Jonathan J. Wylie

Samir Ben Hariz

Ben Hariz, Samir

Wylie, Jonathan J.

Numérisation de Documents Anciens Mathématiques

1/genge-pointory andtees sommesrte ou dequenceograph ofbeen con-sult wase in bothsults to anistrativeC. R. Acad. Sci. Paris, Ser. I 341 (2005) 765–768http://france.elsevier.com/direct/CRASSStatisticsConvergence rates for estimating a change-point with long-randependent sequences✩Samir Ben Hariza, Jonathan J. Wylieba Laboratoire de statistique et processus, département de mathématiques, université du Maine, avenue Olivier-Messiaen,72085 Le Mans cedex 9, Franceb Department of Mathematics, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong KongReceived 31 December 2004; accepted 26 September 2005Available online 9 November 2005Presented by Philippe G. CiarletAbstractWe consider a (possibly) long-range dependent sequence with a shift in the mean. We estimate the location of the chausing a cumulative sum estimator. The 1/n convergence rate typical of the independent case is also achieved for short-memlong-memory sequences.To cite this article: S. Ben Hariz, J.J. Wylie, C. R. Acad. Sci. Paris, Ser. I 341 (2005). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméVitesses de convergence pour l’estimation de rupture pour des suites fortements dépendantes. Nous considérons une suiéventuellement fortement dépendante, avec un saut dans sa moyenne. Nous estimons le temps de rupture à partir dpartielles. La vitesse de convergence 1/n, typique pour des suites indépendantes, est aussi obtenu pour des suites de coulongue mémoire.Pour citer cet article : S. Ben Hariz, J.J. Wylie, C. R. Acad. Sci. Paris, Ser. I 341 (2005). 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.1. IntroductionThere is a wide range of important applications in which a change in the marginal distribution of a random semust be detected. The problem has been widely studied in the literature, and we refer the reader to the monCsörg̋o and Horvath [2] for a comprehensive review on the subject.In the case in which the sequence is independent, both parametric and nonparametric methods havesidered. For example, in the nonparametric setting the problem was considered by Carlstein [1]. This resubsequently improved by Dumbgen [3] and later by Ferger [4] who obtained the exact rates of convergencprobability and in an almost surely sense. Several works are concerned with the generalization of these reweakly dependent setting.✩ The work described in this Note was fully supported by a grant from the Research Grants Council of the Hong Kong Special AdmiRegion, China (Project No. CityU 1220/02P).E-mail addresses:samir.ben_hariz@univ-lemans.fr (S. Ben Hariz), mawylie@cityu.edu.hk (J.J. Wylie).1631-073X/$ – see front matter 2005 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2005.10.003766 S. Ben Hariz, J.J. Wylie / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 765–768alized ineion in thefferenceokoszkained ratestimatords to zerolimitingces that(2). Wefunctionsestrate for.-ct rate indifferentIn recent years the importance of long-memory or long-range dependent (LRD) processes has been renumerous applications, especially financial and telecommunication data. A second-order stationary sequenc(Xi)i0, is generally said to be LRD if∞∑i=1∣∣Cov(X0,Xi)∣∣ = ∞. (1)Otherwise it is said to be short-memory or short-range dependent (SRD). The change-point test and estimatLRD setting was considered by Giraitis et al. [5] using a nonparametric method for the case in which the dibetween the distribution functions before and after the jump tends to zero with increasing sequence length. Kand Leipus [7] considered the change in the mean for dependent observations for LRD sequences. They obtain