Holder norm test statistics for epidemic change

Charles Suquet

Description

Niveau: Supérieur, Master, Bac+5
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

Sujets

• Vilnius University
• Suquet
• Statistical hypothesis testing

Informations

• Publié par: rasum
• Nombre de lectures: 24
• Langue: English

Extrait

H¨oldernormteststatisticsforepidemicchange ∗

Alfredasc ˇ aRsuaksak

Vilnius University and Institute of Mathematics and Informatics

Department of Mathematics, Vilnius University

Naugarduko 24, Lt-2006 Vilnius

Lithuania

CharlesSuquet

Universit´edesSciencesetTechnologiesdeLille

Math´ematiquesAppliq´s,FR.E.CNRS2222 uee .

Bˆat.M2,U.F.R.deMathe´matiques

F-59655 Villeneuve d’Ascq Cedex France

Abstract

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 log γ n,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,

H¨oldernorm,partialsumsprocesses,selfnormalization.

Mathematics Subject Classifications (2000): 62E20, 62G10, 60F17.

∗ Research supported by a cooperation agreement CNRS/LITHUANIA (4714)

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1

Introduction

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 sampleX 1 , X 2 , . . . , X n , we want to test the standard null hypothesis of constant mean

(H 0 ):X 1 , . . . , X n all have the same mean denoted byµ 0 , against the epidemic alternative (H A ):there are integers1< k ∗ < m ∗ < nand a constantµ 1 6=µ 0 such thatEX i =µ 0 + (µ 1 −µ 0 )1 {k ∗ <i≤m ∗ } ,i= 1,2, . . . , n. Writingl ∗ :=m ∗ −k ∗ for the length of the epidemic, we assume throughout the paper that bothl ∗ andn−l ∗ go 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) = X X k ,0< t≤n. k≤t WhenAis a set of integers,S(A) will denote P i∈A X i  instance we suggest to. For use with 0< α <1/2  S(j)−S(i)−S(n)(j/n−i/n)  UI(n, α) := 1≤i m < a j x ≤n  (j/n−i/n)  1−(j−i)/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 timesk ∗ andm ∗ are known. Suppose moreover under (H 0 ) that the (X i , 1≤i≤n) are independent identically distributed with finite variance σ 2  (. UnderH A ), suppose that the (X i , i∈I n ) and the (X i , i∈I c n ) are separately independent identically distributed with the same finite varianceσ 2 , where I n :=  i∈ {1, . . . , n};k ∗ < i≤m ∗  ,I c n :={1, . . . , n} I n . 2

Then we simply have a two sample problem with known variances. It is then natural to accept (H 0 ) for small values of the statistics|Q|and to reject it for large ones, where S(I n )−l ∗ S(n)/n S(I c n )−(n−l ∗ )S(n)/n =− Q ( : l ∗ ) 1/2 (n−l ∗ ) 1/2 .

After some algebra,Qmay be recast as Q= (l ∗ (n (n − ) 1 l / ∗ 2 )) 1/2  S(I n )− l n ∗ S(n) h (1−l ∗ /n) 1/2 + (l ∗ /n) 1/2 i As the last factor into square brackets ranges between 1 and 2 1/2 , we may drop it and so replace|Q|by the statistics ∗ R ( := l ∗ (n (n − ) 1 l / ∗ 2 )) 1/2  S(I n )− l n ∗ S(n)  =n −1/2  S(I n )− n l S(n)   l n ∗  1− n l ∗  1/2 .

Introducing now the notation

k 0≤k t k =t n,k := n ,≤n, enables us to rewriteRas − R=n 1/2  S  ( ( m t m ∗ )−S(k ∗ )−S(n)(t m ∗ ∗ ) −  t 1/ k ∗ 2 )  . ∗ −t k ∗ )  1−(t m ∗ −t k Now in the more realistic situation wherek ∗ andm ∗ are unknown it is rea- sonnable to replaceRby taking the maximum over all possible indexes fork ∗ and m ∗  leads to consider. This  S(j)−S(i)−S(n)(t j −t i )  UI(n,1/2) := 1≤ m i< a j x ≤n  (t j −t i )  1−(t j −t i )  1/2 . It is worth to note that the same statistics arises from likelihood arguments in the special case where the observationsX i  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, theX i ’s are not supposed to be Gaussian. Moreover it seems fair to pay something in terms of normalization when passing fromRton −1/2 UI(n,1/2). Intuitively the cost should depend on the moment assumptions made about the X i  discuss this, ’s. Tolet us introduce the polygonal partial sums processξ n defined

