Testing epidemic changes of infinite dimensional parameters

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Testing epidemic changes of infinite dimensional parameters 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 Laboratoire P. Painleve, UMR CNRS 8524 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 of random elements in a Banach space, we introduce new test statistics 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. We present applications to detect epidemic changes in distribution function or characteristic function of real valued observations as well as changes in covariance matrixes of random vectors. Some keywords: change point, epidemic alternative, functional central limit theorem, Holder norm, partial sums processes 1 Introduction A central question in the area of change point detection is testing for changes in the mean of a sample. Indeed many change point problems may be reduced to this basic setting, see e.g. Brodsky and Darkhovsky [1]. Here we present a new illustration of this general approach. Starting from the detection of epidemic changes in the mean of Banach space valued random elements, we construct new tests to detect changes in the distribution function or the characteristic function of real valued observations as well as changes in covariance matrixes of

holderian weight

detect epidemic

gaussian random

alternative

test statistics

bit stronger then

dyadic increment

sums process

sequence n?1

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Publié par

rasum

Nombre de lectures

Langue

English

Testing

epidemic

changes of inﬁnite dimensional parameters

AlfredasskasckauˇaR Vilnius University and Institute of Mathematics and Informatics Department of Mathematics, Vilnius University Naugarduko 24, Lt-2006 Vilnius Lithuania

CharlesSuquet Universite´desSciencesetTechnologiesdeLille LaboratoireP.Painleve´,UMRCNRS8524 Bˆat.M2,U.F.R.deMath´ematiques F-59655 Villeneuve d’Ascq Cedex France

Abstract To detect epidemic change in the mean of a sample of sizenof random elements in a Banach space, we introduce new test statistics 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. We present applications to detect epidemic changes in distribution function or characteristic function of real valued observations as well as changes in covariance matrixes of random vectors. Some keywords:change point, epidemic alternative, functional central limit theorem,H¨oldernorm,partialsumsprocesses

1 Introduction

A central question in the area of change point detection is testing for changes in the mean of a sample. Indeed many change point problems may be reduced to this basic setting, see e.g. Brodsky and Darkhovsky [1]. Here we present a new illustration of this general approach. Starting from the detection of epidemic changes in the mean of Banach space valued random elements, we construct new tests to detect changes in the distribution function or the characteristic function of real valued observations as well as changes in covariance matrixes of random vectors. LetBbe a separable Banach space with a normkxkand dual spaceB0with duality denoted byf(x), f∈B0, x∈B that. SupposeX1, . . . , Xnare random

elements inBwith meansm1, . . . ,mn want to test the standardrespectively. We null hypothesis of a constant mean

(H0):m1=∙ ∙ ∙=mn against the so called epidemic alternative (HA):there are integers1< k∗< m∗< nsuch that m1=m2=∙ ∙ ∙=mk∗=mm∗+1=∙ ∙ ∙=mn, mk∗+1=∙ ∙ ∙=mm∗andmk∗6=mk∗+1. The study of epidemic change models (withB=R) goes back to Levin and Kline [8] who proposed test statistics based on partial sums. Using a maximum likelihood approach, Yao [14] suggested some weighted versions of these statistics whichmaybeviewedassomediscreteH¨oldernormofthepartialsumsprocess, seealsoCs¨org˝oandHorv´ath[3]andthereferencestherein. In[12],westudyalargeclassofstatisticsobtainedbydiscretizingHo¨lder norms of the partial sums process (withB=R). One important feature ofHo¨lderianweightingisthedetectionofshortepidemics.Roughlyspeak-ing,theuseofHo¨lderiantestsallowsthedetectionofepidemicswhoselength l∗:=m∗−k∗is at least of the order of lnγnwithγ >1, while the same test statisticswithoutH¨olderianweightdetectsonlyepidemicssuchthatn−1/2l∗ goes to inﬁnity. Among the test statistics suggested in [12], the statistics DI(n, ρ) built on the dyadic increments of partial sums are of special interest because their limiting distribution is explicitly computable. The aim of this new con-tribution is to investigate asymptotical behavior of DI(n, ρ) in the setting ofB valued random elementsXi’s. The paper is organized as follows. Section 2 introduce the dyadic increment test statistics and present general results on their asymptotical behavior under (H0) or under (HA). Several applications are proposed in Section 3, involv-ing epidemic changes in distribution function (through Kolmogorov-Smirnov or Cram´er-vonMisesversionsofDI(n, ρ)), in characteristic function, in the co-variance of a random vector. Section 4 gathers the relevant background on Ho¨lderianfunctionalcentrallimittheoremandtheproofs.

