A Nonparametric Test of Serial Independence for Time Series and Residuals

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1 Journal of Multvarate Analyss 79, (2001) do jmva , avalable onlne at httpwww.dealbrary.com on A Nonparametrc Test of Seral Independence for Tme Seres and Resduals Klan Ghoud Unverste du Que bec a Tros-Rve res, Tros-Rve res, Que bec, Canada E-mal ghouduqtr.uquebec.ca Reg J. Kulperger Unversty of Western Ontaro, London, Ontaro, Canada E-mal rjkfsher.stats.uwo.ca and Bruno Re mllard Unverste du Que bec a Tros-Rve res, Tros-Rve res, Que bec, Canada E-mal remllaruqtr.uquebec.ca Receved November 17, 1998; publshed onlne June 19, 2001 Ths paper presents nonparametrc tests of ndependence that can be used to test the ndependence of p random varables, seral ndependence for tme seres, or resduals data. These tests are shown to generalze the classcal portmanteau statstcs. Applcatons to both tme seres and regresson resduals are dscussed Academc Press AMS 1990 subject classfcatons 62G10; 60F05; 62E20. Key words and phrases ndependence; seral ndependence; emprcal processes; pseudo-observatons; resduals; weak convergence; Crame rvon Mses statstcs. 1. INTRODUCTION Testng for ndependence s very mportant n statstcal applcatons. These tests arse n many dfferent settngs, n partcular when checkng the dependence of p random varables, one usually carres out an ndependence test. Such a test s also requred when verfyng that consecutve observatons of a tme seres are ndependent. Fnally when checkng the hypotheses of most lnear models, one often needs to test seral ndependence of the error terms X Copyrght 2001 by Academc Press All rghts of reproducton n any form reserved.

2 192 GHOUDI, KULPERGER, AND RE MILLARD The problem of testng the ndependence of p random varables s qute old. The frst tests were based on correlaton measures (Kendall, Spearman). More powerful tests are based on the emprcal dstrbuton functon, and were consdered by Hoeffdng (1948), Blum et al. (1961), and Cotterll and Cso rgo (1982, 1985). In a tme seres settng, one s more nterested n testng seral ndependence, that s one would lke to verfy f consecutve observatons U,..., U + p&1 are ndependent. Ths problem receved consderable attenton n the lterature. It s usually tackled usng Portmanteau statstcs based on the autocorrelaton functons (see Brockwell and Davs, 1991; Kulperger and Lockhart, 1998). Recently, Skaug and Tjo% sthem (1993) proposed a test for seral parwse ndependence based on the emprcal dstrbuton functon. Ther work generalzes Hoeffdng (1948) to seral ndependence. Interestng extensons of ths test can be found n Hong (1998) and Hong (2000). Delgado (1996) used a Blum, Kefer, and Rosenblatt statstc n the seral ndependence context. He showed that the process converges weakly, but that the lmtng process s not very useful when tryng to tabulate crtcal values of test statstcs. Portmanteau type statstcs are also used when checkng seral ndependence of the errors of a lnear model. The Durbn Watson test s the standard dagnostc for seral ndependence of the errors of lnear regresson and s also based on some measure of the correlaton between the errors (see Jobson, 1991). Ths paper develops nonparametrc tests of ndependence and seral ndependence that can be appled n ether of the above three cases. In other words, the tests proposed here apply when testng the ndependence of p random varables or the seral ndependence of tme seres data and resduals. These tests are Crame rvon Mses or KolmogorovSmrnov functonals of some emprcal processes. Ths paper shows that under the ndependence (seral ndependence) hypothess these emprcal processes converge to Gaussan lmts wth qute convenent covarance functons. It s also shown that f the U 's have contnuous dstrbuton functon then the lmtng dstrbutons of the test statstcs do not depend on the underlyng law of the U 's. Ths holds when testng ndependence of p random varables, seral ndependence of tme seres data and seral ndependence of resduals of a classcal lnear regresson. In other cases such as resduals of an autoregressve model, the lmtng dstrbuton depends, n general, on the law of the U 's. The test of the ndependence of p random varables shall be called Settng 1 whle that of the seral ndependence n tme seres data s called Settng 2. Test of seral ndependence for resduals or resdual-lkes observatons s referred to as Settng 3, and usng the termnology of Ghoud and Re mllard (1998a), ths shall also be called the pseudo-observatons stuaton.

