes. Preprint MAP5 2008-16, revised 2010. Available at http://hal.archives-ouvertes.fr/hal-00333356/fr/. Zbl1271.62222
  12. [12] D. R. Cox. Regression models and life-tables (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 34 (1972) 187–220. Zbl0243.62041MR341758
  13. [13] D. M. Dabrowska. Nonparametric regression with censored survival time data. Scand. J. Statist. 14 (1987) 181–197. Zbl0641.62024MR932943
  14. [14] D. M. Dabrowska. Uniform consistency of the kernel conditional Kaplan–Meier estimate. Ann. Statist. 17 (1989) 1157–1167. Zbl0687.62035MR1015143
  15. [15] I. Daubechies. Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41 (1988) 909–996. Zbl0644.42026MR951745
  16. [16] M. Delecroix, O. Lopez and V. Patilea. Nonlinear censored regression using synthetic data. Scand. J. Statist. 35 (2008) 248–265. Zbl1164.62024MR2418739
  17. [17] G. Grégoire. Least squares cross-validation for counting processes intensities. Scand. J. Statist. 20 (1993) 343–360. Zbl0795.62031MR1276698
  18. [18] C. Heuchenne and I. Van Keilegom. Location estimation in nonparametric regression with censored data. J. Multivariate Anal. 98 (2007) 1558–1582. Zbl1122.62024MR2370107
  19. [19] R. Hochmuth. Wavelet characterizations for anisotropic Besov spaces. Appl. Comput. Harmon. Anal. 12 (2002) 179–208. Zbl1003.42024MR1884234
  20. [20] J. Huang. Efficient estimation of the partly linear additive Cox model. Ann. Statist. 27 (1999) 1536–1563. Zbl0977.62035MR1742499
  21. [21] M. Jacobsen. Statistical Analysis of Counting Processes. Lecture Note in Statistics 12. Springer, New York, 1982. Zbl0518.60065MR676128
  22. [22] A. F. Karr. Point Processes and Their Statistical Inference. Marcel Dekker, New York, 1986. Zbl0733.62088MR851982
  23. [23] C. Lacour. Adaptive estimation of the transition density of a Markov chain. Ann. Inst. H. Poincaré Probab. Statist. 43 (2007) 571–597. Zbl1125.62087MR2347097
  24. [24] C. Lacour. Estimation non paramétrique adaptative pour les chaînes de Markov et les chaînes de Markov cachées. PhD thesis, 2007. Available at http://www.math.u-psud.fr/~lacour/etudes/. 
  25. [25] M. LeBlanc and J. Crowley. Adaptive regression splines in the Cox model. Biometrics 55 (1999) 204–213. Zbl1059.62668
  26. [26] G. Li and H. Doss. An approach to nonparametric regression for life history data using local linear fitting. Ann. Statist. 23 (1995) 787–823. Zbl0852.62037MR1345201
  27. [27] O. B. Linton, J. P. Nielsen and S. Van de Geer. Estimating the multiplicative and additive hazard functions by kernel methods. Ann. Statist. 31 (2003) 464–492. Zbl1040.62089MR1983538
  28. [28] R. S. Liptser and A. N. Shiryayev. Theory of Martingales. Mathematics and its Applications (Soviet Series) 49. Kluwer Academic, Dordrecht, 1989. Zbl0728.60048MR1022664
  29. [29] P. Massart. Concentration Inequalities and Model Selection. Lecture Notes in Mathematics 1896. Springer, Berlin, 2007. Zbl1170.60006MR2319879
  30. [30] I. W. McKeague and K. J. Utikal. Inference for a nonlinear counting process regression model. Ann. Statist. 18 (1990) 1172–1187. Zbl0721.62087MR1062704
  31. [31] Y. Meyer. Ondelettes sur l’intervalle. Rev. Mat. Iberoamericana 7 (1991) 115–133. Zbl0753.42015MR1133374
  32. [32] S. M. Nikol’skii. Approximation of Functions of Several Variables and Imbedding Theorems. Springer, New York, 1975. Zbl0185.37901
  33. [33] H. Ramlau-Hansen. Smoothing counting process intensities by means of kernel functions. Ann. Statist. 11 (1983) 453–466. Zbl0514.62050MR696058
  34. [34] P. Reynaud-Bouret. Adaptive estimation of the intensity of nonhomogeneous Poisson processes via concentration inequalities. Probab. Theory Related Fields 126 (2003) 103–153. Zbl1019.62079MR1981635
  35. [35] P. Reynaud-Bouret. Penalized projection estimators of the Aalen multiplicative intensity. Bernoulli 12 (2006) 633–661. Zbl1125.62027MR2248231
  36. [36] C. J. Stone. Optimal rates of convergence for nonparametric estimators. Ann. Statist. 8 (1980) 1348–1360. Zbl0451.62033MR594650
  37. [37] W. Stute. Conditional empirical processes. Ann. Statist. 14 (1986) 638–647. Zbl0594.62038MR840519
  38. [38] W. Stute. Distributional convergence under random censorship when covariables are present. Scand. J. Statist. 23 (1996) 461–471. Zbl0903.62045MR1439707
  39. [39] M. Talagrand. The Generic Chaining. Springer, Berlin, 2005. Zbl1075.60001MR2133757
  40. [40] H. Triebel. Theory of Function Spaces. III. Monographs in Mathematics 100. Birkhäuser, Basel, 2006. Zbl1104.46001MR2250142
  41. [41] A. Tsybakov. Introduction à l’estimation non-paramétrique. Springer, Berlin, 2004. Zbl1029.62034MR2013911
  42. [42] S. van de Geer. Exponential inequalities for martingales, with application to maximum likelihood estimation for counting processes. Ann. Statist. 23 (1995) 1779–1801. Zbl0852.60019MR1370307

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