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Statistics & Probability Letters 76 (2006) 1514–1521 DMA, Ecole Normale Superieure de Paris, 45 rue d'Ulm, 75230 Paris Cedex 05, France therefore a certain structure that produces concentration inequalities. More precisely, the inequalities ARTICLE IN PRESS 0167-7152/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2006.03.012 E-mail address: . Keywords: Exponential inequalities; Counting process; Supremum of centered integrals 1. Introduction Counting processes can model a large number of biomedical situations, (see Andersen et al., 1993). In all these problems, the intensity of the process has to be estimated. If we want to use the penalized model selection method of estimation developed by Birge and Massart (see Birge and Massart, 2001; Massart, 2000b for instance), some very sharp exponential inequalities have to be available. Birge's and Massart's framework is usually the white noise model or the i.i.d. n-sample framework. There is Received 24 March 2004; received in revised form 20 February 2006; accepted 1 March 2006 Available online 18 April 2006 Abstract Talagrand [1996. New concentration inequalities in product spaces.

  • h2s dls

  • let ?zt?tx0

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  • classical scalar

  • inegalite de bennett pour les maxima de processus empiriques

  • positive real

  • exponential inequalities

  • exist positive


Publié le : mardi 19 juin 2012
Lecture(s) : 31
Source : math.unice.fr
Nombre de pages : 8
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ARTICLE IN PRESS
Statistics & Probability Letters 76 (2006) 1514–1521
Compensator and exponential inequalities counting processes
Abstract
www.elsevier.com/locate/stapro
for some suprema of
P. Reynaud-Bouret ´ DMA, Ecole Normale Supe´rieure de Paris, 45 rue d’Ulm, 75230 Paris Cedex 05, France Received 24 March 2004; received in revised form 20 February 2006; accepted 1 March 2006 Available online 18 April 2006
Talagrand [1996. New concentration inequalities in product spaces. Invent. Math. 126 (3), 505–563], Ledoux [1996. On Talagrand deviation inequalities for product measures. ESAIM: Probab. Statist. 1, 63–87], Massart [2000a. About the constants in Talagrand’s concentration inequalities for empirical processes. Ann. Probab. 2 (28), 863–884], Rio [2002. Une ine´ galit ´e de Bennett pour les maxima de processus empiriques. Ann. Inst. H. Poincare´ Probab. Statist. 38 (6), 1053–1057. En l’honneur de J. Bretagnolle, D. Dacunha-Castelle, I. Ibragimov] and Bousquet [2002. A Bennett concentration inequality and its application to suprema of empirical processes. C. R. Math. Acad. Sci. Paris 334 (6), 495–500] have obtained exponential inequalities for suprema of empirical processes. These inequalities are sharp enough to build adaptive estimation procedures Massart [2000b. Some applications of concentration inequalities. Ann. Fac. Sci. Toulouse Math. (6) 9 (2), 245–303]. The aim of this paper is to produce these kinds of inequalities when the empirical measure is replaced by a counting process. To achieve this goal, we first compute the compensator of a suprema of integrals with respect to the counting measure. We can then apply the classical inequalities which are already available for martingales Van de Geer [1995. Exponential inequalities for martingales, with application to maximum likelihood estimation for counting processes. Ann. Statist. 23 (5), 1779–1801]. r2006 Elsevier B.V. All rights reserved.
MSC:60E15; 60G55
Keywords:Exponential inequalities; Counting process; Supremum of centered integrals
1. Introduction
Counting processes can model a large number of biomedical situations, (seeAndersen et al., 1993). In all these problems, the intensity of the process has to be estimated. If we want to use the penalized model selection methodofestimationdevelopedbyBirg´eandMassart(seeBirg ´e and Massart, 2001;Massart, 2000bfor instance), some very sharp exponential inequalities have to be available. Birge´ ’s and Massart’s framework is usually the white noise model or the i.i.d.n-sample framework. There is therefore a certain structure that produces concentration inequalities. More precisely, the inequalities
Email address:Patricia.Reynaud-Bouret@ens.fr.
0167-7152/$ - see front matterr2006 Elsevier B.V. All rights reserved. doi:10.1016/j.spl.2006.03.012