3 NONPARAMETRIC TEST OF INDEPENDENCE 193 The dea behnd the constructon n the tme seres settng s qute smlar to that proposed by Skaug and Tjo% sthem (1993) and Delgado (1996). It uses the famous method of tme delay, whch s well known n the chaotc tme seres lterature. The constructon together wth few defntons are gven n the next secton. Secton 3 presents the propertes of the lmtng processes. It s shown, n partcular, that these processes have very convenent covarance functons and that they admt attractve representatons n terms of Brownan drums. Secton 4 defnes test statstcs used n ths work. It s shown that for Crame rvon Mses type statstcs, one obtans a closed form for the asymptotc dstrbuton. The CornshFsher asymptotc expanson (Abramowtz and Stegun, 1964) and Imhof's characterstc functon nverson algorthm (Imhof (1961)) are used to tabulate the dstrbuton and the quantles of these statstcs. Secton 5 dscusses applcatons of these statstcs. A frst subsecton consders the case where U 's are tme seres data. It uses a specal alternatve to provde a power study. The second subsecton provdes a smulaton study comparng the power of the tests dscussed n Secton 4 wth Delgado's (1996) test. Secton 6 s devoted to the proofs of the results stated wthn ths work. 2. DEFINITIONS AND RESULTS Ths secton defnes and states the results for the asymptotc behavor of the emprcal processes used to develop the test procedures. Frst a characterzaton of the ndependence of p random varables s provded. Then the secton gets dvded nto three subsectons. Each subsecton presents one of the three partcular cases descrbed earler. Let U 1,..., U p be p2 random varables. For 1 j p, let K ( j) denotes the margnal dstrbuton functon of U j and for any t=(t (1),..., t ( p) )nr p, let K p (t)=p[u 1 t (1),..., U p t ( p) ] be the jont dstrbuton functon of U 1,..., U p. Now for any A/I p =[1,..., p], and any t # R p, set + A (t)= (&1) A"B K p (t B ) j # A"B K ( j) (t ( j) ), where A denotes the number of elements n A, where by conventon, > < =1 and where (t B ) s the vector wth components (t B ) () = {t(),, # B; # I p "B. Then one can state the followng characterzaton of the ndependence of U 1,..., U p.

4 194 GHOUDI, KULPERGER, AND RE MILLARD Proposton 2.1. A/[1,..., p]. U 1,..., U p are ndependent f and only f + A #0, for all 2.1. Testng the Independence of p Random Varables. Ths secton deals wth the classcal problem of testng the ndependence of p random varables. Ths problem receved consderable attenton n the lterature. Here one s manly nterested n tests based on the emprcal dstrbuton functon. Such tests were consdered by Hoeffdng (1948), Blum et al. (1961), and Cotterll and Cso rgo (1982, 1985). In partcular, f one lets = =(= (1),..., = ( p) ); =1,..., n be a random sample of R p valued random varables, t s desred to test the ndependence of the components = (1),..., = ( p). Blum et al. (1961) proposed the followng emprcal process p ; n, p (t)=- n {K n, p(t)& =1 K () n (t() ) =, where K n, p s the jont emprcal dstrbuton functon and where K () n s the th emprcal margnal dstrbuton functon. It s shown that, except for the case p=2, the asymptotc covarance functon of the process ; n, p s not very convenent. In other words t does not provde a nce way to produce crtcal values of some useful test statstcs. Here alternatve processes are proposed. The dea behnd the ntroducton of these processes comes from the characterzaton of ndependence gven n Proposton 2.1. To ths end, for any A/I p =[1,..., p] and any t=(t (1),..., t ( p) )#R p let R n, A (t)=- n (&1) A"B K n, p (t B ) # A"B K () n (t() ). Usng the multnomal formula (8) n Secton 6, the above reduces to R n, A (t)= 1 - n n =1 (&1) A"B k # B I[= (k) t (k) ] K ( j ) n (t ( j ) ) j # A"B = 1 - n n =1 k # A [I[= (k) t (k) ]&K (k) n (t(k) )], where I denotes the ndcator functon. The asymptotc behavour of these processes s stated next. Theorem 2.1. Let = 1,..., = n be ndependent and dentcally dstrbuted random vectors and suppose that K (k), the margnal dstrbuton functon of = (k) (1) 1, s contnuous for k=1,..., p. Then f = 1,..., =(p) 1 are ndependent, the

