The Mann-Whitney U-statistic for α-dependent sequences

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1 The Ma-Whitey U-statistic for α-depedet sequeces Jérôme Dedecker a ad Guillaume Saulière b arxiv: v [math.st] 2 Nov 206 November 22, 206 a Uiversité Paris Descartes, Sorboe Paris Cité, Laboratoire MAP5 (UMR 845). jerome.dedecker@parisdescartes.fr b Uiversité Paris Sud, Istitut de Recherche biomédicale et d Epidémiologie du Sport (IRMES), Istitut Natioal du Sport de l Expertise et de la Performace (INSEP). Guillaume.SAULIERE@isep.fr Abstract We give the asymptotic behavior of the Ma-Whitey U-statistic for two idepedet statioary sequeces. The result applies to a large class of short-rage depedet sequeces, icludig may o-mixig processes i the sese of Roseblatt [5]. We also give some partial results i the lograge depedet case, ad we ivestigate other related questios. Based o the theoretical results, we propose some simple correctios of the usual tests for stochastic domiatio; ext we simulate differet (o-mixig) statioary processes to see that the corrected tests perform well. Itroductio ad mai result Let (X i ) i Z ad (Y i ) i Z be two idepedet statioary sequeces of real-valued radom variables. Let m = m be a sequece of positive itegers such that m as. The Ma- Whitey U-statistic may be defied as U = U,m = m m Xi <Y j. We shall also use the otatios π = P(X < Y ), H Y (t) = P(Y > t) ad G X (t) = P(X < t). It is well kow that, if (X i ) i Z ad (Y i ) i Z are two idepedet sequeces of idepedet ad idetically (iid) radom variables, the U may be used to test π = /2 agaist π /2 (or π > /2, π < /2). Whe π > /2 we shall say that Y stochastically domiates X i a j=

2 weak sese. This property of weak domiatio is true for istace if Y stochastically domiates X, that is if H Y (t) H X (t) for ay real t, with a strict iequality for some t 0. But it also holds i may other situatios, for istace if X ad Y are two Gaussia radom variables with E(X) < E(Y ), whatever the variaces of X ad Y. I this paper, we study the asymptotic behavior of U uder some coditios o the α- depedece coefficiets of the sequeces (X i ) i Z ad (Y i ) i Z. Let us recall the defiitio of these coefficiets. Defiitio.. Let (X i ) i Z be a strictly statioary sequece, ad let F 0 = σ(x k, k 0). For ay o-egative iteger, let where F is the distributio fuctio of X 0. α,x () = sup E ( X x F 0 ) F (x), x R We shall also always require that the sequeces are 2-ergodic i the followig sese. Defiitio.2. A strictly statioary sequece (X i ) i Z is 2-ergodic if, for ay bouded mesurable fuctio f from R 2 to R, ad ay o-egative iteger k lim f(x i, X i+k ) = E(f(X 0, X k )) almost surely. (.) Note that the almost sure limit i (.) always exists, by the ergodic theorem. The property of 2-ergodicity meas oly that this limit is costat. This property is weaker tha usual ergodicity, which would implies the same property for ay bouded fuctio f from R l to R, l N {0}. Our mai result is the followig theorem. Theorem.. Let (X i ) i Z ad (Y i ) i Z be two idepedet statioary 2-ergodic sequeces of real-valued radom variables. Assume that α,x (k) < ad Let (m ) be a sequece of positive itegers such that lim α,y (k) <. (.2) m = c, for some c [0, ). (.3) The the radom variables (U π) coverge i distributio to N (0, V ), with V = Var(H Y (X 0 )) + 2 Cov(H Y (X 0 ), H Y (X k )) + c ( Var(G X (Y 0 )) ) Cov(G X (Y 0 ), G X (Y k )). (.4)

3 Note that the result of Theorem. has bee obtaied by Serflig [6], provided that the more restrictive coditio k 0 ( ) α(k) < ad α(k) = O k log k is satisfied by the two sequeces (X i ) i Z ad (Y i ) i Z, where the α(k) s are the usual α-mixig coefficiets of Roseblatt [5]. (.5) The otio of α-depedece that we cosider i the preset paper is much weaker tha α- mixig i the sese of Roseblatt [5]. Some properties of these α-depedece coefficiets are stated i the book by Rio [4], ad may examples are give i the two papers [7], [8]. Let us recall oe of these examples. Let X i = k 0 a k ε i k (.6) where (a k ) k 0 l 2, ad the sequece (ε i ) i Z is a sequece of idepedet ad idetically distributed real-valued radom variables, with mea zero ad fiite variace. If the cumulative distributio fuctio F of X 0 is γ-hölder for some γ (0, ], the α,x () C ( k a 2 k ) γ γ+2 for some positive costat C (this follows from the computatios i Sectio 6. of [8]). particular, if a k is geometrically decreasig, the so is α,x () (whatever the idex γ). Note that, without extra assumptios o the distributio of ε 0, such liear processes have o reasos to be mixig i the sese of Roseblatt. For istace, it is well kow that the liear process X i = k 0 is ot α-mixig (see for istace []). geometrically decreasig. ε i k 2 k+, where P(ε = /2) = P(ε = /2) = /2, By cotrast, for this particular example, α,x () is To coclude, let us also metio the paper [5] where it is proved that the coefficiets α,x () ca be computed for a large class of dyamical systems o the uit iterval (see Subsectio 6.2 of the preset paper for more details). Remark.. Other types of depedece are cosidered i the papers by Dewa ad Prakasa Rao [] ad Dehlig ad Fried [9]. The paper [] deals with the U-statistics U for two idepedet samples of associated radom variables. The secod paper [9] deals with geeral U-statistics of fuctios of β-mixig sequeces, for either two idepedet samples, or for two I 3