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by linear interpolation between the points  t k , S(k)  . Then UI(n,1/2) appears as the discretization through the grid (t k ,0≤k≤n) of the functionalT 1/2 (ξ n ) where T 1/2 (x) := 0< s s u <t p <1  x(t)−  ( x t ( − s) s) −  1  x −( 1() t− − s x ) (  0  ) 1  / ( 2 t−s)  .(1) ThisfunctionaliscontinuousintheH¨olderspaceH 1/ o 2 of functionsx: [0,1]→R such that|x(t+h)−x(t)|=o(h 1/2 ), uniformly int finite dimensional dis-. Obviously tributions ofn −1/2 σ −1 ξ n converge to those of a standard Brownian motionW. How- ever Lvy’s theorem on the modulus of uniform continuity ofWimplies thatH 1/ o 2 has too strong a topology to support a version ofW. Son −1/2 ξ n cannot converge in distribution toσWin the spaceH 1/ o 2  forbid us to obtain limiting distribution. This forT 1/2 (n −1/2 ξ n ) by invariance principle inH o 1/2  Fortu-via continuous mapping. nately,Ho¨lderianinvariancesprinciplesdoexistfor,roughlyspeaking,allthescale ofH¨olderspacesH ρo of functionsxsuch that|x(t+h)−x(t)|=o  ρ(h)  , uniformly int, provided that the weight functionρsatisfies lim h↓0 ρ(h)(hlog|h|) −1/2 =∞. This type of invariance principles goes back to Lamperti [6] who studied the case ρ(h) =h α (0< α <1/2). Acomplete characterization in terms of moments  assumptions onX 1 recently by the authors in the general case (seewas obtained Theorem 11 below). This leads us to replaceT 1/2 (n −1/2 ξ n ) byT α (n −1/2 ξ n ) ob- tained substituting the denominator in (1) by (t−s) α  1−(t−s)  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 ofh α . Together with UI we consider the class of DI (dyadic increments) statistics, which includes particularly DI(n, α) = 1≤j m ≤l a o x g 2 n 2 r jα m ∈ a D x j  S(nr)− 2 1 S(nr+n2 −j )− 1 2 S(nr−n2 −j )  , where D j is the set of dyadic numbers of the leveljand log 2 denotes 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 theX i ’s, it is easy to see that even stochastic boundedness of either ofn −1/2 UI(n, α) orn −1/2 DI(n, α) yields that ofn −1/2+α max 1≤i≤n |X i |. Hence, necessarily P(|X 1 |> t) =O  t −p(α)  , wherep(α) = (1/2− α) −1  heavier is the weight. So,ρ(h), stronger are the required moment assumptions

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to obtain the convergence ofn −1/2 UI(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 lengthl ∗ is such thatn 1/2 =o(l ∗ ). For 0< α <1/2, UI(n, α) detects epidemics withn δ =o(l ∗ ) whereδ= (1− 2α)/(2−2α). With the weightρ(h) =h 1/2 log β (c/h),β >1/2, UI(n, ρ) detects epidemics such that log 2β n=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

n2 −j DI(n, α, γ) = 1≤j m ≤γ a lo x g 2 n 2 jα r m ∈ a D j x  S(nr)− 2 1 S(nr+n2 −j )− 2 1 S(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 pointst k by the randomv k :=V k 2 /V n 2 , whereV k 2 =X 12 +∙ ∙ ∙+X k 2 , k= 1, . . . , n,

)(v j −v i )  SUI(n, α) = 1≤ m i< a j x ≤n  S  ( ( j v j )− − S v i ) (i  1 ) − −S ( ( v j n −v i )  α . WhenX 1 is symmetric, the convergence in distribution ofV n −1 SUI(n, α) requires only the membership ofX 1 in the domain of attraction of normal law, otherwise the existence of E|X 1 | 2+ε for 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.

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UIand

DIstatistics and their asymptotics

All the test statistics studied in this paper may be viewed as discretizations of someHo¨ldernormsorsemi-norms.Thefollowingsubsectioncontainstherelevant background.