5 NONPARAMETRIC TEST OF INDEPENDENCE 195 processes (R n, A ) A/Ip converge n D(R p ) to ndependent mean zero Gaussan processes R A havng covarance functons gven by C A (s, t)=cov[r A (s), R A (t)] = [mn[k () (s () ), K () (t () )]&K () (s () ) K () (t () )]. # A In fact one can easly verfy that the processes R n, A are related to the process ; n, p through the followng representatons and ; n, p (t)= R n, A (t)= B/I p ; B >1 R n, B (t B ) (&1) A"B ; n, p (t B ) # I p "B # A"B K () n (t() ), K () n (t() ). As a corollary one gets the asymptotc behavour of the ; n, p gven n Blum et al. (1961). The nverse s also true, that s the above theorem can be obtaned usng the results of Blum et al. (1961). Ths does not shorten the proof, therefore a drect proof s presented n Secton 6. Test statstcs based on these processes are ntroduced n Secton 4. In partcular, asymptotcs of Crame rvon Mses and KolmogorovSmrnov type statstcs shall be dscussed. The presentaton of these tests s delayed to Secton 4, n order to treat the three settngs together, that s p random varables, seral ndependence n tme seres and seral ndependence wth pseudo-observatons Testng Seral Independence of Tme Seres Data. Here the tme seres framework s presented. Let [U ] 1 be a statonary and ergodc tme seres. Let p2 be a fxed nteger. For any nteger 1n& p+1, set = =(U, U +1,..., U + p&1 ) and for any t=(t (1),..., t ( p) )#R p, defne K n, p (t)= 1 n n& p+1 =1 Let K p be the dstrbuton functon of (= (1) 1 I[= (1) t (1),..., = ( p) t ( p) ]. p),..., =( 1 ), that s, K p (t)=p(= (1) 1 t(1),..., = ( p) 1 t( p) )=P(U 1 t (1),..., U p t ( p) ). As n the prevous secton, for any set A/I p =[1, 2,..., p], let R n, A (t)=- n (&1) A"B K n, p (t B ) # A"B K () n (t() ) (1)

6 196 GHOUDI, KULPERGER, AND RE MILLARD It follows that R n, A =0 f A 1 and that the coeffcent of > # A K () n (t() ) n R n, A s (&1) A &1 ( A &1). One can also rewrte R n, A (t)= 1 - n = 1 - n n& p+1 =1 n& p+1 =1 k # A (&1) A"B k # B [I[= (k) t (k) ]&K (k) n (t(k) )]. I[= (k) t (k) ] K ( j ) n (t( j ) ) j # A"B In ths settng, t s clear that a translate B=A+k of a gven set A generates bascally the same process R n, A. Therefore, wthout loss of generalty, one can restrct the attenton to the processes R n, A where A # A p =[A/I p ;1#A, and A >1]. Let K denotes the common dstrbuton functon of the U 's. The asymptotc behavour of R n, A s stated n the followng theorem. Theorem 2.2. If K s contnuous and f the U 's are ndependent, then the processes (R n, A ) A # Ap converge n D(R p ) to ndependent mean zero Gaussan processes R A havng covarance functons gven by C A (s, t)=cov[r A (s), R A (t)]= [mn[k(s () ), K(t () )]&K(s () ) K(t () )]. # A Skaug and Tjo% sthem (1993) based ther test for seral ndependence on Cramervon Mses functonal of the above process R n, A where A s of the form [1, k] for some 2k p. Ths shows that ther test s only for seral parwse ndependence and not seral ndependence n general. But t was argued that for many alternatves, ths test s more powerful then tests based on the jont dstrbuton of p varables. Delgado (1996) used the followng process p ; n, p (t)=k n, p (t)& =1 K () n (t() ). Once agan ths process s related to our processes va the representatons Next, set ; n, p (t)= R n, A (t)= B/I p ; B >1 ; p (t)= R n, B (t B ) (&1) A"B ; n, p (t B ) B/I p ; B >1 R B (t B ) # I p "B # I p "B # A"B K () n (t() ), K () n (t () ). K(t () ),