4 adjacet samples from the same time series (see Subsectio 5.2 for a extesio of our results to this more difficult cotext). The class of uctios of β-mixig sequeces cotais also may examples, with a large itersectio with the class of α-depedet sequeces. The advatage of our approach is that it leads to coditios that are optimal i a precise sese (see Sectio 3). O aother had, it seems more difficult to deal with geeral U-statistics i our cotext (some kid of regularity is eeded, for istace we could certaily deal with U-statistics based o fuctios which have bouded variatio i each coordiates). 2 Testig the stochastic domiatio from two idepedet time series The statistic (U /2) caot be used to test H 0 : π = /2 versus oe of the possible alteratives, because uder H 0 the asymptotic distributio of (U /2) depeds of the ukow quatity V defied i (.4). I the ext Propositio, we propose a estimator of V uder some slightly stroger coditios o the depedece coefficiets. Defiitio 2.. Let (X i ) i Z be a strictly statioary sequece of real-valued radom variables, ad let F 0 = σ(x k, k 0). Let f x (z) = z x F (x), where F is the cumulative distributio fuctio of X 0. For ay o-egative iteger, let { } α 2,X () = max α,x (), sup E (f x (X i )f y (X j ) F 0 ) E (f x (X i )f y (X j )). x,y R,j i Propositio 2.. Let (X i ) i Z ad (Y i ) i Z be two idepedet statioary sequeces, ad let (m ) be a sequece of positive ietegers satisfyig (.3). Assume that α 2,X (k) < ad α 2,Y (k) <. (2.) Let ad H m (t) = m m Yj >t ad G (t) = j= Xi <t, ˆγ X (k) = k (H m (X i ) H m )(H m (X i+k ) H m ), ˆγ Y (l) = m l (G (Y j ) Ḡ)(G (Y j+l ) Ḡ), m where H m = H m(x i ) ad mḡ = m j= G (Y j ). Let also (a ) ad (b ) be two sequeces of positive itegers tedig to ifiity as teds to ifiity, such that a = o( ) ad b = 4 j=

5 o( m ). The V = ˆγ X (0) + 2 a coverges i L 2 to the quatity V defied i (.4). ˆγ X (k) + m (ˆγ Y (0) + 2 b l= ˆγ Y (l) Combiig Theorem. ad Propositio 2., we obtai that, uder H 0 : π = /2, if V > 0, the radom variables coverge i distributio to N (0, ). T = (U /2) max{v, 0} Remark 2.. The coditio (2.) is ot restrictive: i all the atural examples we kow, the coefficiets α 2,X (k) behave as α,x (k) (this is the case for istace for the liear process (.6)). Note that, if we assume (2.) istead of (.2), the ergodicity is o loger required i Theorem.. Fially, followig the proof of Theorem 4. page 89 i [2], oe ca also build a cosistet estimator of V uder the coditio α,x (k) < ad k ) (2.2) α,y (k) <, (2.3) k which is ot directly comparable to (2.) (for istace, the first part of (2.) is satisfied if α 2,X (k) = O(k (log k) a ) for some a >, while the first part of (2.3) requires α,x (k) = O(k (log k) a ) for some a > 2). Remark 2.2. The choice of the two sequeces (a ) ad (b ) is a delicate matter. If the coefficiets α 2,X (k) decrease very quickly, the a should icrease very slowly (it suffices to take a 0 i the iid settig). O the cotrary, if α 2,X (k) = O(k (log k) a ) for some a >, the the terms i the covariace series have o reaso to be small, ad oe should take a close to to estimate may of these covariace terms. A data-drive criterio for choosig a ad b is a iterestig (but probably difficult) questio, which is far beyod the scope of the preset paper. However, from a practical poit of view, there is a easy way to proceed: oe ca plot the estimated covariaces ˆγ X (k) s ad choose a (ot too large) i such a way that ˆγ X (0) + 2 a ˆγ X (k) should represet a importat part of the covariace series Var(H Y (X 0 )) + 2 Cov(H Y (X 0 ), H Y (X k )). 5

6 Similarly, oe may choose b from the graph of the ˆγ Y (l) s. As we shall see i the simulatios (Sectio 6), if the decays of the true covariaces γ X (k) ad γ Y (l) are ot too slow, this provides a easy ad reasoable choice for a ad b. 3 Log rage depedece I this sectio, we shall see that the coditio (.2) caot be weakeed for the validity of Theorem.. More precisely, we shall give some examples where (.2) fails, ad the asymptotic distributio of a (U π) (if ay) ca be o Gaussia. Let us begi with the boudary case, where α 2,X (k) = O((k + ) ) ad α 2,Y (k) = O((k + ) ). (3.) I that case, oe ca prove the followig result. Propositio 3.. Let (X i ) i Z ad (Y i ) i Z be two idepedet statioary sequeces. Assume that (3.) is satisfied, ad let (m ) be a sequece a positive itegers satisfyig (.3). The:. For ay r (2, 4), there exists a positive costat C r such that, for ay x > 0, ( ) lim sup P / log (U π) > x C r x. r 2. There are some examples of statioary Markov chais (X i ) i Z ad (Y i ) i Z for which the sequece / log (U π) coverges i distributio to a o-degeerate ormal distributio. We cosider ow the case where α,x (k) < ad α,y (k) = O((k + ) (p ) ), for some p (, 2). (3.2) This meas that (X i ) i Z is short-rage depedet, while (Y i ) i Z is possibly log-rage depedet. I that case, we shall prove the followig result. Propositio 3.2. Let (X i ) i Z ad (Y i ) i Z be two idepedet statioary sequeces. Assume that (3.2) is satisfied for some p (, 2), ad that (X i ) i Z is 2-ergodic. Let (m ) be a sequece a positive itegers such that The: lim m 2(p )/p = c, for some c [0, ). (3.3) 6

