Penalized projection estimators of the Aalen multiplicative intensity PATR IC IA REYNAUD BOURET
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Penalized projection estimators of the Aalen multiplicative intensity PATR IC IA REYNAUD-BOURET Departement de Mathematiques et Applications, Ecole Normale Superieure, 45 Rue d'Ulm, 75230 Paris Cedex 05, France. E-mail: We study the problem of nonparametric, completely data-driven estimation of the intensity of counting processes satisfying the Aalen multiplicative intensity model. To do so, we use model selection techniques and, specifically, penalized projection estimators for a random inner product. For histogram estimators, under some assumptions on the process, we obtain adaptive results for the minimax risk. In general, for more intricate (predictable) models, we only obtain oracle inequalities. The study is complemented by some simulations in the right-censoring model. Keywords: adaptive estimation; counting processes; model selection; multiplicative intensity model; penalized projection estimators 1. Introduction 1.1. The bibliographical context Counting processes with Aalen multiplicative intensity are a generalization of temporal Poisson processes. They can model a large variety of situations (especially in biology and medicine). Let (, F , P) be a probability triple and (F t, t > 0) be a filtration. A counting process N ? (Nt) t>0 satisfies the Aalen multiplicative intensity model with predictable process Y ? (Yt) t>0 (see Andersen et al.
Voir
- aalen multiplicative
- yt ?
- estimators
- process
- intensity model
- right-censoring model
- patient dies
- positive random variable
- model selection
Bernoulli12(4), 2006, 633–661
Penalized projection estimators of the Aalen multiplicative intensity
PAT R I C I A R E Y N A U D - B O U R E T ´ D´epartementdeMathe´matiquesetApplications,EcoleNormaleSupe´rieure,45Rued’Ulm, 75230 Paris Cedex 05, France. E-mail: Patricia.Reynaud-Bouret@ens.fr
We study the problem of nonparametric, completely data-driven estimation of the intensity of counting processes satisfying the Aalen multiplicative intensity model. To do so, we use model selection techniques and, specifically, penalized projection estimators for a random inner product. For histogram estimators, under some assumptions on the process, we obtain adaptive results for the minimax risk. In general, for more intricate (predictable) models, we only obtain oracle inequalities. The study is complemented by some simulations in the right-censoring model.
Keywords:adaptive estimation; counting processes; model selection; multiplicative intensity model; penalized projection estimators
1. Introduction
1.1. The bibliographical context
Counting processes with Aalen multiplicative intensity are a generalization of temporal Poisson processes. They can model a large variety of situations (especially in biology and medicine). Let (,F,P) be a probability triple and (Ft,t>0) be a filtration. A counting processN¼(Nt)t>0satisfies the Aalen multiplicative intensity model with predictable processY¼(Yt)t>0(see Andersenet al.1993) if
d¸t¼Yts(t)dt, (1:1) where (¸t)t>0is the compensator of (Nt)t>0with respect to (Ft,t>0), (Yt)t>0a non-negative predictable process andsa deterministic function. When the process (Yt)t>0is constant, (Nt)t>0is a temporal Poisson process with intensityswith respect to the measure Ydt. Let us give some other examples. Let (Nt¼1X<t)t>0whereXis a positive random variable with densityf. This process satisfies (1.1) withYt¼1X>tand with s(t)¼f(t)=P(X>t), thehazard rateofX; ifXrepresents a patient’s lifetime,s(t) represents the probability that the patient dies just aftertgiven that he is alive at timet. Observations of lifetimes may sometimes be censored. This is the case when a patient drops out of a hospital study. The time of death is not observed, but we know that the patient was still alive when he left the study. This situation is modelled by some other positive random variableUwhich is independent ofXand the observations are the
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variablesT¼X^UandD¼1T¼X. This model is known as theright-censoring model with independent censorship. Then the processNt¼D1T<thas an Aalen multiplicative intensity (1.1) whereYt¼1T>tandsis the hazard rate ofX. We may also have ann-sample of counting processes,N1,. . .,Nn, satisfying (1.1) (corresponding tondifferent patients, for instance). Their predictable processes are denoted byY1,. . .,Yn. They have the same intensitys. Then we can define theaggregated process Nwith predictable processYby
n n Nt¼XNtiandYt¼XYit, for allt>0: i¼1i¼1
(1:2)
This aggregated process also satisfies (1.1) with the sames. For instance, in the right-censoring model, the processYis a non-increasing process with integer values and withY0¼n,nbeing the number of observations. The numberYt represents the number of events which will happen aftert, whether these events are real observed deaths or departures. Many other examples of processes with multiplicative intensity are mentioned in Andersenet al.(1993). For instance, if (Xt)t>0is a Markov process with finite state space, the counting process (Nhjt,t>0), whereNjhtrepresents the number of transitions fromhto jby timet, has a multiplicative intensity of the form (1.1) wheresis the transition intensity fromhtojand whereYis defined by (Yt¼1X(t)¼h)t>0. We may have ann-sample of independent and identically distributed counting processes corresponding to each individual Markov process. In this situation, we can look at the aggregated processes (1.2) whereYis still integer-valued and upper-bounded byn: at timet,Ytrepresents the number of individuals in stateh. This situation models, for instance, the transition from healthy to diseased (Andersenet al.1993: Example I.3.10). There are also cases where the process cannot be divided into individual processes, and so cannot be written as in (1.2). This is the case for the model of the number of matings of Drosophilaflies (Andersenet al.1993: Example III.1.10). However, this model satisfies the multiplicative intensity property (1.1) with aYwhich still corresponds to a bounded number of events which may happen after timet. The purpose of this paper is to estimateson [0,] using observations of (Nt)0<t<and (Yt)0<t<. Our aim is to do so in a nonparametric adaptive way with as few assumptions on sto stay within the most general framework, but, as mentionedas possible. We also try later, we need some extra assumptions on the process itself (aggregated or not, for instance) depending on the type of estimator (piecewise constant or predictable). Many papers consider the problem of the estimation ofsin the general Aalen multiplicative intensity model. Ramlau-Hansen (1983) proved consistency and asymptotic normality results for some kernel estimators with fixed bandwidth. Gre´goire (1993) gives a data-driven criterion for choosing the bandwidth of the Ramlau-Hansen estimators by cross-validation. He also proves consistency and asymptotic normality results. Other possible estimators are maximum likelihood estimators on certain sieves, whose rates of convergence were studied by van de Geer (1995). Antoniadis (1989) chooses the sieve by penalization,