7 NONPARAMETRIC TEST OF INDEPENDENCE 197 As a corollary one gets the asymptotc behavor of the ; n, p gven n Lemma 1 of Delgado (1996). The nverse s also true, as llustrated n the remark followng Theorem 2.1. Corollary 2.1. If K s contnuous and f the U 's are ndependent, then the process ; n, p converges n D(R p ) to a contnuous Gaussan process ; p wth covarance functon 1 ; gven by 1 ; (x, y)=k p (x p 7 y p )&K p (x p ) K p ( y p ) p&1 + k=1 K k (x k ) K k (({ p&k y) k )[K p&k ( y p&k 7 ({ k x) p&k ) &K p&k ( y p&k ) K p&k (({ k x) p&k )] p&1 + k=1 K k ( y k ) K k (({ p&k x) k )[K p&k (x p&k 7 ({ k y) p&k ) &K p&k (x p&k ) K p&k (({ k y) p&k )], x, y # R p, where for any d1, K d s the dstrbuton functon of (U 1,..., U d ) and where a7 b s the vector wth components [mn(a (1), b (1) ),..., mn(a (k), b (k) )], a, b # R k and ({ j x) k =(x ( j+1),..., x ( j+k) ). As noted by Delgado the asymptotc covarance functon of ; n, p s not convenent for the tabulaton of crtcal values of Crame rvon Mses functonals. In hs paper Delgado proposed the use of a permutaton method to approxmate these crtcal values. However, one should be very careful when usng smulatons to tabulate crtcal values of tests of ndependence, snce any smulaton procedure uses pseudo-random varables havng some knd of seral dependence. The effect of usng smulaton could be neglgble n relatvely small samples, as ponted by Delgado (1996) and the results n Secton 4. However, theoretcally one should be able to detect ths knd of seral dependence at least for very large samples. Thus usng smulaton wll result n mscalculatng the crtcal values. It s therefore essental to have an alternate method for evaluatng these crtcal values. Ths s exactly the man am of ths paper. As shown n Sectons 3 and 4, the covarance functons of the processes R n, A are qute easy to handle. In fact, explct form of ther egenvalues and egenfunctons wll be gven. Remark 2.1. If the dstrbuton functon K s known, one could use the statstcs R n, A obtaned by replacng K () n by K n (1). For example, ths would be the case f one wshes to verfy f sequence of observatons s a sequence of ndependent and dentcally dstrbuted random varables wth unform margnals. As noted n Secton 6, the processes R n, A and R n, A are asymptotcally equvalent.

8 198 GHOUDI, KULPERGER, AND RE MILLARD 2.3. Testng Seral Independence wth Pseudo-Observatons. Ths secton deals wth the pseudo-observatons stuaton. To be precse, let [X ] 1 be an X valued tme seres. Let H be a functon from X to an nterval T of R and consder the seres [U =H(X )] 1. Suppose that the seres of U 's s statonary and ergodc and that the dstrbuton functon K of U 1 s contnuous. The am s to test f U..., U + p&1 are ndependent. If H s known, ths reduces to the tme seres settng dscussed n the prevous secton. On the other hand, f H s unknown and s estmated by some functon H n and U s estmated by U =H n (X ), then ths s called the pseudo-observatons case. Even though the U 's depend on n, no subscrpt s added for the sake of smplcty. Resduals are just a specal case of pseudo-observatons. Emprcal processes based on pseudo-observatons lke the U 's are consdered by Barbe et al. (1996) and Ghoud and Re mllard (1998a, 1998b). Ths secton redefnes the processes R n, A, A # A p for ths settng and studes ther asymptotc behavor. For each =1,..., n& p+1 set = =(U,..., U + p&1 ) and e =(U,..., U + p&1). Let K n, p be the emprcal dstrbuton functon of the e 's and let (k) K n ; k=1,..., p be the kth emprcal margnal dstrbuton. Then, for ths settng, the process R n, A (t) s gven by and R n, A (t)= 1 - n = 1 - n n& p+1 =1 n& p+1 =1 k # A One also defnes the processes (&1) A"B k # B I[e (k) t (k) ( j ) ] K n (t( j ) ) j # A"B [I[e (k) t (k) (k) ]&K n (t(k) )]. (2) p ; n, p(x)=- n {K n, p(t)& =1 p ; * n, p (x)=- n {K n, p(t)& K =1 () n (t() ) =, K(t () ) =. The process ; * n, p s a specal case of the emprcal process based on pseudoobservatons studed by Ghoud and Remllard (1998b). In partcular ther Theorem 2.1 appled to ths context yelds the asymptotc behavor of ; * n, p. Before the precse statement of ths result, we ntroduce the followng condtons. (R1) There exsts some postve contnuous functon r X R such that nf x # X r(x)>0 and E[r(X)] s fnte. Further let C r be a closed subset