7 . There exists a positive costat C p such that, for ay x > 0, lim sup P ( (U π) > x ) C p x. p 2. There are some examples of statioary Markov chais (X i ) i Z ad (Y i ) i Z for which the sequece (U π) coverges i distributio to Z + cs, where Z is idepedet of S, Z is a Gaussia radom variable, ad S is a stable radom variable of order p. Remark 3.. Note that we do ot deal with the case where the two samples are log-rage depedet; i that case the degeerate U-statistic from the Hoeffdig decompositio (see (4.2)) is o loger egligible, which gives rise to other possible limits. See the paper [0] for a treatmet of this difficult questio i the case of fuctios of Gaussia processes. 4 Proofs 4. Proof of Theorem. The mai igrediet of the proof is the followig propositio: Propositio 4.. Let (X i ) i Z ad (Y i ) i Z be two idepedet statioary sequeces. Let also f(x, y) = x<y H Y (x) G X (y) + π. (4.) The ( E m ) 2 m f(x i, Y j ) 6 m j= i=0 m j=0 mi{α,x (i), α,y (j)}. Let us admit this propositio for a while, ad see how it ca be used to prove the theorem. Startig from (4.), we easily see that (U π) = m m j= f(x i, Y j ) + (H Y (X i ) π) + m m (G X (Y j ) π). (4.2) j= This is the well kow Hoeffdig decompositio of U-statistics; the first term o the right-had side of (4.2) is called a degeerate U-statistic, ad is hadled via Propositio 4.. Ideed, it follows easily from this propositio that ( ) 2 E m f(x i, Y j ) 6 m m j= i=0 (i + )α,x (i) + 6 m 7 m (j + )α,y (j). (4.3) j=0

8 Sice (.2) holds, the α,x () = o( ) ad α,y () = o(m ) (because these coefficiets decrease) ad by Cesaro s lemma lim i=0 (i + )α,x (i) = 0 ad lim m m m (j + )α,y (j) = 0. From (4.3) ad Coditio (.3), we ifer that m f(x i, Y j ) coverges to 0 i L 2 as. (4.4) m j= It remais to cotrol the two last terms o the right-had side of (4.2). We first ote that these two terms are idepedet, so it is eough to prove the covergece i distributio for both terms. The fuctios H Y ad G X beig mootoic, the cetral limit theorem for statioary ad 2-ergodic α-depedet sequeces (see for istace [5]) implies that ad (H Y (X i ) π) coverges i distributio to N (0, v ), where j=0 v = Var(H Y (X 0 )) + 2 m (G X (Y j ) π) coverges i distributio to N (0, v 2 ), where m j= v 2 = Var(G X (Y 0 )) + 2 Combiig (4.2), (4.4), (4.5) ad (4.6), the proof of Theorem. is complete. Cov(H Y (X 0 ), H Y (X k )), (4.5) Cov(G X (Y 0 ), G X (Y k )). (4.6) It remais to prove Propositio 4.. We start from the elemetary iequality ( ) 2 E m f(x i, Y j ) 4 m m E(f(X m 2 m 2 i, Y j )f(x k, Y l )). (4.7) j= To cotrol the term E(f(X i, Y j )f(x k, Y l )), we first work coditioally o Y = (Y i ) i Z. Note first that, sice Y is idepedet of X, E(f(X i, Y j ) Y) = 0. Sice f(x, y) 2, j= k=i E(f(X i, Y j )f(x k, Y l ) Y = y) 2 E(f(X k, y l ) F i ), (4.8) where F i = σ(x k, k i). Now, by defiitio of f, E(f(X k, y l ) F i ) E( Xk <y l F i ) G X (y l ) + E(H Y (X k ) F i ) π 2α,X (k i). (4.9) 8 l=j

9 From (4.8) ad (4.9), it follows that E(f(X i, Y j )f(x k, Y l )) E(f(X i, Y j )f(x k, Y l ) Y) 4α,X (k i). (4.0) I the same way E(f(X i, Y j )f(x k, Y l )) 4α,Y (l j). (4.) Propositio 4. follows from (4.7), (4.0) ad (4.). 4.2 Proof of Propositio 2. We proceed i two steps. Step. Let γx(k) = k (H Y (X i ) π)(h Y (X i+k ) π), γy (l) = m l (G X (Y j ) π)(g X (Y i+k ) π), m ad ote that E(γ X(k)) = k Cov(H Y (X 0 ), H Y (X k )) ad E(γ Y (l)) = k Cov(G X(Y 0 ), G X (Y l )). (4.2) I this first step, we shall prove that V = γ X(0) + 2 a j= γ X(k) + m (γ Y (0) + 2 b l= γ Y (l) coverges i L 2 to V. Clearly, by (4.2), it suffices to show that Var(V ) coverges to 0 as. We oly deal with the secod term i the expressio of V, the other oes may be hadled i the same way. Lettig Z i = H Y (X i ) π, ad γ k = Cov(H Y (X 0 ), H Y (X k )) we have ( a ) Var γx(k) = a a k l E((Z 2 i Z i+k γ k )(Z j Z j+l γ l )). (4.3) We must examie several cases. O Γ = {j i + k}, l= j= E((Z i Z i+k γ k )(Z j Z j+l γ l )) 2α 2,X (j i k). ), Cosequetly 2 a E((Z i Z i+k γ k )(Z j Z j+l γ l )) Γ 2 2 a k l l= j= a a l= ( ) α 2,X (j) j 0 Ca2 (4.4) 9

10 for some positive costat C. O Γ 2 = {i j < i + k} {i + k j + l}, E((Z i Z i+k γ k )(Z j Z j+l γ l )) = E((Z i Z i+k γ k )Z j Z j+l ) 2α,X (j + l i k). Cosequetly 2 a a k l l= j= E((Z i Z i+k γ k )(Z j Z j+l γ l )) Γ2 2 2 O Γ 3 = {i j} {i + k > j + l}, a a l= ( ) α,x (j) j 0 Ca2. (4.5) E((Z i Z i+k γ k )(Z j Z j+l γ l )) = E((Z j Z j+l γ l )Z i Z i+k ) 2α,X (i + k j l). Cosequetly 2 a E((Z i Z i+k γ k )(Z j Z j+l γ l )) Γ3 2 2 a k l l= j= a a l= j= ( ) α,x (i) From (4.4), (4.5) ad (4.6), settig Γ = {i j} = Γ Γ 2 Γ 3, oe gets 2 a a k l l= j= i 0 E((Z i Z i+k γ k )(Z j Z j+l γ l )) Γ 3Ca2 Of course, iterchagig i ad j, the same is true o Γ c = {i > j}, ad fially Ca2. (4.6) 2 a a k l l= j= E((Z i Z i+k γ k )(Z j Z j+l γ l )) 6Ca2 (4.7) From (4.3), (4.7) ad the fact that a = o( ), we ifer that a γ X (k) coverges i L2 to k>0 Cov(H Y (X 0 ), H Y (X k )). Sice the other terms i the defiitio of V ca be hadled i the same way, it follows that V coverges i L 2 to V. Step 2. I this secod step, i the expressio of γ X (k), we replace H Y (X i ) by H m (X i ), H Y (X i+k ) by H m (X i+k ), ad π by H m. Similarly, i the expressio of γ Y (l), we replace G X(Y j ) by G (Y j ), G X (Y j+l ) by G (Y j+l ), ad π by Ḡ. If we ca prove that these replacemets i the expressio of V Let are egligible i L 2, the coclusio of Propositio 2. will follow. γ,x (k) = k (H m (X i ) π)(h Y (X i+k ) π). 0