probability for the cumulative sum (CUSUM) change-point estimator and give a rate of consistency of the esthat gets worse as the strength of the dependence increases. The problem with a jump in the mean that tenwas considered by Horvath and Kokoszka [6]. They proved the consistency of the estimator and gave thedistribution.In this Note we will develop a unified framework for estimating a change in the mean of stationary sequencan be either SRD or LRD. We will give rates of convergence for the widely used CUSUM estimator defined inshow that the rate of convergence of the CUSUM estimator is independent of the decay rate of the correlationand so, under very weak conditions, independent of the dependence structure. So the 1/n rat in the independent cais also achieved for both LRD and SRD sequences.2. Main resultsLet (Xi)i=1,...,n be a stationary sequence that can be either independent, SRD or LRD. We assume thatE(X2) <∞ and without loss of generality we takeE(X) = 0. The correlation function of the sequence is given by:r(i) =Cor(X0,Xi). Let (Yi)i=1,...,n be a sequence with a shift in the mean, defined byYi = Xi + δ1inθ0 where 0< θ0 < 1is the location of the change-point. Given a sequence,(Yi)i=1,...,n, we aim to estimate the change-pointθ0 using thefollowing family of estimators:θn = 1nmin(argmax1k<n{|Uk|}), (2)whereUk =(k(n − k)n)1−γ (1kk∑i=1Yi − 1n − kn∑i=k+1Yi)andγ is a parameter satisfying 0 γ < 1. We now state the main result of the Note:Theorem 2.1. Let (Xi)i=1,...,n be a stationary sequence with correlation functionr( ). Assume that there exiconstantsB > 0 andα > 0 such that|r(i)|  Bi−α for all i. Then we havelimx→+∞ limn→+∞ P(n|θn − θ0| > x) = 0 (3)whereθn is defined in(2).We note that under a very weak condition on the decay of the correlation function, we obtain the sameevery value ofα. We recall that ifr(i) ∼ Bi−α, thenα > 1 corresponds to the SRD case andα < 1 to the LRD caseSo the 1/n rate is achieved for both SRD and LRD sequences.The second result deals with the case in which the jump size,δ = δn, depends onn. As mentioned in the introduction, this has already been studied by Horváth and Kokoszka [6] where the authors obtained the exaprobability (see Lemma 4.5 therein), and the limiting distribution for Gaussian sequences. Here under slightlyconditions on the size of the jump we obtain the same rate in probability for more general sequences.S. Ben Hariz, J.J. Wylie / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 765–768 767nt,enerallyl-onskerfmhe overallyhhrem 1 inCorollary 2.2. Under the conditions of Theorem(3). Suppose|δn| → 0 and either|δn|nα/2 → ∞ if γ < 1 − α/4,|δn|nα/2(lnn)−1 → ∞ if γ = 1− α/4 or |δn|n2(1−γ ) → ∞ if γ > 1− α/4, thenlimx→+∞ limn→+∞ P(n|δn|2/α|θn − θ0| > x) = 0. (4)Before presenting the proof, we give a qualitative explanation of why the rate is independent ofα. We definetk := k/n, thenUk := Un(tk) whereUn is expressed as a sum of its mean and a centered random component:Un(t) := n1−γ(t (1− t))γ(δg(t) + Bn(t)). (5)Here, the mean component is given byg(t) = (1− θ0)t1tθ0 + θ0(1− t)1t>θ0 and the centered random componeBn, is given byBn(t) := Wn(t) − tWn(1) and Wn(t) := 1n[nt]∑i=1Xi.Under appropriate conditions on the underlying sequence, the sequence of functionsWn(·) (when suitably nor-malized) converges, in a weak sense, to either a Brownian motion, a fractional Brownian motion or more ga Hermite process. The fact that the rate of convergence for the estimator is independent ofα is due to the cancelation of two opposing effects: the rate of convergence of partial sums, and the size of fluctuations of the Dline Wn(t). Assume for simplicity thatr(i) ∼ Ci−α. Roughly speaking, in the LRD case(α < 1), Wn(t), and