Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

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

F. Comte, S. Gaïffas, and A. Guilloux

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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.


Dans ce travail, nous proposons un estimateur original de l’intensité conditionnelle d’un processus de comptage marqué, c’est-à-dire d’un processus de comptage dépendant de covariables. Nous utilisons une méthode de sélection de modèle et nous obtenons pour notre estimateur, une borne non asymptotique du risque quadratique sur un compact. Nous vérifions ensuite que l’estimateur atteint automatiquement une vitesse de convergence sur des classes fonctionnelles de régularité anisotropique fixée mais inconnue. Enfin, nous démontrons une borne inférieure qui garantit l’optimalité de la vitesse obtenue. Une brève illustration de la façon dont fonctionne l’estimateur dans le contexte de l’estimation du taux de risque instantané conditionnel est fournie pour conclure.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 47, Number 4 (2011), 1171-1196.

First available in Project Euclid: 6 October 2011

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Zentralblatt MATH identifier

Primary: 62N02: Estimation 62G05: Estimation

Marker-dependent counting process Conditional intensity Model selection Adaptive estimation Minimax and nonparametric methods Censored data Conditional Hazard function


Comte, F.; Gaïffas, S.; Guilloux, A. Adaptive estimation of the conditional intensity of marker-dependent counting processes. Ann. Inst. H. Poincaré Probab. Statist. 47 (2011), no. 4, 1171--1196. doi:10.1214/10-AIHP386.

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  • [1] P. K. Andersen, O., Borgan, R. D. Gill and N. Keiding. Statistical Models Based on Counting Processes. Springer, New York, 1993.
  • [2] Y. Baraud. A Bernstein-type inequality for suprema of random processes with an application to statistics. Bernoulli (2010). To appear.
  • [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.
  • [4] A. Barron, L. Birgé and P. Massart. Risk bounds for model selection via penalization. Probab. Theory Related Fields 113 (1999) 301–413.
  • [5] J. Beran. Nonparametric regression with randomly censored survival data. Technical report, Dept. Statist., Univ. California, Berkeley, 1981.
  • [6] L. Birgé and P. Massart. Minimum contrast estimators on sieves: Exponential bounds and rates of convergence. Bernoulli 4 (1998) 329–375.
  • [7] E. Brunel, F. Comte and C. Lacour. Adaptive estimation of the conditional density in presence of censoring. Sankhyā A 69 (2007) 734–763.
  • [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] A. Cohen, I. Daubechies and P. B. Vial. Wavelets on the interval and fast wavelet transforms. Appl. Comput. Harmon. Anal. 1 (1993) 54–81.
  • [10] F. Comte. Adaptive estimation of the spectrum of a stationary Gaussian sequence. Bernoulli 7 (2001) 267–298.
  • [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
  • [12] D. R. Cox. Regression models and life-tables (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 34 (1972) 187–220.
  • [13] D. M. Dabrowska. Nonparametric regression with censored survival time data. Scand. J. Statist. 14 (1987) 181–197.
  • [14] D. M. Dabrowska. Uniform consistency of the kernel conditional Kaplan–Meier estimate. Ann. Statist. 17 (1989) 1157–1167.
  • [15] I. Daubechies. Orthonormal bases of compactly supported wavelets. Comm. Pure Appl. Math. 41 (1988) 909–996.
  • [16] M. Delecroix, O. Lopez and V. Patilea. Nonlinear censored regression using synthetic data. Scand. J. Statist. 35 (2008) 248–265.
  • [17] G. Grégoire. Least squares cross-validation for counting processes intensities. Scand. J. Statist. 20 (1993) 343–360.
  • [18] C. Heuchenne and I. Van Keilegom. Location estimation in nonparametric regression with censored data. J. Multivariate Anal. 98 (2007) 1558–1582.
  • [19] R. Hochmuth. Wavelet characterizations for anisotropic Besov spaces. Appl. Comput. Harmon. Anal. 12 (2002) 179–208.
  • [20] J. Huang. Efficient estimation of the partly linear additive Cox model. Ann. Statist. 27 (1999) 1536–1563.
  • [21] M. Jacobsen. Statistical Analysis of Counting Processes. Lecture Note in Statistics 12. Springer, New York, 1982.
  • [22] A. F. Karr. Point Processes and Their Statistical Inference. Marcel Dekker, New York, 1986.
  • [23] C. Lacour. Adaptive estimation of the transition density of a Markov chain. Ann. Inst. H. Poincaré Probab. Statist. 43 (2007) 571–597.
  • [24] C. Lacour. Estimation non paramétrique adaptative pour les chaînes de Markov et les chaînes de Markov cachées. PhD thesis, 2007. Available at
  • [25] M. LeBlanc and J. Crowley. Adaptive regression splines in the Cox model. Biometrics 55 (1999) 204–213.
  • [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.
  • [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.
  • [28] R. S. Liptser and A. N. Shiryayev. Theory of Martingales. Mathematics and its Applications (Soviet Series) 49. Kluwer Academic, Dordrecht, 1989.
  • [29] P. Massart. Concentration Inequalities and Model Selection. Lecture Notes in Mathematics 1896. Springer, Berlin, 2007.
  • [30] I. W. McKeague and K. J. Utikal. Inference for a nonlinear counting process regression model. Ann. Statist. 18 (1990) 1172–1187.
  • [31] Y. Meyer. Ondelettes sur l’intervalle. Rev. Mat. Iberoamericana 7 (1991) 115–133.
  • [32] S. M. Nikol’skii. Approximation of Functions of Several Variables and Imbedding Theorems. Springer, New York, 1975.
  • [33] H. Ramlau-Hansen. Smoothing counting process intensities by means of kernel functions. Ann. Statist. 11 (1983) 453–466.
  • [34] P. Reynaud-Bouret. Adaptive estimation of the intensity of nonhomogeneous Poisson processes via concentration inequalities. Probab. Theory Related Fields 126 (2003) 103–153.
  • [35] P. Reynaud-Bouret. Penalized projection estimators of the Aalen multiplicative intensity. Bernoulli 12 (2006) 633–661.
  • [36] C. J. Stone. Optimal rates of convergence for nonparametric estimators. Ann. Statist. 8 (1980) 1348–1360.
  • [37] W. Stute. Conditional empirical processes. Ann. Statist. 14 (1986) 638–647.
  • [38] W. Stute. Distributional convergence under random censorship when covariables are present. Scand. J. Statist. 23 (1996) 461–471.
  • [39] M. Talagrand. The Generic Chaining. Springer, Berlin, 2005.
  • [40] H. Triebel. Theory of Function Spaces. III. Monographs in Mathematics 100. Birkhäuser, Basel, 2006.
  • [41] A. Tsybakov. Introduction à l’estimation non-paramétrique. Springer, Berlin, 2004.
  • [42] S. van de Geer. Exponential inequalities for martingales, with application to maximum likelihood estimation for counting processes. Ann. Statist. 23 (1995) 1779–1801.