11 The ( γx(k) γ,x (k) 2 H m (X i ) H Y (X i ) 2 = H m (x) H Y (x) 2 2P X (dx)) /2, (4.8) the last equality beig true because (X i ) i Z ad (Y i ) i Z are idepedet. Now, ( ) H m (x) H Y (x) 2 2 Var( Y0 >x) + 2 Cov( Y0 >x, Yk >x) 2 k=0 α,y(k). (4.9) m m From (4.8) ad (4.9), we ifer that there exist a positive costat C such that a γ X(0) γ,x (0) + 2 (γx(k) γ,x (k)) C a. m By coditio (.3) ad the fact that a = o( ), this last quatity teds to 0 as. The Let ow 2 γ 2,X (k) = k (H m (X i ) H m )(H Y (X i+k ) π). γ,x (k) γ 2,X (k) 2 H m H Y (X i ) + 2 H Y (X i ) π. (4.20) 2 Proceedig as i (4.9) for the first term o the right-had side of (4.20), ad otig that H Y (X i ) π k=0 α,x(k), we get that there exist a positive costat C 2 such that γ a,x(0) γ 2,X (0) + 2 (γ,x (k) γ 2,X (k)) Agai, this last quatity teds to 0 as. 2 C 2a + C 2a. m Fially, all the other replacemets may be doe i the same way, ad lead to egligible cotributios i L 2. Hece ad the result follows from Step. lim V V 2 = 0

12 4.3 Proof of Propositio 3. We start from (4.2) agai, but with the ormalizatio / log istead of. Sice (3.) holds, lim sup i=0 (i + )α,x (i) < ad lim sup m From Iequality (4.3) ad Coditio (.3), we ifer that m log m m (j + )α,y (j) <. j=0 m f(x i, Y j ) coverges to 0 i L 2 as. (4.2) j= Let us prove Item. From (4.2) ad (4.2), ( ( ) lim sup P / log (U π) > x lim sup P (H Y (X i ) π) > x ) log /2 ( m + lim sup P (G X (Y i ) π) > x ) log /2. (4.22) m Sice (3.) holds, it follows from Propositio 5. i [6] that: for ay r (2, 4), there exists a positive costat c,r such that, for ay x > 0 ( lim sup P (H Y (X i ) π) > x ) log /2 j= c,r x r. Sice (.3) holds, the same is true for the secod term o the right-had side of (4.22) (for a positive costat c 2,r ), ad Item follows. Let us prove Item 2. We use a result by Gouëzel [2] about itermittet maps. Let θ /2 be the map described i Subsectio 6.2 of the preset paper. As explaied i Subsectio 6.2, there exists a uique absolutely cotiuous θ /2 -ivariat measure ν /2 o [0, ], ad (see the commets o the case α = /2 i Sectio.3 of Gouëzel s paper), o the probability space ([0, ], ν /2 ), ( HY (θ/2) k ν /2 (H Y ) ) coverges i distributio to a o-degeerate ormal distributio provided H Y (0) ν(h Y ) (which is true for istace if Y has distributio ν /2, sice i that case H Y (0) = ad ν(h Y ) = /2). As fully explaied i Subsectio 6.2, there exists a statioary Markov chai (X i ) i Z such 2

13 that the distributio of (X,..., X ) is the same as (θ /2,..., θ /2) o the probability space ([0, ], ν /2 ). Take (Y i ) i Z a idepedet copy of the chai (X i ) i Z. For such chais, both (H Y (X i ) π) ad m (G X (Y j ) π) m j= coverge to the same o-degeerate ormal distributio (ote that π = /2 i that case). Moreover, as recalled i Subsectio 6.2, there exist two positive costats A, B such that This completes the proof of Item 2. A k + α 2,X(k) = α 2,Y (k) B k Proof of Propositio 3.2 We start from (4.2) agai. At the ed of the proof, we shall prove that if (3.2) ad (3.3) are satisfied, the (4.4) holds. Let us ow prove Item. From (4.2) ad (4.4), lim sup P ( (U π) > x ) lim sup P ( (H Y (X i ) π) > x ) /2 + lim sup P ( m m (G X (Y i ) π) > x ) /2. (4.23) Sice the first part of (3.2) is satisfied, it follows from the cetral limit theorem for statioary ad 2-ergodic α-depedet sequeces that ( lim P (H Y (X i ) π) > x ) /2 = P(Z > x/2), (4.24) where Z is a Gaussia radom variable. From Coditio (3.2) ad Remark 3.3 i [6], we ifer that if (3.3) holds, the there exists a positive costat κ p such that, for ay x > 0, ( m lim sup P (G X (Y i ) π) > x ) /2 κ p m x. (4.25) p j= Item follows from (4.23), (4.24) ad (4.25). Let us prove Item 2. j= For the sequece (X i ) i Z, take a sequece of iid radom variables uiformly distributed over [0, ]. To fid the sequece (Y i ) i Z we use agai a result by Gouëzel [2] about itermittet maps. Let θ /p be the map described i Subsectio 6.2 of the preset 3