2 The class of DI test statistics

Let us denote by Djthe set of dyadic numbers in [0,1] of levelj, i.e. D0={0,1},Dj=(2l−1)2−j; 1≤l≤2j−1, j≥1. Write forr∈Dj,j≥0, r−:=r−2−j, r+:=r+ 2−j .

For a functionx: [0,1]→B, we shall denote

λr(x) :=x(r)−21(x(r+) +x(r−)),

r∈Dj,

j≥1

(1)

andλr(x) =x(r) in the special caser= 0,1. Consider partial sums S(0) = 0, S(u) =XXk,0< u <∞. k≤u

We deﬁne also Sn(a, b) :=S(nb)−S(na) =XXk,0≤a < b≤1 na<k≤nb

and

Sn(t) =Sn(0, t) =XXk,0≤t≤1. k≤nt

Dyadic increments statistics DI(n, ρ) depend on a weight functionρ: [0,1]→R and are deﬁned by

DI(n, ρ m 1) := 1≤j≤aloxgnρ(2−j)rm∈aDxjkλr(Sn)k =211≤mjlaoxgnρ(12−j)rm∈aDxjXXk−X+Xk.(2) nr−<k≤nr nr<k≤nr In this paper, we write “log” for the logarithm with basis 2 (log(2j) =j) and “ln” for the natural logarithm (ln(et) =t we assume that). Throughoutρ belongs to the following classR.

Deﬁnition 1.LetRbe the class of non decreasing functionsρ: [0,1]→R, positive on (0,1], such thatρ(0) = 0 and satisfying

i) for some 0< α≤1/2, and some functionLwhich is normalized slowly varying at inﬁnity,

ρ(h) =hαL(1/h),0< h≤1; ii)θ(t) :=t1/2ρ(1/t) isC1on [1,∞);

(3)

iii) there is aβ >1/2 and somea >0, such thatθ(t) ln−β(t) is non decreasing on [a,∞).

For a random elementXin a separable Banach spaceBsuch that for every f∈B0,Ef(X) = 0 andEf2(X)<∞, its covariance operatorQ=Q(X) is the linear bounded operator fromB0toBdeﬁned byQf=Ef(X)X,f∈B0. A random elementX∈B(or covariance operatorQ) is said to bepregaussian if there exists a mean zero Gaussian random elementY∈Bwith the same covariance operator asX all for, i.e.f, g∈B0,Ef(X)g(X) =Ef(Y)g(Y). Since the distribution of a centered Gaussian random element is deﬁned by its covariance structure, we denote byYQa zero mean Gaussian random element with covariance operatorQ.

For any pregaussian covarianceQthere exists aB-valued Brownian motion WQwith parameterQ centered Gaussian process indexed by [0, i.e. a,1] with independent increments such thatWQ(t)−WQ(s) has the same distribution as |t−s|1/2YQ. A random elementXis said to satisfy the central limit theorem inB(denoted X∈CLT(B)) if the sequencen−1/2(X1+∙ ∙ ∙+Xn) converges in distribution in B, whereX1, . . . , Xnare independent copies ofX. Necessarily then (see [7]),X is mean zero, pregaussian and satisﬁes the moment condition tli→m∞tPkX1k> t1/2= 0.(4)

However the central limit theorem forXcannot be characterized in general in terms of the only integrability ofXbecause the geometry ofBis involved in the problem. To complete these preliminaries, we recall here the following classical esti-mate (see [7], (3.5) p.59) for the tail of a mean zero Gaussian random element YinB. P(kYk ≥u)≤4 exp8E−kuY2k2, u >0.(5) In order to investigate the asymptotic behavior of the statistics DI(n, ρ) we consider a null hypothesis a little bit stronger thenH0. Namely, (H00):X1, . . . , Xn meanare i.i.d. zero pregaussian random elements inBwith covarianceQ.