9 NONPARAMETRIC TEST OF INDEPENDENCE 199 of the Banach space of all contnuous functons f from X to R such that & f & r =sup x # X f(x)r(x) s fnte. Assume that there exsts a contnuous verson H n of H n such that - n &H n(x)&h n (x)& r converges n probablty to zero. (R2) Suppose also that for any f # C r wth g= fr, and any contnuous on R, 01, the processes n, j, b g (s, t) = 1 - n n& p+1 =1 _ (g(x + j)) I[= ( j ) t ( j ) +sr(x + j )] I[= (k) t (k) ] k{ j &E {(g(x j)) I[= ( j ) 1 t( j ) +sr(x j )] k{ j I[= (k) 1 t(k) ] =&, are such that for any compact subset C of R and for s # R, sup n, j, b g (s- n, t)& n, j, b g (0, t) (3) t # C converge n probablty to zero. Fnally suppose that f n (t)= n,1,1 (0, t) and H n =- n(h n&h), then ( n, H n ) converges n C(R d )_C r to a process (, H). (R3) The support T of K s an nterval of R, K admts a densty k(}) on T whch s bounded on every compact subset of T and that there exsts a verson of the condtonal dstrbuton of X> k{ j I[= (k) 1 t(k) ] gven = ( j ) =H(X 1 j)=t ( j ), denoted by P j, t, such that for any f = f 0 +%r wth f 0 # C r and % # R, the mappngs t [ + j (t, f )=k j (t ( j ) ) E { f(x j) are contnuous on T. k{ j )= I[= (k) 1 t(k) ] = ( j ) 1 =t( j, Fnally suppose that for any compact subset C of T, lm M M sup P s (r(x)>u) du=0. s # C Wth these notatons Theorem 2.1 of Ghoud and Remllard (1998b) appled to ths context may be restated as follows

10 200 GHOUDI, KULPERGER, AND RE MILLARD Theorem 2.3. If [U ] 1 s a statonary and ergodc tme seres and f U 1,..., U p are ndependent, then f condtons R1R3 are satsfed the process (; * n, p ) converges n D(R p ) to As a corollary one gets. d ; * p (t)=; p (t)& j=1 + j (t, H). Corollary 2.2. Under the condtons of Theorem 2.3 the process R n, A (t) converges n D(R p ) to R A(t)= (&1) A"B ; * p (t B ) # A"B K(t () ). As an example t wll be shown how these results apply to the lnear regresson resduals. Lnear Regresson Resduals. Consder a classcal lnear regresson model, Y=a+b$Z+U, where Y # R, Z # R d and where Z and U are ndependent. To apply the results of ths paper to ths case, note that n ths context X=(Y, Z) and U=H(X)=H(Y, Z)=Y&a&b$Z. One also has H n (x)=h n ( y, z)=y&a n &b$ n z where a n and b n could be taken as the least square estmate of a and b respectvely. r( y, x)=r(z)=1+&z&, C r = [a+b$z; a # R and b # R d ] and H( y, z)=a+b$z where (A, B, ; p ) s the jont weak lmt of (- n(a n &a), - n(b n &b), ; n, p ). The + j (t, H)'s reduce to + j (t, H)=k(t ( j ) ) k{ j K(t (k) )[A+B$E(Z)]. The applcaton of Corollary 2.2 to ths settng yelds the followng proposton. Proposton 2.2. Suppose that the desgn matrx s not sngular, E(&Z& 2 ) s fnte and U admts a support T that s an nterval of R and a contnuous bounded densty on ths support. Then f a n and b n are the least square estmates of a and b, the processes R n, A, defned by (2) for A # A p, converge to the ndependent centered Gaussan processes [R A ] A # Ap gven n Theorem 2.2. Remark 2.2. Note that for general lnear models the lmtng process does not necessarly smplfy to R A, n partcular, for the autoregressve model, Y &+=,(Y &1 &+)+U, the process R A s not equal to R A.In