14 paper. As explaied is Subsectio 6.2, there exists a uique absolutely cotiuous θ /p -ivariat measure ν /p o [0, ], ad (see Theorem.3 i [2]), o the probability space ([0, ], ν /p ), m /p m ( GX (θ/p) k ν /p (G X ) ) coverges i distributio to a o-degeerate stable distributio of order p provided G X (0) ν(g X ) (this coditio is satisfied here, because G X (0) = 0 ad ν /p (G X ) > 0). As fully explaied i Subsectio 6.2, there exists a statioary Markov chai (Y i ) i Z such that the distributio of (Y,..., Y ) is the same as (θ /p,..., θ /p) o the probability space ([0, ], ν /p ). For such a chai, m /p m (G X (Y j ) π) j= coverges i distributio to a o-degeerate stable distributio of order p. Moreover, as recalled i Subsectio 6.2, there exist two positive costats A, B such that This completes the proof of Item 2. A (k + ) p α,y(k) B (k + ) p. It remais to prove (4.4). From Propositio 4. we get that ( ) 2 ( ) E m m f(x i, Y j ) 6 mi{α,x (i), α,y (j)}. (4.26) m m Note that j= i=0 j=0 m 6 mi{α,x (i), α,y (j)} 6α,X (i), (4.27) m j=0 ad that, for ay positive iteger i, sice α,y (j) 0 as j, by Cesaro s lemma. m 6 lim mi{α,x (i), α,y (j)} = 0, (4.28) m m j=0 domiated covergece theorem that ( lim E m provig (4.4). Sice i>0 α,x(i) <, it follows from (4.26), (4.27), (4.28), ad the ) 2 m f(x i, Y j ) = 0, j= 4

15 5 Other related results 5. Testig the stochastic domiatio by a kow distributio We briefly describes here a procedure for testig the (weak) domiatio by a kow distributio µ. Let H(t) = µ((t, )), ad let (X i ) i Z be a statioary 2-ergodic sequece of real-valued radom variables. Let π = E(H(X 0 )), ad defie If H = H(X i ). α,x (k) <, (5.) the the radom variables ( H π) coverge i distributio to N (0, V ), where V = Var(H(X 0 )) + 2 Cov(H(X 0 ), H(X k )). (5.2) Now, we wat to test H 0 : π = /2 (o relatio of weak domiatio betwee µ ad P X ) agaist oe of the possible alteratives: H : π /2 (oe of the two distributios weakly domiates the other oe), H : π > /2 (µ weakly domiates P X ), H : π < /2 (P X weakly domiates µ). Agai, oe caot use directly the statistics ( H /2) to test H 0 : π = /2 versus oe of the possibles alteratives, because uder H 0 the asymptotic distributio of ( H /2) depeds of the ukow quatity V defied i (5.2). To estimate V, we take for some sequece a = o( ), where V, = ˆγ(0) + 2 a ˆγ(k) ˆγ(k) = k (H(X i ) H )(H(X i+k ) H ). Now, as i Propositio 2., if (5.) holds for α 2,X (k) istead of α,x (k), the v coverges i L 2 to v. It follows that, uder H 0 : π = /2, if v > 0, the radom variables ( T H /2) = max{v,, 0} coverges i distributio to N (0, ). 5

16 5.2 The case of two adjacet samples from the same time series Let (X i ) i Z be a statioary sequece of real-valued radom variables. Let m = m be a sequece of positive itegers such that m as. We cosider here the U-statistic U = U,m = m +m j=+ Xi <Y j, where Y j = f(x j ), f beig a mootoic fuctio. This statistic ca be used to test if there is a chage i the time series after the time, ad more precisely if Y + weakly domiates X. Let π = P(X < f(x )), where X is a idepedet copy of X. Let also H Y (t) = P(f(X ) > t) ad G X (t) = P(X < t). Fidig the asymptotic behavior of U is a slightly more difficult problem tha for two idepedet samples, because we caot use the idepedece to cotrol the degeerate U- statistic as i Propositio 4.. However, it ca be doe by usig a more restrictive coefficiet tha α 2,X. Defiitio 5.. Let (X i ) i Z be a strictly statioary sequece of real-valued radom variables, ad let F 0 = σ(x k, k 0). Let P be the law of X 0 ad P (Xi,X j ) be the law of (X i, X j ). Let P Xk F 0 be the coditioal distributio of X k give F 0, ad let P (Xi,X j ) F 0 be the coditioal distributio of (X i, X j ) give F 0. Defie the radom variables b(k) = sup PXk F 0 (f x ), b(i, j) = x R sup P (Xi,X j ) F 0 (f x f y ) P (Xi,X j ) (f x f y ), (x,y) R 2 where the fuctio f x has bee defied i Defiitio 2.. Defie ow the coefficiets { } β,x (k) = E(b(k)) ad β 2,X (k) = max β,x (k), sup E((b(i, j))) i>j k. Of course, by defiitio of the coefficiets, we always have that α 2,X (k) β 2,X (k). coefficiets β 2,X are weaker tha the usual β-mixig coefficiets of (X i ) i Z. May examples of o-mixig process for which β 2,X ca be computed are give i [8]. For all the examples that we metio i the preset paper, the coefficiet β 2,X (k) is of the same order tha the coefficiet α 2,X (k) (see the paper [4] for the example of the itermittet maps described i Subsectio 6.2). Theorem 5.. Let (X i ) i Z be a statioary sequece of real-valued radom variables. Assume that The β 2,X (k) <. (5.3) 6