Let (YQ, YQ,r, r∈D) be a collection of non degenerate independent mean zero Gaussian random elements with covarianceQ with. Setθas in Deﬁni-tion 1 ii) DI(ρ, Q) := sjup√2θ2(1j)tm∈aDjxkYQ,rk.(6) ≥1

Due to the deﬁnition of the classRit can be shown (see Theorem 13 below), that the random variable DI(ρ, Q) is well deﬁned for anyρ∈ R. Theorem 2.If under(H00),X1∈CLT(B)and for everyA >0 limtPkX1k> Aθ(t)= 0,(7) t→∞

then

n−1/2DI(n, ρ)−−D−→DI(ρ, Q). n→∞

(8)

Asθ(t) =t1/2ρ(1/t), Condition (7) is clearly stronger than (4) which is implicitly included in the assumption “X1∈CLT(B)”. Ifρ(h) =ρα(h) = hα, h∈[0,1], where 0< α <1/2, then Condition (7) reads − PkX1k ≥t=ot−p(α)withp(α) =21−α1.

In the case whereρ(h) =ρα,β(h) =h1/2lnβ(c/h), h∈[0,1] withβ >1/2, Condition (7) is equivalent to

Eexp{dkX1k1/β}<∞,for alld >0. For a discussion of the condition “X1∈CLT(B Let)”, we refer to [7]. us mention simply here that if the spaceBis either of type 2 or has Rosenthal’s property (e.g.Bis any separable Hilbert space,B=Lp(S, µ) with 2≤p <∞etc.) then Condition (7) yieldsX1∈CLT(B). IfBis of cotype 2 (e.g.B=Lp(S, µ),1≤ p≤2), then “X1∈CLT(B)” follows fromX1being pregaussian. Due to the independence ofYQ,rthe limiting distribution functionFQ,ρ(u) of the dyadic increments statistic is completely speciﬁed by the distribution function ΦQ(u) :=PkYQ,1k ≤u√2, u≥0. Namely, we have

Proposition 3.Ifρ∈ R, then the distribution ofDI(ρ, Q)is absolutely con-tinuous and its distribution functionFQ,ρis given by ∞ FQ,ρ(u) :=PDI(ρ, Q)≤u=YΦQθ(2j)u2j−1, u≥0.(9) j=1

The convergence in (9) is uniform on any interval[ε,∞),ε >0.

For practical applications we sumarized in the next proposition some esti-mates of the tail of distribution functionFQ,ρ. Denote forJ≥0 J F(JQ,)ρ(u) =YΦQθ(2j)u2j−1, u≥0 j=1

and X2jjθ22 cJ:=γin>f0nγ+ 8j=J∞+1θ2(2 ) exp−γ(4j)o. Proposition 4.For eachρ∈ R,FQ,ρsatisﬁes the following estimates.

i) For eachu >0 1−FQ,ρ(u)≤4 exp8c0E|−u|Y2Q||2.

ii) For eachJ≥1andu >0 1−4 exp8cJE||−Yu2Q||2FQ(ρ,J)(u)≤FQ,ρ(u)≤FQ(,ρJ)(u).

(10)

(11)

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We end this section by examining a consistency of rejecting (H00) versus the epidemic alternative (H0A) for large values of DI(n, ρ), where :={k∗+ 1, . . . , m∗} (H0A)Xk=(Xmc0k+Xk0fiifkk∈∈IIncn:={1, . . . , n} \In wheremc6= 0 may depend onnand theXk0’s satisfy (H00). =− Theorem 5.Letρ∈ R. Under(H0A), writel∗:m∗k∗for the length of epidemics and assume that ck nli→m∞n1/2ρhn(khmn)∞,wherehn:= minnln∗; 1−nl∗o.(13) =

Then

− n1/2DI(n, ρ)−p−r∞−→. n→∞

To discuss Condition (13), consider for simplicity the case wheremcdoes not depend onn. Whenρ(h) =hα, (13) allows us to detectshort epidemicssuch ∗ thatl=o(n) andl∗n−δ→ ∞, whereδ= (1−2α)(2−2α)−1. Symmetrically − one can detectlong epidemicssuch thatn−l∗=o(n) and (n−l∗)nδ→ ∞. Whenρ(h) =h1/2lnβ(c/h) withβ >1/2, (13) is satisﬁed provided that hn=n−1lnγn, withγ >2β. This leads to detection of short epidemics such thatl∗=o(n) andl∗ln−γn→ ∞as well as of long ones verifyingn−l∗=o(n) and (n−l∗) ln−γn→ ∞.

