Holder norm test statistics for epidemic change

Holder norm test statistics for epidemic change


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Holder norm test statistics for epidemic change? Alfredas Racˇkauskas Vilnius University and Institute of Mathematics and Informatics Department of Mathematics, Vilnius University Naugarduko 24, Lt-2006 Vilnius Lithuania Charles Suquet Universite des Sciences et Technologies de Lille Mathematiques Appliquees, F.R.E. CNRS 2222 Bat. M2, U.F.R. de Mathematiques F-59655 Villeneuve d'Ascq Cedex France Abstract To detect epidemic change in the mean of a sample of size n, we introduce new test statistics UI and DI based on weighted increments of partial sums. We obtain their limit distributions under the null hypothesis of no change in the mean. Under alternative hypothesis our statistics can detect very short epidemics of length log? n, ? > 1. Using self-normalization and adaptiveness to modify UI and DI, allows us to prove the same results under very relaxed moment assumptions. Trimmed versions of UI and DI are also studied. Keywords: change point, epidemic alternative, functional central limit theorem, Holder norm, partial sums processes, selfnormalization. Mathematics Subject Classifications (2000): 62E20, 62G10, 60F17. ?Research supported by a cooperation agreement CNRS/LITHUANIA (4714) 1

  • using continuous functionals

  • change point

  • line process

  • statistics can

  • parameter changes

  • ?n

  • test statistics

  • ?n cannot

  • when x1

  • self-normalization



Publié par
Nombre de lectures 24
Langue English
Signaler un problème
Vilnius University and Institute of Mathematics and Informatics
Department of Mathematics, Vilnius University
Naugarduko 24, Lt-2006 Vilnius
Math´ematiquesAppliq´s,FR.E.CNRS2222 uee .
F-59655 Villeneuve d’Ascq Cedex France
To detect epidemic change in the mean of a sample of sizen, we introduce
new test statistics UI and DI based on weighted increments of partial sums.
We obtain their limit distributions under the null hypothesis of no change in
the mean. Under alternative hypothesis our statistics can detect very short epidemics of length logγn,γ >1. Using self-normalization and adaptiveness
to modify UI and DI, allows us to prove the same results under very relaxed
moment assumptions. Trimmed versions of UI and DI are also studied.
Keywords:change point, epidemic alternative, functional central limit theorem,
Mathematics Subject Classifications (2000): 62E20, 62G10, 60F17.
Research supported by a cooperation agreement CNRS/LITHUANIA (4714)
An important question in the large area of change point problems involves testing the null hypothesis of no parameter change in a sample versus the alternative that parameter changes do take place at an unknown time. For a survey we refer to the booksbyBrodskyandDarkhovsky[4]orCs¨org˝oandHorv´ath[5].Inthispaper we suggest a class of new statistics for testing change in the mean under so called epidemic alternative. More precisely, given a sampleX1, X2, . . . , Xn, we want to test the standard null hypothesis of constant mean