17 Let (m ) be a sequece of positive itegers satisfyig (.3). distributio to N (0, V ), with The (U π) coverge i V = Var(H Y (X 0 )) + 2 Cov(H Y (X 0 ), H Y (X k )) + c ( Var(G X (f(x 0 ))) + 2 ) Cov(G X (f(x 0 )), G X (f(x k ))). (5.4) For the estimatio of V, oe ca prove the followig aalogue of Propositio 2.. Propositio 5.2. Let (X i ) i Z be a idepedet statioary sequece, ad let Y j = f(x j ) where f is a mootoic fuctio. Let (m ) be a sequece of positive itegers satisfyig (.3). Assume that (5.3) is satisfied. Let ad H m (t) = m +m j=+ Yj >t ad G (t) = ˆγ X (k) = k (H m (X i ) H m )(H m (X i+k ) H m ), ˆγ Y (l) = m Xi <t, +m l j=+ (G (Y j ) Ḡ)(G (Y j+l ) Ḡ), where H m = H m(x i ) ad mḡ = +m j=+ G (Y j ). Let also (a ) ad (b ) be two sequeces of positive itegers tedig to ifiity as teds to ifiity, such that a = o( / log()) ad b = o( m / log(m )). The V = ˆγ X (0) + 2 a coverges i L 2 to the quatity V defied i (5.4). ˆγ X (k) + m (ˆγ Y (0) + 2 b l= ˆγ Y (l) Combiig Theorem 5. ad Propositio 5.2, we obtai that, uder H 0 : π = /2, if V > 0, the radom variables coverge i distributio to N (0, ). T = (U /2) max{v, 0} ) (5.5) The proof of Propositio 5.2 follows the same lies as that of Propositio 2., so we do ot give the details here. The oly mai chage cocers Step 2, Iequalities (4.8) ad (4.9), where we used the idepedece betwee the samples. Here istead, oe ca use the fact that 2 sup m H (x) H(x) C (log(m)) 2 k=0 β,x(k). x R m 2 7

18 This ca be easily proved by followig the proof of Propositio 7. i Rio [4] up to (7.8) ad by usig the computatios i the proof of Theorem 2. i [3]. The proof of Theorem 5. follows also the lie of that of Theorem.; the oly o-trivial chage cocers Propositio 4.. Here, we have to deal with L = +m f(x i, Y j ) m j=+ where f is defied i (4.). We wat to prove that E(L 2 ) teds to 0 as teds to ifiity. As i the proof of Propositio 4., we start from the elemetary iequality E(L 2 ) 4 +m +m E(f(X 2 m 2 i, Y j )f(x k, Y l )). (5.6) j=+ k=i l=j Let (X i ) i Z be a idepedet copy of (X i ) i Z ad Y i = f(x i ), ad write E(f(X i, Y j )f(x k, Y l )) = E(f(X i, Y j )f(x k, Y l )) E(f(X i, Y j )f(x k, Y l )) + E(f(X i, Y j )f(x k, Y l )). (5.7) Now, usig the idepedece, the secod term o the right-had side of (5.7) ca be hadled exactly as i Propositio 4.. It remais to deal with the first term i the right-had side of (5.7). Note that E(f(X i, Y j )f(x k, Y l )) E(f(X i, Yj )f(x k, Yl )) E(f(x i, Y j )f(x k, Y l ) X i = x i, X k = x k ) E(f(x i, Y j )f(x k, Y l )) P (Xi,X k )(dx i, dx k ) (5.8) For ay fixed x, the fuctio y h x (y) = f(x, y) is a bouded variatio fuctio whose variatio is bouded by 2, ad E(h x (X i )) = 0. Proceedig as i Lemma of [7], oe ca the prove that E(f(x i, Y j )f(x k, Y l ) X i = x i, X k = x k ) E(f(x i, Y j )f(x k, Y l )) P (Xi,X k )(dx i, dx k ) 4β 2,X (j k). (5.9) Cosiderig separately the two cases j k > /4 or j k /4, we get the upper boud 2 m 2 +m +m j=+ k=i l=j E(f(X i, Y j )f(x k, Y l ) E(f(X i, Y j )f(x k, Y l )) 4 m + 4 m k> /4 β 2,X (k) (5.0) from which we easily derive that E(L 2 ) teds to 0 as teds to ifiity (sice (.3) ad (5.3) are satisfied). 8

19 6 Simulatios 6. Example : fuctios of a auto-regressive process I this sectio, we first simulate (Z,..., Z ), accordig to the simple AR() equatio Z k+ = 2 (Z k + ε k+ ), where Z is uiformly distributed over [0, ], ad (ε i ) i 2 is a sequece of iid radom variables with distributio B(/2), idepedet of Z. Oe ca check that the trasitio Kerel of the chai (Z i ) i is K(f)(x) = ( ( x ) ( )) x + f + f, ad that the uiform distributio o [0, ] is the uique ivariat distributio by K. Hece, the chai (Z i ) i is strictly statioary. It is well kow that the chai (Z i ) i is ot α-mixig i the sese of Roseblatt [5] (see for istace []). However, oe ca prove that the coefficiets α 2,X of the chai (Z i ) i are such that α 2,Z (k) 2 k (ad the same upper boud is true for β 2,Z (k), see for istace Sectio 6. i [8]). the Let ow Q µ,σ 2 be the iverse of the cumulative distributio fuctio of the law N (µ, σ 2 ). Let X i = Q µ,σ 2 (Z i). The sequece (X i ) i is also a statioary Markov chai (as a ivertible fuctio of a statioary Markov chai), ad oe ca easily check that α 2,X (k) = α 2,Z (k). By costructio, X i is N (µ, σ 2 )- distributed, but the sequece (X i ) i is ot a Gaussia process (otherwise it would be mixig i the sese of Roseblatt). Next, we simulate iid radom variables (Y,..., Y m ) with commo distributio N (µ 2, σ 2 2). Note that, if µ = µ 2, the hypothesis H 0 : π = /2 is satisfied ad the radom variables T defied i (2.2) coverge i distributio to N (0, ). We shall study the behavior of T for σ 2 = 4, σ 2 2 = ad differet choices of µ, µ 2, a ad b. As explaied i Sectio 2, the quatities a ad b may be chose by aalyzig the graph of the estimated auto-covariaces ˆγ X (k) ad ˆγ Y (l). For this examples, these graphs suggest a choice of a = 3 or 4 ad b = 0 (see Figure ). 9