11 NONPARAMETRIC TEST OF INDEPENDENCE 201 fact, even for the smple case of A=[1, j], t follows from (3.7) n Ghoud and Re mllard (1998b) that R [1, j](t)=r [1, j] (t)+k(t ( j ) ) G(t (1) ), j&2 8, where G(s)=E(= 1 I[= 1 s]) and 8 s a random varable representng the lmt of - n(, n&,). Note also that for ; n, p, even n the lnear regresson settng, the extra term n the lmt does not smplfy. 3. PROPERTIES OF THE LIMITING PROCESSES Ths secton shows that the lmtng processes admt very convenent covarance functons and that they can be represented n term of the process ; p defned n Corollary 2.1 or more approprately n terms of Brownan drums. The covarance functons C A, A # A p are very easy to use. In fact C A s the product of A covarance functons of Brownan brdges. That s, the egenvalues and the egenfunctons of C A are qute easy to obtan and are summarzed n the next proposton. Proposton 3.1. Let k= A. Then the covarance functon C A admts egenvalues and egenfunctons gven by k * j1,..., j k = l=1 respectvely, for ( j 1,..., j k )#N k. k ( j l?) &2 and f j1,..., j k (t 1,..., t k )= l=1 sn( j l?t l ), Next consder the representaton of the process R A. The frst result s straghtforward and s stated n the next proposton. Proposton 3.2. If K s a contnuous dstrbuton functon then R A (t)= (&1) A"B ; p (t B ) # A"B K(t () ). For the second representaton, recall that by Theorem 2.2, the processes (R A ) A # Ap are all ndependent wth covarance functon C A, A # A p. Ths representaton shows that these processes can also be wrtten n terms of

12 202 GHOUDI, KULPERGER, AND RE MILLARD Wener sheets. For, let W be a Wener sheet on [0, 1] p, that s a mean zero contnuous Gaussan process on [0, 1] p wth covarance functon gven by p E[W(s) W(t)]= =1 s () 7 t (), s, t #[0,1] p. Next, defne the Wener drums or Brownan drums D A, A # A p, n terms of W by D A (s)= (&1) A"B W(s B ) k # A"B s (k), A # A p, where s () B =s () f # B and 1 otherwse. To smplfy the statement of the results, assume K s contnuous and consder the followng rescaled verson of the processes R A. D A(s)=R A [K &1 (s (1) ),..., K &1 (s ( p) )], A # A P. The representaton of D A(s) s gven n the followng proposton. Proposton 3.3. The jont law of (D A ) A # Ap s the same as that of (D A) A # Ap. Moreover D A (s)=w(s A )&E[W(s A ) K A ], where K A s the sgma-algebra generated by the values of the Wener sheet on the boundary of [0, 1] A, that s, K A =_[W(t); t () =0 or 1, for some # A]. It s thus justfed to call D p a Wener drum or a Brownan drum, snce t vanshes on the boundary of [0, 1] p. The proof of the proposton requres two steps and s gven n Secton TEST STATISTICS Ths secton studes test statstcs based on the processes consdered earler. In fact for the three settngs of Subsectons 2.1, 2.2, and 2.3 one can ntroduce Crame rvon Mses or KolmogorovSmrnov type statstcs usng the processes R n, A 's. It wll be shown that the lmtng dstrbuton of the Crame rvon Mses statstcs s n general easy to obtan.