(H0):X1, . . . , Xnall have the same mean denoted byµ0, against the epidemic alternative (HA):there are integers1< k< m< nand a constantµ16=µ0such thatEXi=µ0+ (µ1µ0)1{k<im},i= 1,2, . . . , n. Writingl:=mkfor the length of the epidemic, we assume throughout the paper that bothlandnlgo to infinity withn. In this paper we follow the classical methodology to build test statistics by using continuous functionals of a partial sums process. Set S(0) = 0, S(t) =XXk,0< tn. kt WhenAis a set of integers,S(A) will denotePiAXi instance we suggest to. For use with 0< α <1/2 S(j)S(i)S(n)(j/ni/n)UI(n, α) :=1im<ajxn(j/ni/n)1(ji)/nα. Thisisafunctional,continuousinsomeH¨oldertopology,oftheclassicalDonsker-Prokhorov polygonal line process. The statistics UI(n,0) was suggested by Levin and Kline (1985), see [5]. To motivate the definition of such test statistics, let us assume just for a moment that the changes timeskandmare known. Suppose moreover under (H0) that the (Xi, 1in) are independent identically distributed with finite variance σ2 (. UnderHA), suppose that the (Xi, iIn) and the (Xi, iIcn) are separately independent identically distributed with the same finite varianceσ2, where In:=i∈ {1, . . . , n};k< im,Icn:={1, . . . , n} In. 2
Then we simply have a two sample problem with known variances. It is then natural to accept (H0) for small values of the statistics|Q|and to reject it for large ones, where S(In)lS(n)/n S(Icn)(nl)S(n)/n =Q(:l)1/2(nl)1/2.
After some algebra,Qmay be recast as Q=(l(n(n)1l/2))1/2S(In)lnS(n)h(1l/n)1/2+ (l/n)1/2i As the last factor into square brackets ranges between 1 and 21/2, we may drop it and so replace|Q|by the statistics R(:=l(n(n)1l/2))1/2S(In)lnS(n)=n1/2S(In)nlS(n)ln1nl1/2.
Introducing now the notation
k0k tk=tn,k:=n,n, enables us to rewriteRas R=n1/2S((mtm)S(k)S(n)(tm)t1/k2). tk)1(tmtk Now in the more realistic situation wherekandmare unknown it is rea-sonnable to replaceRby taking the maximum over all possible indexes forkand m leads to consider. This S(j)S(i)S(n)(tjti)UI(n,1/2) :=1mi<ajxn(tjti)1(tjti)1/2. It is worth to note that the same statistics arises from likelihood arguments in the special case where the observationsXi The asymptoticare Gaussian, see [14]. distribution of UI(n,1/unknown, due to difficulties caused by the denominator2) is (for historical remarks see [5, p.183]). In our setting, theXi’s are not supposed to be Gaussian. Moreover it seems fair to pay something in terms of normalization when passing fromRton1/2UI(n,1/2). Intuitively the cost should depend on the moment assumptions made about the Xi discuss this, ’s. Tolet us introduce the polygonal partial sums processξndefined
by linear interpolation between the pointstk, S(k). Then UI(n,1/2) appears as the discretization through the grid (tk,0kn) of the functionalT1/2(ξn) where T1/2(x) :=0<ssu<tp<1x(t)(xt(s)s)1x(1()tsx)(0)1/(2ts).(1) ThisfunctionaliscontinuousintheH¨olderspaceH1/o2of functionsx: [0,1]R such that|x(t+h)x(t)|=o(h1/2), uniformly int finite dimensional dis-. Obviously tributions ofn1/2σ1ξnconverge to those of a standard Brownian motionW. How-ever Lvy’s theorem on the modulus of uniform continuity ofWimplies thatH1/o2 has too strong a topology to support a version ofW. Son1/2ξncannot converge in distribution toσWin the spaceH1/o2 forbid us to obtain limiting distribution. This forT1/2(n1/2ξn) by invariance principle inHo1/2 Fortu-via continuous mapping. nately,Ho¨lderianinvariancesprinciplesdoexistfor,roughlyspeaking,allthescale ofH¨olderspacesHρoof functionsxsuch that|x(t+h)x(t)|=oρ(h), uniformly int, provided that the weight functionρsatisfies limh0ρ(h)(hlog|h|)1/2=. This type of invariance principles goes back to