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developed byTalagrand (1996),Ledoux (1996),Massart (2000a),Rio (2002)andBousquet (2002)consist of exponential inequalities for suprema of countably many empirical processes. For counting processes, we extensively use the martingale properties. There are already a lot of exponential inequalities for martingales using the exponential semi-martingales approach (seeVan de Geer, 1995; Kallenberg, 1997, Theorem 23.17for instance). We are hence left with the precise computation of the compensator of a supremum of countably many integrals with respect to a counting process to obtain the desired exponential inequalities. In Section 2, we explain how to build the compensator of a supremum. We can then apply classical exponential inequalities to this supremum, which is done in Section 3. In Section 4, we derive a more suitable version for the statistical applications.
2. Compensator of the supremum
LetðO;F;PÞbe a probability triple andðFt;tX0Þbe a filtration satisfying the usual conditions (see Kallenberg, 1997, p. 124for the definition and use of the usual augmentation of a classical filtration). Let ðNtÞbe a counting process adapted toðFt;tX0Þand letðLtÞbe its compensator, i.e. the nondecreasing tX0tX0 LÞis a martingale wit predictable function such thatðMt¼Nt t tX0h respect toðFt;tX0Þ(for precise definitions, see for instance,Bre´ maud, 1981). LetTbe some positive fixed time, eventually infinite. We suppose in the whole paper the following assumption.
Assumption 1.The compensatorðLtÞis absolutely continuous and almost surely finite on½0;T. tX0
LetHa;tÞ;a2Agbe a countable family of predictable processes. We suppose them to be locally tX0 bounded intand uniformly bounded ina. LetðZtÞbe the process defined by tX0 Z  t 8tX0;Zt¼supHa;sdMs. (1) a2A0 therefore an adapted process with bounded variations. For alltpT, letðT;1pipNÞ The processðZtÞtisi t X0 be the ordered jumps ofNbeforet, for there is almost surely a finite number of these jumps. Indeed, it is a consequence of the finiteness of the compensator (Assumption 1) and of Theorem II-8 (a) ofBr ´emaud (1981). Since the compensator is continuous (Assumption 1), the jumps ofZonly happen whenNjumps. We can consequently write: X X ½ZTZT þZtZTZTa.e., (2) 8tpT;Zt¼nti i þ ½ZTii1 Tipt Tipt whereZT0¼Z0¼0:(Zsdenotes the left limit of the processZat times). Reasoning by induction, it is straightforward to prove the following result, using the absolute continuity and Corollary 6.18 ofLieb and Loss (1997).
Lemma 1.AssumeA¼ f1;. . .;kgto be finite and ordered. Let i be a positive integer. Let v be a real number in R v Ti1;Ti½.Then under Assumption1,Zv as;sdLs;where a^ ZTi1¼ TiH^sis the first index where Zsis 1 attained.
We have deliberately taken the left limit to prove the forthcoming proposition. Of course, we could have takenHa^s;s, for this is equal toHa^s;son the intervals between the jumps ofN. Using this lemma we get the following result.
(1).Under Assumption1,ifAis finite, Proposition 1.Let T be a fixed positive number. LetðZtÞtbe defined by X0 then Z Z t t 80ptpT;Zt¼DZðsÞdNsH^as;sdLsa:s:; 0 0
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where  Z  Z ssDZðsÞ ¼supHa;sþHa;udMusupHa;udMu. a2A0a2A0 The compensator ofðZt^TÞis then defined by tX0 Z t^T 8tX0;At¼ ½DZðsÞ Ha^s;sdLs. 0 If,ðAÞex Ais countable,the compensator ofðZt^TÞtX0t tX0ists,is nonnegative,nondecreasing and Z t 80ptpT;ZtAt¼DZðsÞdMs. 0 Proof.Let us first assume thatAis finite. The first integral inZtis exactly the first sum on the right-hand side of (2). In the second sum on the right-hand side of (2), all the differences are between two consecutive jumps and we can use the previous lemma. Moreover,DZðsÞintroduced in the proposition is predictable. The compensator is then obvious. AsDZHa^s;sis nonnegative andLnondecreasing,Ais nonnegative nondecreasing. n IfAis just countable,Ais an increasing union of finite setsBn. Let us denote byZthe supremum overBn n instead ofA. As, for alln,Bnis finite,Zsatisfies the first part of the proposition. But, for alltpT, the R R t t n n predictable processZDZðsÞdNsconverges almost surely toXt¼ZtDZðsÞdNs. The process t0 0 R t ðXtÞis consequently predictable. As a result, the processðADZðsÞdLþX, is the tX0tÞtX0, defined byAt¼0s t compensator o tays nonnegative a fðZt^TÞX0and it s nd nondecreasing as a limit of nonnegative nondecreasing t functions.&
3. Exponential inequalities for supremum