13 NONPARAMETRIC TEST OF INDEPENDENCE 203 Frst, defne the Crame rvon Mses statstcs as T n, A = R2 n, A (t) dk n, p(t), (4) for the frst two settngs and T n, A= R2 n, A (t) dk n, p(t), for the pseudo-observatons settng. Next the KolmogorovSmrnov statstc s gven by S n, A =sup t R n, A (t). (5) To test the ndependence or the seral ndependence one can, n partcular, use the statstcs V n, p = A T n, A, V n, p=max A T n, A or W n, p =max A S n, A, where A ranges over all the subset of I P for the test of ndependence of p random varables and over the class A p for the test of seral ndependence n a tme seres. When dealng wth pseudo-observatons one replaces T n, A by T n, A n the above. Next the asymptotc dstrbuton of T n, A s establshed. But frst, set! k = ( 1,..., k )#N k 1? 2k ( 1 }}} k ) 2 Z2 1,..., k, where the Z 1,..., k 's are ndependent N(0, 1) random varables. The asymptotcs of T n, A are gven next Lemma 4.1. Under the condtons of Theorem 2.1 or Theorem 2.2, the statstc T n, A converges n law to! A. The crtcal values of the asymptotc dstrbuton of! k are easy to compute. In fact ths can be acheved by frst computng the cumulants gven by (6), and then applyng the Cornsh Fsher asymptotc expanson, or by nverson of the characterstc functon. Ths nverson s obtaned by the numercal ntegraton method proposed by Imhof (1961), or the mproved verson of ths algorthm ntroduced by Deheuvels and Martynov (1996). The followng provdes the cumulant of order m of the! k where ( } ) denotes the Remann zeta functon. } m = 2m&1 (m&1)!? 2km (2m) k, (6)

14 204 GHOUDI, KULPERGER, AND RE MILLARD TABLE I Crtcal Values of the Dstrbuton of! k Table I provdes an approxmaton of the cut-off values obtaned from the Cornsh Fsher asymptotc expanson wth the frst sx cumulants. A careful examnaton of the asymptotc dstrbutons of the T n, A 's shows that ther expectatons and ther varances dmnsh consderably as the cardnalty of A ncreases. For example, the mean of! k s equal to 16 k and the asymptotc varance s gven by Var(! k )=290 k. So when usng the statstcs V n, p or V n, p, the bggest contrbuton tends to come from the sets A of small szes. To avod ths problem t s more convenent to work wth a standardzed verson of the statstcs. To be specfc let T* n, A =(T n, A & E(! k ))- Var(! k ) and defne V* n, p = A T* n, A and V * n, p =max A T* n, A, where the range of the sets A s that gven n the defnton of V n, p and V n, p. Lemma 4.1 mples that T* n, A converges n dstrbuton to! k *=(! k &E(! k )) - Var(! k ), that V* n, p converges n dstrbuton to V p * and that V * n, p converges n dstrbuton to V p*, where V p *= A!* A and V p*=max A!* A wth

15 NONPARAMETRIC TEST OF INDEPENDENCE 205 TABLE II Dstrbutons of!* k ; k=2, 3, and 4 and of V * 3 and V * 4 for Settngs 1 and 2!* A ndependent for dfferent A's. Table II provdes the dstrbuton functon of the statstcs! k * and V p*, as approxmated by Imhof's technque. Imhof's technque can also be used to tabulate the dstrbuton of V p *. Unfortunately, no closed form can be gven for the KolmogorovSmrnov statstc. However, for contnuous U 's and for both the ndependence of p random varables and the seral ndependence n a tme seres, S n, A and T n, A are dstrbuton free, therefore one can approxmate ther lmtng dstrbutons by smulatng the lmtng Brownan drum process. Note that one can also smulate sequences of ndependent unform (0, 1) random varables. Table III reports the result of 5000 smulaton of pseudo-random sequences generated usng the KISS algorthm, Marsagla and Zaman (1995). It shows that the effect of the dependence contaned n pseudo-random sequence s neglgble. Ths fact s also llustrated n Table IV, where the smulaton procedure used Splus random number generator, for Crame rvon Mses (4) and TABLE III Sample 0.95 Quantles for V * n,2