Lamperti [6] who studied the case ρ(h) =hα(0< α <1/2). Acomplete characterization in terms of moments assumptions onX1recently by the authors in the general case (seewas obtained Theorem 11 below). This leads us to replaceT1/2(n1/2ξn) byTα(n1/2ξn) ob-tained substituting the denominator in (1) by (ts)α1(ts)α back to. Going the discretization we finally suggest the class UI (uniform increments) of statistics which includes particularly UI(n, α) and similar ones UI(n, ρ) built with a general weightρ(h) instead ofhα. Together with UI we consider the class of DI (dyadic increments) statistics, which includes particularly DI(n, α) =1jmlaoxg2n2rjαmaDxjS(nr)21S(nr+n2j)12S(nrn2j), where Djis the set of dyadic numbers of the leveljand log2denotes the logarithm with basis 2. DI(n, α) and UI(n, α Moreover,) have similar asymptotic behaviors. dyadic increments statistics are of particular interest since their limiting distribu-tions are completely specified (see Theorem 10 below). Due to the independence of theXi’s, it is easy to see that even stochastic boundedness of either ofn1/2UI(n, α) orn1/2DI(n, α) yields that ofn1/2+α max1in|Xi|. Hence, necessarily P(|X1|> t) =Otp(α), wherep(α) = (1/2α)1 heavier is the weight. So,ρ(h), stronger are the required moment assumptions
to obtain the convergence ofn1/2UI(n, ρ the other hand, the interest of a). On heavy weight is in the detection of short epidemics. Indeed (see Theorem 4 below), UI(n,0) can detect only epidemics whose the lengthlis such thatn1/2=o(l). For 0< α <1/2, UI(n, α) detects epidemics withnδ=o(l) whereδ= (12α)/(22α). With the weightρ(h) =h1/2logβ(c/h),β >1/2, UI(n, ρ) detects epidemics such that log2βn=o(l). We consider two ways to preserve the sensitiveness to short epidemics while re-laxing moments assumptions. One is trimming and the other one is self-normalization and adaptive selection of partial sums increments. Trimming leads to a class of statistics, which includes
n2j DI(n, α, γ) =1jmγaloxg2n2jαrmaDjxS(nr)21S(nr+n2j)21S(nr), where 0< γ < construction of partial sums process and the corre-1. Adaptive sponding functional central limit theorem proved in [10] allows to deal with a class of statistics which includes e.g. SUI(n, α) obtained by replacing in UI(n, α) the deterministic pointstkby the randomvk:=Vk2/Vn2, whereVk2=X12+∙ ∙ ∙+Xk2, k= 1, . . . , n,
)(vjvi)SUI(n, α) =1mi<ajxnS((jvj)Svi)(i1)S((vjnvi)α. WhenX1is symmetric, the convergence in distribution ofVn1SUI(n, α) requires only the membership ofX1in the domain of attraction of normal law, otherwise the existence of E|X1|2+εfor someε >0 is sufficient. The paper is organized as follows. In Section 2, definitions of classes of statis-tics are presented and limiting distributions are given. All proofs are deferred to Section 3.
DIstatistics and their asymptotics
All the test statistics studied in this paper may be viewed as discretizations of someHo¨ldernormsorsemi-norms.Thefollowingsubsectioncontainstherelevant background.
Letρ: [0,1]R+ of a continuous function Membershipbe a weight function. x: [0,1]RepscadlreHeo¨nihtHoρmeans roughly that|x(t+h)x(t)|=oρ(|h|)uniformly intfHeold¨oalicalscsesupsresecahTcealss.ρ(h) =hα, 0< α <1. Le´vystheoremonthemodulusofcontinuityoftheBrownianmotionrestrictsthe investigation of invariance principles in this scale to the range 0< α <1/2. The sameresultleadsnaturallytoconsidertheH¨olderspacesbuiltwiththefunctions ρ(h) =h1/2logβ(c/h) forβ >1/include these two cases of practical interest 2. We in the following rather general classR.