20 We shall compute T for differet choices of (, m) (from (50, 00) to (750, 500) with /m =.5). We estimate the three quatities Var(T ), P(T >.645) ad P( T >.96) via a classical Mote-Carlo procedure, by averagig over N = 2000 idepedet trials. If µ = µ 2, we say that P(T >.645) is level (the level of the oe-sided test) ad P( T >.96) is level 2 (the level of the two-sided test). If a, b are well chose, the estimated variace should be close to ad the estimated levels ad 2 should be close to If µ µ 2, we say that P(T >.645) is power (the power of the oe-sided test) ad P( T >.96) is power 2 (the power of the two-sided test). Case µ = µ 2 = 0 ad a = 0, b = 0 (o correctio): ; m 50 ; ; ; ; ; 500 Estimated variace Estimated level Estimated level Here, sice a = 0, we do ot estimate ay of the covariace terms γ X (l); hece, we compute T as if both series (X i ) i ad (Y j ) j were ucorrelated. The result is that the estimated variace is too large (aroud 2. whatever (, m)) ad so are the estimated levels ad 2 (aroud 0.3 for level ad 0.8 for level 2, whatever (, m)). This meas that the-oe sided or two-sided tests build with T will reject the ull hypothesis too ofte. Case µ = µ 2 = 0 ad a = 3, b = 0: ; m 50 ; ; ; ; ; 500 Estimated variace Estimated level Estimated level As suggested by the graphs of the estimated auto-covariaces (see Figure ), the choice a = 3, b = 0 should give a reasoable estimatio of V. This is ideed the case: the estimated variace is ow aroud., the estimated level is aroud 0.06 ad the estimated level 2 aroud Case µ = µ 2 = 0 ad a = 4, b = 0: 20

21 Figure : Estimatio of the γ X (k) s (top) ad γ Y (l) s (bottom) for Example, with µ = µ 2 = 0, = 300 ad m = 200. ; m 50 ; ; ; ; ; 500 Estimated variace Estimated level Estimated level Here, we see that the choice a = 4, b = 0 works well also, ad seems eve slightly better tha the choice a = 3, b = 0. Case µ = 0, µ 2 = 0.2 ad a = 4, b = 0 : I this example, H 0 is ot satisfied, ad π > /2. However, µ 2 is close to µ, ad sice σ = 2 ad σ 2 =, Y does ot stochastically domiate X (there is oly stochastic domiatio 2

22 i a weak sese). We shall see how the two tests are able to see this weak domiatio, by givig some estimatio of the powers. ; m 50 ; ; ; ; ; 500 Estimated variace Estimated power Estimated power As oe ca see, the powers of the oe sided ad two-sided tests are always greater tha 0.05, as expected. Still as expected, these powers icrease with the size of the samples. For = 750, m = 500, the power of the oe-sided test is aroud 0.47 ad that of the two-sided test is aroud Example 2: Itermittet maps For γ i ]0, [, we cosider the itermittet map θ γ from [0, ] to [0, ], itroduced by Liverai, Saussol ad Vaieti [3]: x( + 2 γ x γ ) if x [0, /2[ θ γ (x) = 2x if x [/2, ]. It follows from [3] that there exists a uique absolutely cotiuous θ γ -ivariat probability measure ν γ, with desity h γ. Let us briefly describe the Markov chai associated with θ γ, ad its properties. Let first K γ be the Perro-Frobeius operator of θ γ with respect to ν γ, defied as follows: for ay fuctios u, v i L 2 ([0, ], ν γ ) (6.) ν γ (u v θ γ ) = ν γ (K γ (u) v). (6.2) The relatio (6.2) states that K γ is the adjoit operator of the isometry U : u u θ γ actig o L 2 ([0, ], ν γ ). It is easy to see that the operator K γ is a trasitio kerel, ad that ν γ is ivariat by K γ. Let ow (ξ i ) i be a statioary Markov chai with ivariat measure ν γ ad trasitio kerel K γ. It is well kow that o the probability space ([0, ], ν γ ), the radom vector (θ γ, θ 2 γ,..., θ γ ) is distributed as (ξ, ξ,..., ξ ). Now it is proved i [5] that there exist two positive costats A, B such that A ( + ) ( γ)/γ α 2,ξ() B ( + ) ( γ)/γ (this is also true for the coefficiet β 2,ξ (), see [4]). Moreover, the chai (ξ i ) i is ot α-mixig i the sese of Roseblatt [5]. 22

23 Let X i (x) = θ i γ (x) ad Y j (y) = θ j γ 2 (y). O the probability space ([0, ] [0, ], ν γ ν γ2 ) the two sequeces (X i ) i ad (Y j ) j are idepedet. Note that, if γ = γ 2, the hypothesis H 0 : π = /2 is satisfied ad the radom variables T defied i (2.2) coverge i distributio to N (0, ). We shall study the behavior of T for differet choices of γ, γ 2, a ad b. As explaied i Sectio 2, the quatities a ad b may be chose by aalyzig the graph of the estimated auto-covariaces ˆγ X (k) ad ˆγ Y (l). As i Subsectio 6., we shall compute T for differet choices of (, m) (from (50, 00) to (750, 500) with /m =.5). We estimate the three quatities Var(T ), P(T >.645) ad P( T >.96) via a classical Mote-Carlo procedure, by averagig over N = 2000 idepedet trials. If γ = γ 2, we say that P(T >.645) is level (the level of the oe-sided test) ad P( T >.96) is level 2 (the level of the two-sided test). If a, b are well chose, the estimated variace should be close to ad the estimated levels ad 2 should be close to If γ γ 2, we say that P(T >.645) is power (the power of the oe-sided test) ad P( T >.96) is power 2 (the power of the two-sided test). Note that we caot simulate exactly (X i ) i ad (Y j ) j, sice the distributio ν /4 is ukow. We start by pickig X ad Y idepedetly over [0, 0.05] (i order to start ear the eutral fixed poit 0) ad the geerate X i = θγ i (X ) ad Y j = θγ j 2 (Y ) for i >, j >. We shall see that the tests are robust to this (slight) lack of statioarity. Case γ = γ 2 = /4 ad a = 0, b = 0 (o correctio): ; m 50 ; ; ; ; ; 500 Estimated variace Estimated level Estimated level Here agai, sice a = b = 0, we compute T as if both series (X i ) i ad (Y j ) j were ucorrelated. It leads to a disastrous result, with a estimated variace aroud 4.8, a estimated level aroud 0.2, ad a estimated level 2 aroud This meas that the-oe sided or twosided tests build with T will reject the ull hypothesis much too ofte. Case γ = γ 2 = /4 ad a = 4, b = 4: 23