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2.1

TheH¨olderianframework

Letρ: [0,1]→R +  of a continuous function Membershipbe a weight function. x: [0,1]→RepscadlreHeo¨nihtH o ρ means roughly that|x(t+h)−x(t)|=o  ρ(|h|)  uniformly intfHeold¨oalicalscsesupsresecahTcealss.ρ(h) =h α , 0< α <1. Le´vy’stheoremonthemodulusofcontinuityoftheBrownianmotionrestrictsthe investigation of invariance principles in this scale to the range 0< α <1/2. The sameresultleadsnaturallytoconsidertheH¨olderspacesbuiltwiththefunctions ρ(h) =h 1/2 log β (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< α≤1/2, and some functionL, positive on[1,∞)and normal- ized slowly varying at infinity, ρ(h) =h α L(1/h),0< h≤1;

ii)θ(t) =t 1/2 ρ(1/t)isC 1 on[1,∞); iii) there is aβ >1/2and somea >0, such thatθ(t) log −β (t)is non decreasing on[a,∞).

Let us recall thatLis normalized slowly varying at infinity if and only if for everyδ >0,t δ L(t) is ultimately increasing andt −δ L(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) :=h α log β (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 byH o ρ caerepso¨dltheH o = H ρ :{x∈C[0,1]; δ li → m 0 ω ρ (x, δ) = 0},

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where ω ρ (x, δ) := sup |x(t ρ ) (t − −s x ) (s)| . s,t∈[0,1], 0<t−s<δ H ρo is a separable Banach space for its native norm

kxk ρ :=|x(0)|+ω ρ (x,1). Under technical assumptions, satisfied by anyρinR, the spaceH ρ o may be endowed with an equivalent normkxk ρ seq built on weighted dyadic increments ofx. Let us denote by D j the set of dyadic numbers in [0,1] of levelj, i.e. D 0 ={0,1},D j =  (2l−1)2 −j ; 1≤l≤2 j−1  , j≥1. We write D (resp. D ∗ ) for the sets of dyadic numbers in [0, (01] (resp.,1]) D := j= ∞ ∪ 0 D j ,D ∗ := D {0}.

Put forr∈D j ,j≥0,

r − :=r−2 −j , r + :=r+ 2 −j .

For any functionx: [0,1]→R, define its Schauder coefficientsλ r (x) by − λ r (x) :=x(r) x(r + +) 2 x(r − ) , r∈D j , j≥1 (2) and in the special casej= 0 byλ 0 (x) :=x(0),λ 1 (x) :=x(1). Whenρbelongs to R, we have on the spaceH o ρ the equivalence of norms (see [11] and [12]) 1 kxk ρ ∼ kxk ρ seq := s j u ≥0 p ρ(2 −j ) r m ∈ a D x j |λ r (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< s t− u s p <1 ρ  (t | −s B(t )1( ) − −B ( ( t s) − | s))  (3)

and DI(ρ) = s j u ≥1 p ρ 1 2( −j ) r m ∈ a D x j  W(r)− 2 1 W(r + )− 2 1 W(r − )  =kBk s ρ eq .(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.

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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 k ρ andk k s ρ eq , 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) :=ρ  h(1−h)  ,0≤h≤1.

Forρ∈ R, define (recalling thatt k =k/n, 0≤k≤n), t j −t i )  UI(n, ρ) = 1≤i m < a j x ≤n  S(j)−S(i % ) (t j − − S t ( i ) n)(

and DI(n, ρ) = 1< m 2 j a ≤ x n ρ 1 2( −j ) r m ∈ a D x j  S(nr)− 2 1 S(nr + )− 2 1 S(nr − )  . To obtain limiting distribution for these statistics we shall work with a stronger null hypothesis, namely

(H 0 0 ):X 1 , . . . , X n are independent identically distributed random vari- ables with mean denotedµ 0 . Theorem 3.Under(H 0 0 ), assume thatρ∈ Rand for everyA >0,

Then

limtP(|X 1 | t> Aθ= 0. t→∞ ( ))

σ −1 n −1/2 UI(n, ρ)−− D −→UI(ρ) n→∞

and D σ −1 n −1/2 DI(n, ρ)−−−→DI(ρ), n→∞ whereσ 2 = var(X 1 )andUI(ρ),DI(ρ)are defined by (3) and (4) respectively.

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(5)

(6)

(7)