Definition 1.We denote byRthe class of non decreasing functionsρ: [0,1]R+ satisfying
i) for some0< α1/2, and some functionL, positive on[1,)and normal-ized slowly varying at infinity, ρ(h) =hαL(1/h),0< h1;
ii)θ(t) =t1/2ρ(1/t)isC1on[1,); iii) there is aβ >1/2and somea >0, such thatθ(t) logβ(t)is non decreasing on[a,).
Let us recall thatLis normalized slowly varying at infinity if and only if for everyδ >0,tδL(t) is ultimately increasing andtδL(t) is ultimately decreasing [3, Th. 1.5.5]. The main practical examples we have in mind may be parametrized by
ρ(h) =ρ(h, α, β) :=hαlogβ(c/h).
We writeC[0,1] for the Banach space of continuous functionsx: [0,1]R endowed with the supremum normkxk:= sup{|x(t)|;t[0,1]}. Definition 2.Letρbe a real valued non decreasing function on[0,1], null and right continuous at0. We denote byHoρcaerepso¨dltheH o = Hρ:{xC[0,1];δlim0ωρ(x, δ) = 0},
where ωρ(x, δ) := sup|x(tρ)(tsx)(s)|. s,t[0,1], 0<ts<δ Hρois a separable Banach space for its native norm
kxkρ:=|x(0)|+ωρ(x,1). Under technical assumptions, satisfied by anyρinR, the spaceHρomay be endowed with an equivalent normkxkρseqbuilt on weighted dyadic increments ofx. Let us denote by Djthe set of dyadic numbers in [0,1] of levelj, i.e. D0={0,1},Dj=(2l1)2j; 1l2j1, j1. We write D (resp. D) for the sets of dyadic numbers in [0, (01] (resp.,1]) D :=j=0Dj,D:= D {0}.
Put forrDj,j0,
r:=r2j, r+:=r+ 2j.
For any functionx: [0,1]R, define its Schauder coefficientsλr(x) by λr(x) :=x(r)x(r++)2x(r), rDj, j1 (2) and in the special casej= 0 byλ0(x) :=x(0),λ1(x) :=x(1). Whenρbelongs to R, we have on the spaceHoρthe equivalence of norms (see [11] and [12]) 1 kxkρ∼ kxkρseq:= sju0pρ(2j)rmaDxj|λr(x)|. LetW= (W(t), t[0,1]) be a standard Wiener process andB= (B(t), t[0,1]) the corresponding Brownian bridgeB(t) =W(t)tW(1),t[0,1]. Consider forρinR, the following random variables UI(ρ) :=0<stusp<1ρ(t|sB(t)1()B((ts)|s))(3)
and DI(ρ) = sju1pρ12(j)rmaDxjW(r)21W(r+)21W(r)=kBksρeq.(4) These variables serve as limiting for uniform increment (UI) and dyadic increment (DI) statistics respectivly. No analytical form seems to be known for the distribu-tion function of UI(ρ), whereas the distribution of DI(ρ) is completely specified by Theorem 10 below.
Remark 1.atrettnodoWeeptrehrpnihtsiapowregulaoblemoflredytirlo¨H norms associated to the weightsρ(h) =L(1/h)withLslowly varying. this case In nothing seems known about equivalence of the normsk kρandk ksρeq, which plays akeyrˆoleinourproofsofHo¨lderianinvarianceprinciples.Theextensionofour results to this boundary case would require other technics and seems to be of limited practical scope.
2.2 StatisticsUI(n, ρ)andDI(n, ρ)
To simplify notation put
%(h) :=ρh(1h),0h1.
Forρ∈ R, define (recalling thattk=k/n, 0kn), tjti)UI(n, ρ) =1im<ajxnS(j)S(i%)(tjSt(i)n)(
and DI(n, ρ) =1<m2jaxnρ12(j)rmaDxjS(nr)21S(nr+)21S(nr). To obtain limiting distribution for these statistics we shall work with a stronger null hypothesis, namely
(H00):X1, . . . , Xnare independent identically distributed random vari-ables with mean denotedµ0. Theorem 3.Under(H00), assume thatρ∈ Rand for everyA >0,
limtP(|X1| t> Aθ= 0. t→∞( ))
σ1n1/2UI(n, ρ)−−D−→UI(ρ) n→∞
and D σ1n1/2DI(n, ρ)−−−→DI(ρ), n→∞ whereσ2= var(X1)andUI(ρ),DI(ρ)are defined by (3) and (4) respectively.