24 Figure 2: Estimatio of the γ X (k) s (top) ad γ Y (l) s (bottom) for Example 2, with γ = γ 2 = /4, = 300 ad m = 200. ; m 50 ; ; ; ; ; 500 Estimated variace Estimated level Estimated level As suggested by the graphs of the estimated auto-covariaces (see Figure 2), the choice a = 4, b = 4 should give a reasoable estimatio of V. We see that the procedure is much better tha if o correctio is made: the estimated variace is ow aroud.3, the estimated level is aroud ad the estimated level 2 is aroud Case γ = γ 2 = /4 ad a = 5, b = 5: 24

25 ; m 50 ; ; ; ; ; 500 Estimated variace Estimated level Estimated level For a = 5, b = 5, the results seem slightly better tha for a = 4, b = 4. Eve for large samples, the estimated level 2 is still larger tha the expected level: aroud 0.07 istead of 0.05 (but 0.07 has to be compared to 0.36 which is the estimated level 2 without correctio). Recall that the coefficiets α 2,X () ad α 2,Y () are exactly of order /3, which is quite slow. The procedure ca certaily be improved for larger, m by estimatig more covariace terms (i accordace with Propositio 2.). Case γ = 0.25, γ 2 = 0. ad a = 5, b = 4. We first estimate the quatity π = P(X < Y ) via a realizatio of m m Xi <Y j. j= For = m = 30000, this gives a estimatio aroud for π, meaig that Y weakly domiates X. Let us see how the tests based o T ca see this departure from H 0 : π = /2. Based o the estimated covariaces, a reasoable choice is a = 5, b = 4. This choice leads to the followig estimated powers (based o N = 2000 idepedet trials). ; m 50 ; ; ; ; ; 500 Estimated variace Estimated power Estimated power As oe ca see, the powers of the oe sided ad two-sided tests are always greater tha 0.05, as expected. Still as expected, these powers icrease with the size of the samples. For = 750, m = 500, the power of the oe-sided test is aroud 0.24 ad that of the two-sided test is aroud 0.7. This has to be compared with the estimated levels whe T is cetered o istead of 0.5 (recall that is the previous estimatio of π o very large samples): i that case, for = 750, m = 500 ad a = 5, b = 4, oe gets a estimated level = 0.059, ad a estimated level 2 = 0.068, which are both close to 0.05 as expected. 25

26 6.3 Itermittets maps: the case of two adjacet samples Here, we shall illustrate the result of Subsectio 5.2. As i Subsectio 6.2, we first simulate X accordig to the uiform distributio over [0, 0.05], ad the geerate X i = θγ i (Z ), but ow for the idexes i {2,..., + m}. For j { +,..., + m}, we take Y j = f(x j ) for a give mootoic fuctio f. I our first experimet, we simply take f =Id, so that Y j = X j. Hece the assumptio H 0 : π = /2 is satisfied. For the secod experimet, we take f(x) = x 4/5, so that f(x ) > X almost surely. Hece π > /2. Let T be defied i (5.5). We follow the same scheme as i Subsectio 6.2, ad we estimate the three quatities Var(T ), P(T >.645) ad P( T >.96) via a classical Mote-Carlo procedure, by averagig over N = 2000 idepedet trials. Case γ = /4, f =Id ad a = 0, b = 0 (o correctio): ; m 50 ; ; ; ; ; 500 Estimated variace Estimated level Estimated level Here, as i the correspodig two-sample case (see the table for γ = γ 2 = /4), the ucorrected tests lead to a disastrous result, with a estimated level aroud 0.25 ad a estimated level 2 aroud Case γ = /4, f =Id ad a = 5, b = 5: ; m 50 ; ; ; ; ; 500 Estimated variace Estimated level Estimated level The corrected tests with the choice a = b = 5 give a much better result tha if o correctio is made. Oe ca observe, however, that the estimated levels are still too high ( 8%) for moderately large samples (up to = 450, m = 300). The results seem less good tha i the correspodig two-sample case (see the table for γ = γ 2 = /4), which is of course ot surprisig. Case γ = /4, f(x) = x 4/5 ad a = 5, b = 5: 26

27 ; m 50 ; ; ; ; ; 500 Estimated variace Estimated power Estimated power As oe ca see, the powers of the oe sided ad two-sided tests are always greater tha 0.05, as expected. Still as expected, these powers icrease with the size of the samples. For = 750, m = 500, the power of the oe-sided test is aroud 0.55 ad that of the two-sided test is aroud Ackowledgmets. We thak Herold Dehlig for helpful commets o a prelimiary versio of the paper. Refereces [] R. C. Bradley (986), Basic properties of strog mixig coditios, Depedece i probability ad statistics. A survey of recet results. Oberwolfach, 985. E. Eberlei ad M. S. Taquu editors, Birkäuser, [2] S. Dédé, Théorèmes limites foctioels et estimatio de la desité spectrale pour des suites statioaires. Phd thesis, Uiversité Pierre et Marie Curie, [3] J. Dedecker, A empirical cetral limit theorem for itermittet maps, Probab. Theory Related Fields 48 (200) [4] J. Dedecker, H. Dehlig, ad M. S. Taqqu, Weak covergece of the empirical process of itermittet maps i L 2 uder log-rage depedece, Stochastics ad Dyamics 5 (205) 29 pages. [5] J. Dedecker, S. Gouëzel ad F. Merlevède, Some almost sure results for ubouded fuctios of itermittet maps ad their associated Markov chais, A. Ist. Heri Poicaré Probab. Stat. 46 (200) [6] J. Dedecker ad F. Merlevède, A deviatio boud for α-depedet sequeces with applicatios to itermittet maps, to appear i Stochastics ad Dyamics (206). [7] J. Dedecker ad C. Prieur, New depedece coefficiets. Examples ad applicatios to statistics, Probab. Theory Related Fields 32 (2005) [8] J. Dedecker ad C. Prieur, A empirical cetral limit theorem for depedet sequeces, Stochastic Process. Appl. 7 (2007)

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