henceBn(t), converges to zero with a rate of the ordern−α/2. So asα decreases, the size of the random component oUkbecomes larger, making the estimation more difficult. However, asα decreasesWn(t), and henceBn(t), become moreregular in the sense thatn−α/2|Wn(t)−Wn(t ′)| has size of order|t − t ′|1−α/2. Hence, local fluctuations in the randocomponent ofUk are reduced and so estimation becomes easier. These two effects cancel each other and trate is independent ofα.3. Sketch of the proofIn what followsK1,K2, . . . denote positive constants that are independent of. We will take α < 1, since anycorrelation function that satisfies the conditions of the theorem forα  1 will also satisfy the conditions for ansmaller value ofα. Without loss of generality we also assume that the size of the jump is positive (δ > 0).We define the following setsSn,j ={θ : 2j < rn|θ − θ0|  2j+1},wherern is a positive sequence to be chosen later. Let 0< η < min(θ0,1 − θ0)/2 andJ = J (n,η) be chosen sucthat 2J < rnη  2J+1. For the sake of brevity, we definew(θ) := (θ(1 − θ))γ andh(θ) := g(θ)/w(θ). Let θn be amaximum of{|Un(t)|, t ∈ Gn}, whereGn := {k/n, 1 k < n}. ThenP(rn|θn − θ0| > 2M)J∑j=MP(θn ∈ Sn,j ) + P(|θn − θ0| > η). (6)Sinceθn is a maximum,|Un(θn)|  |Un(θ0)|. In order to controlP(θn ∈ Sn,j ) the only difficulty arises when botUn(θn) andUn(θ0) are positive, in which case we have to boundP(Bn(θn)w(θn)− Bn(θ0)w(θ0) δ(h(θ0) − h(θn))). (7)This in turn can be reduced to consideringP(supθ∈Sn,j∣∣(Bn(θ) − Bn(θ0))∣∣  δCw(θ0)2j−1rn).To bound the above expression, we need the following maximal inequality which is a special case of TheoMoricz [8].768 S. Ben Hariz, J.J. Wylie / C. R. Acad. Sci. Paris, Ser. I 341 (2005) 765–768a1.1 in–310.–81.–314.Lemma 3.1. Assume that there exists constantsB > 0 andα < 1 such that|r(i)|  Bi−α for all i. Then there existsconstantD = D(B,α) such thatE(max1kn∣∣∣∣∣k∑i=1Xi∣∣∣∣∣)2 DE(X2)n2−α. (8)Using (6) and (8) yieldsP(rn|θn − θ0| > 2M) K5δ−2JE(X2)n−α + K6δ−2E(X2)(rnn)α J∑j=M2−jα + P(|θn − θ0| > η). (9)According to the results of Kokoszka and Leipus [7], limn→∞ P(|θn − θ0| > η) = 0. Taking rn = n and lettingngo to∞ yieldslimn→∞ P(n|θn − θ0| > 2M) K6δ−2E(X2)∞∑j=M2−jα.Letting M tend to infinity completes the proof of (3). The corollary is just a simple consequence of CorollaryKokoszka and Leipus [7] and (9).References[1] E. Carlstein, Nonparametric change-point estimation, Ann. Statist. 16 (1988) 188–197.[2] M. Csörg̋o, L. Horváth, Limit Theorems in Change-Point Analysis, Wiley, Chichester, 1997.[3] L. Dumbgen, The asymptotic behavior of some nonparametric change-point estimators, Ann. Statist. 19 (1991) 1471–1495.[4] D. Ferger, Exponential and polynomial tailbounds for change-point estimators, J. Statist. Plann. Inference 92 (2001) 73–109.[5] L. Giraitis, R. Leipus, D. Surgailis, The change-point problem for dependent observations, J. Statist. Plann. Inference 53 (1996) 297[6] L. Horváth, P. Kokoszka, The effect of long-range dependence on change-point estimators, J. Statist. Plann. Inference 64 (1997) 57[7] P. Kokoszka, R. Leipus, Change-point in the mean of dependent observations, Statist. Probab. Lett. 40 (1998) 385–393.[8] F. Moricz, Moment inequalities and the strong laws of large numbers, Z. Wahrscheinlichkeitstheorie Verw. Gebiete 35 (4) (1976) 299

Convergence rates for estimating a change-point with long-range dependent sequences

Comptes Rendus Mathématique

https://comptes-rendus.academie-sciences.fr/mathematique/item/10.1016/j.crma.2005.10.003.pdf

Convergence rates for estimating a change-point with long-range dependent sequences

Abstract

Similar works

Full text

Available Versions

Numérisation de Documents Anciens Mathématiques

Comptes Rendus Mathématique