We first present some well-known facts about exponential inequalities for martingales in our particular framework. R t LetðHbe a locally bounded predictable process andðZtÞbe defined byZ¼HdMfor alltX0. tÞtX0tX0t0s s u Letfbe defined byfðuÞ ¼eu1 for allu. Let  Z t 8tX0;Et¼explZtfðlHsÞdLs. 0 R T lHs LetTbe a positive real number. LetIbe an interval such that for alllinI,edLsis almost surely finite. 0 is thatðEs a super-Under Assumption 1, a consequence of Theorem VI-2 of(1981)Bre´ maud t^TÞtX0i martingale and that for all stopping timet(tpT),EðEtÞis less than 1. This implies that for alllinI,  ! Z T le 8e40;PsupZtXepeexpfðlHsÞdLs. ½0;T01 The next result is a consequence of the previous inequality. It could also be seen as an application to this special framework ofVan de Geer (1995)except that the absolute values are inside the integral in (3) when applying van de Geer’s results. Proposition 2.LetðZtÞbe the process: tX0 Z t 8tX0;Zt¼HsdMs, 0 i predictable process. Let T be a positive real number. If there exist c and v po whereðHtÞ0sitive constantss a , tX such that Z T k! k k2 8kX2;dv, (3) HsLspc   02
ARTICLE IN PRESS P. ReynaudBouret / Statistics & Probability Letters 76 (2006) 1514–1521
then under Assumption1,  ! pffiffiffiffiffiffiffi 8uX0;PsupZtX2vuþcupexpðuÞ. ½0;T
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There exists a simpler form whenðHtÞis bounded. This form is well known; it can be found inShorack tX0 and Wellner (1986), for instance.
Corollary 1.With the notations of Proposition2,if there exist b and v positive constants such that,on½0;T, ð Þ Ht thas values in½b;band X0 Z T 2 HdLspv   s 0 then under Assumption1,  ! pffiffiffiffiffiffiffi b 8uX0;PsupZtX2vuþupexpðuÞ. ½0;T3
ered supremum,ðZt In Section 2, we have written the centAtÞ0ptp, as an integral of a predictable process T with respect to the centered counting measure. To find concentration inequalities for the supremum, it is now sufficient to apply the previous results toHs¼DZðsÞ.
be a counting process satisfying Assumption1.LetHÞ;a2Agbe a countable Theorem 1.LetðNtÞtX0a;t tX0 family of predictable processes. Let Z  t 8tX0;Zt¼supHa;sdMs. a2A0 Let T be a positive real number. LetðAÞbe the compensator of thðZ,defined by Proposition t tX0e processt^TÞtX0 1. (a)If there exist positive constants b and v such that the Has have values in½b;bon½0;Tand such that R T 2 sup½HdLspv,then 0a2Aa;s  ! pffiffiffiffiffiffiffi 1 8uX0;PsupðZtAtÞX2vuþbupexpðuÞ. ½0;T3
(b)If there exist positive constants c and v,such that Z  T k! k k2 8kX2;supjHa;sjdLspc v 0a2A2 then  ! pffiffiffiffiffiffiffi 8uX0;PsupðZtAtÞX2vuþcupexpðuÞ. ½0;T
Let us compare this result to the inequalities successively obtained byTalagrand (1996),Ledoux (1996), Massart (2000a),Rio (2002)andBousquet (2002)by looking at the counting process as an empirical measure and at the compensator as an expectation. At first glance, it seems that this new inequality is in some sense stronger: we can manage random (predictable) functions and one has also a ‘‘moment’’ version (see (b)), which does not assume an absolute bound on the family of functions to integrate (or to sum in the i.i.d. framework). One can remark that a ‘‘moment’’ version of Talagrand’s inequality in the i.i.d framework has recently been proved (Massart, 2005is just a refinement due to the martingale structure but this). The presence of the sup ½0;Tdoes not affect the orders of magnitude.
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However, we lose something important with respect to Talagrand’s inequality. Let us compare in each case what is usually called the ‘‘variance term’’,v. In Talagrand’s inequality,vcan be seen as Z  v¼sup VarHadPn, a2A where theHa’s are here deterministic functions and where dPnis the empirical measure. The supremum is outside the integral. But in Theorem 1(a), it lies inside the integral and it is consequently of bigger order. This phenomenon has already been underlined bySamson (2000). He recovers Talagrand’s inequality for F-mixing, apart from this exchange between the supremum and the sum. For the Poisson processes,Wu (2000)andHoudre´ and Privault (2002)use martingales approach to derive exponential inequalities for very general functionals of the process. When we apply these inequalities to the supremum, this exchange also appears in the variance term. To have supremum on the left-hand side in the Poisson case (Reynaud-Bouret, 2003), we need some techniques using the infinitely divisible property of the Poisson process. Consequently, it seems that the exchange between supremum and sum (or integral) can only be done when there exists an independence property in the framework. For general counting processes, we have not been able to prove such results.
4. Statistical applications
Let us now describe the statistical framework for which these inequalities are made and let us give another corollary which is ready to be used in practice.