3 Examples

3.1 Testing change of distribution function

As an example of applications of Theorem 2, here we consider change-point problem for distribution function of a random sample inRunder epidemic alternative. LetZ1, . . . , Znbe real valued random variables with distribution functionsF1, . . . , Fnrespectively. Consider the null hypothesis (H0) :F1=∙ ∙ ∙=Fn=F and the following epidemic alternative: (HA):there are integers1< k∗< m∗< nsuch that F1=F2=∙ ∙ ∙=Fk∗=Fm∗+1=∙ ∙ ∙=Fn, Fk∗+1=∙ ∙ ∙=Fm∗andFk∗6=Fk∗+1. Tests constructed in Lombard [9] and Gombay [5] are based on rank statistics. Theorem 2 suggests tests based on the dyadic increments of empirical process. For simplicity we consider only the case of continuous functionF we. Then can restrict to the uniform empirical process built on the sample (U1, . . . , Un), whereUk=F(Zk).

Deﬁne κn(s, t) :=X1{Ui≤t}−t t, s,∈[0,1]. i≤ns Forr∈Dj,j≥1, deﬁne random functionsλr(κn) by λr(κn)(t) :=κn(r, t)−21κn(r−, t) +κn(r+, t), t∈[0,1] which for practical computations may be conveniently recast as 2λr(κn)(t) =X1{Ui≤t}−t−X1{Ui≤t}−t, t∈[0,1]. nr−<i≤nr nr<i≤nr+

3.1.1 Cramer-von Mises type DI statistics ´

TheCram´er-vonMisestypedyadicincrementsstatisticsaredeﬁnedby 2 CMDI(n, ρ) =1≤jm≤aloxgnρ2(21−j)rm∈aDxjλr(κn)2,

(14)

where λr(κn)22Z10 :=λr(κn)(t)2dt. Its limiting distribution is completely deﬁned by the limiting distribution of the classicalCram´er-vonMisesstatistic,namely,bythedistributionfunction L2(u) =PZ10B2(t) dt≤u, u≥0,

whereB(t), t∈[0, The distribution function1] is a standard Brownian bridge. L2(uis well known and several of its representations are available (see [13]).) Theorem 6.Ifρ∈ Rthen nli→m∞Pn−1CMDI(n, ρ)≤u=L2,ρ(u), for eachu >0, where

∞−1 L2,ρ(u) =YhL22θ2(2j)ui2j. j=1

Proof.The result is a straightforward application of Theorem 2 to the random elementsX1, . . . , Xnwith values in the spaceL2(0,1) deﬁned by Xk(t) =1{Uk≤t}−t, t∈[0,1], k= 1, . . . , n.(15) It is well known that these random elements satisfy the central limit theorem with Brownian bridge as the limiting Gaussian element. The moment condition (7) is fulﬁled since theXks are bounded. ’

For practical uses, the following estimate of the tails ofL2,ρcan be helpfull. Deﬁne foru >0 J L(2J,ρ)(u) =YhL22θ2(2j)ui2j−1. j=1

Proposition 7.

i) Ifu >0then 1−L2,ρ(u)≤4 exp−4c30u. where the constantc0is deﬁned by (10). ii) For eachu >0andJ≥1 h1−4 exp4−c3JuiL(2J,ρ)(u)≤L2,ρ(u)≤L(2ρ,J)(u). Proof.The result is a straightforward adaptation of Proposition 4, withYQ=B, a standard Brownian bridge, using the elementary fact thatEkBk22= 1/6. 3.1.2 Kolmogorov-Smirnov type DI statistics

The Kolmogorov-Smirnov type dyadic increments statistics are deﬁned by KSDI(n, ρ) =1≤jm≤aloxgnρ(21−j)rm∈aDxjλr(κn)∞, where := supλr(κn)(t). λr(κn)∞0≤t≤1 LetL∞(u) be the limiting distribution of the classical Kolmogorov-Smirnov statistic, L∞(u) =P0m≤ta≤x1|B(t)| ≤u, u≥0. This distribution function is well known and has several representations (see [13]).

Theorem 8.Ifρ∈ Rthen nli→m∞Pn−1/2KSDI(n, ρ)≤u=L∞,ρ(u), for eachu >0, where

∞i2j−1. L∞,ρ(u) =YhL∞√2θ(2j)u j=1

The proof of this theorem is not a straightforward corollary of Theorem 2 and is postponed to Subsection 4.5 below.

Voir