4.1. Statistical background
2 These exponential inequalities are useful to provide exponential deviations for the followingw-type statistics. LetTbe a fixed positive real number and letfhl;l2mgbe a finite family of predictable processes. We set Z 2 X T 2 w T¼hlðtÞdMt. (4) 0 l2m This quantity naturally appears if we estimate the signalsby penalized model selection in the white noise framework (seeMassart, 2001Birge´ and ). One has a model i.e. a finite dimensional linear subspace with orthonormal basisfj;l2mgfor the classical scalar product on½0;T. The classical projection estimator on l this subspace satisfies that theL2-distance between the least-square estimator and the true orthogonal 2 2 gi projection ofs,jsm^smj, is awTven by (4), withjinstead ofhl(i.e. deterministic functions) and with l 2 dW, the white noise, instead of dM. In this case, this quantity obeys a realw-distribution. The deviations of this quantity have to be controlled to prove the adaptive properties of the model selection procedure. In the 2 white noise framework, they use the exponential inequalities available forw-distributions. If we estimate the densitysfrom an-sample by penalized model selection, we can still consider the same model as before. As previously, the least-square estimator,s^m, is an unbias estimation of the classical 2 2 ype statistics whe i.e. an orthonormal projection,sm. The distancejsms^mjis also aw-t rehl¼jl, deterministic basis of the model for the classical scalar product. In this case, dMis replaced by the centered empiricalmeasure.Inthiscontext,Birg´eandMassartuseTalagrandsinequalitytoprovidecontrolonthe 2 w-type statistics (and Massart, 1997Birg ´e ). If we estimate the intensitysof a Poisson processNby penalized model selection, we can keep the same 2 2 procedure and notations. The distancejsms^mjis still aw-type statistics where dMis the centered Poisson process, and wherej;l2mgrepre amily ofLð½0;T;dtÞ. In this fhl¼lsents an orthonormal deterministic f2 case, we can use the concentration inequality ofReynaud-Bouret (2003)(see Proposition 3) to control these distances, which gives the same order of magnitude as Talagrand’s inequality in then-sample framework. Generalizing the Poisson process, we can be interested by the Aalen multiplicative intensity model where the satisfies dL¼Y sðtÞdtcess, with a predic ðYÞ. For instance compensator ofðNtÞt ttable and known prot tX0 tX0 the right-censoring model (seeAndersen et al., 1993, for a complete description) has an Aalen multiplicative
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intensity. We can estimate the deterministic functionsusing the observations of the processesNandYby penalized model selection (seeReynaud-Bouret, 2002). In this case, we are using a random scalar product R t fðsÞgðsÞYsdsinstead of the classical one. In this context, we cannot exactly use the same model as before but 0 d, if;l2mgis an orthonormal family ofLð½0;1;dtÞ( we can use predictablehl’s. Indeefjl2typically pffiffiffiffi histograms or Fourier basis),fhl¼j=Y;l2mgbecomes an orthonormal predictable family for the l random product (whenYtis positive). The subspace generated by thehl’s is the used model. Our aim is now to provide forwan exponential inequality which is ready to be used for the statistical T applications.
4.2. An inequality which is ready for immediate application
2 In order to provide a concentration inequality forw, we can remark that T  ! Z X t tX0;w¼sa hðs 8tupl lÞdMs(5) P 0 2 a¼1l2m l l2m 2 is the square root ofw. We can consequently use Theorem 1 on a countable dense subset of the unit ball of t pffiffiffiffim R. But as we do not know in practice the tor ofw, we may prefer compa compensatring it toCtwhere Z X t 2 8tX0;Ct¼hlðsÞdLs, (6) 0 l2m 2 is the compensator ofw. This leads to the forthcoming result. t
Corollary 2.Let T be a fixed positive real number. Letwbe defined by(4).Then,for any positive number u,with T u probability larger than12e, pffiffiffiffiffipffiffiffiffiffiffiffi wCTp3 2vuþbu, T where
CTis defined by(6), v¼jCTj1;and P 2 2  8spT;hðsÞpb: l2ml
Proof.LetuX0. First, we can interpretwas a supremum (see (5)). Moreover, letBbe a countable dense t m subset of the unit ball ofR. We can say that  ! Z t X w¼supa tlhlðsÞdMs. a2B0 l2m P a h. We o We can therefore apply Proposition 1(a) withHa¼l2ml lbtain thatðwÞhas a compensator t^T tX0 ðAtÞand that tX0  ! pffiffiffiffiffiffiffi b u ðw PsuptAtÞX2vuþupe. ½0;T3
We can replace theHabyHato obtain that  ! pffiffiffiffiffiffiffi b u Psupjw tAtjX2vuþup2e. ½0;T3
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