16 206 GHOUDI, KULPERGER, AND RE MILLARD TABLE IV Sample 0.95 Quantles for Lags k=1,2,3 KolmogorovSmrnov (5) functonals n the tme seres settng, for sets A of the form [1, 1+k] where k=1, 2 and 3. k s often called the lag. These smulatons were done for dfferent sample szes n=100, n=200 and n=400 and wth 2500 Monte Carlo replcates. Observe that the results for the Crame rvon Mses statstcs compares very well wth the asymptotc quantles gven n Table I. One also notces from the smulaton results that the asymptotcs take effect for reasonable sample szes and does so more quckly for the Crame rvon Mses statstc than the KolmogorovSmrnov statstc. One also notces that the crtcal values are ndeed consstent wth R n, A beng dentcally dstrbuted for dfferent A. It s also seen that the Crame rvon Mses statstc s qute consstent across dfferent sample szes n, but that the KolmogorovSmrnov statstc has crtcal values that change a small amount as the sample sze n ncreases. Ths s qute consstent wth the results for KolmogorovSmrnov statstc n the usual settng where a fnte sample correcton s often used; see, for example, Stephens (1986). Moreover, a careful examnaton of these results show that, as expected, the processes for dfferent lags are ndependent and dentcally dstrbuted Gaussan processes. Ths parwse ndependence of the processes occurs for moderate sample sze n (n the range of 100). Note that by Proposton 2.2, all the results stated above for the seral ndependence n the tme seres settng wll apply to the test of seral ndependence for the resduals of lnear regresson models. In fact, when workng wth the resduals of a classcal lnear regresson model, the lmtng processes are exactly the same as those obtaned for the tme seres settng. In partcular Tables I, II and IV apply to these resduals. 5. POWER STUDIES Ths secton presents two power studes. The frst s a comparson wth the classcal portmanteau statstcs. The second dscusses the performance of the tests presented earler compared to that of Delgado (1996). The next

17 NONPARAMETRIC TEST OF INDEPENDENCE 207 subsecton ntroduces Portmanteau processes as a specal case of the processes ntroduced n Secton 2. Then t presents a smulaton study for the power of the statstcs, ntroduced n Secton 4, n detectng a product alternatve Portmanteau Processes. The classcal portmanteau statstc s based on sample autocorrelatons. Its samplng dstrbuton s based on the fact that f the data comes from an..d. sequence, the normalzed sample autocorrelatons are asymptotcally ndependent standard normal random varables. Ths secton shows that by properly choosng the set A, the processes R n, A are n fact emprcal processes based on lags and that the classcal Portmanteau statstcs are functonals of these processes. Theorem 2.2 shows that these processes are asymptotcally ndependent Gaussan processes. Some functonal of these processes, such as Crame rvon Mses and KolmogorovSmrnov statstcs, are dstrbuton free. In ths sense t wll be shown that these emprcal processes play the role of a generalzed Portmanteau process. Ths secton presents some uses for ths process. It also gves a power study, for tests based on these processes, aganst a product process alternatve whch conssts of a 1-dependent sequence wth zero lag 1 covarance. Assume one dsposes of a statonary and ergodc sequence of random varables [U ] 1 wth common dstrbuton K, the dea of the portmanteau statstcs for p=2 s to consder pars of random varables (U, U +k ) at lag k. In fact, to smplfy the presentaton, only the case p=2 wll be dscussed here. Now, set A k =[1, k+1] and consder the process R n, Ak. From the prevous results one concludes that f the U 's are ndependent then the processes R n, Ak converge to ndependent Gaussan processes wth common covarance functon C(s 1, s 2, t 1, t 2 )=[K(s 1 ) 7K(t 1 )&K(s 1 ) K(t 1 )][K(s 2 ) 7 K(t 2 )&K(s 2 ) K(t 2 )]. In the classcal settng of the portmanteau statstcs, the lag k covarance s gven by c(k)=cov(u, U +k ), and ts sample estmate s obtaned va c n (k)= 1 n&k n&k (U &U )(U +k &U ), =1 where U s the sample mean. Frst note that the normalzng factor n&k can be replaced wth n wthout affectng the asymptotc of c n (k). Wth ths modfcaton one easly obtans - nc n (k)= & & R n, Ak (t 1, t 2 ) dt 1 dt 2.

18 208 GHOUDI, KULPERGER, AND RE MILLARD It s well known that under the ndependence hypothess, the c n (k)'s converge to ndependent N(0, { 2 ) dstrbutons, where { 2 = _ & & 2 [K(s 1 ) 7 K(t 1 )&K(s 1 ) K(t 1 )]ds 1 dt 1& =_ 4, where _ 2 =Var(U 1 ) and where the last equalty follows from Hoeffdng's Lemma (see Block and Fang, 1988). As dscussed n the prevous secton one mght consder other more powerful statstcs such as those of the Crame rvon Mses or the KolmogorovSmrnov types dscussed earler Detecton of a Product Process Alternatve. To get an dea about the power of these tests, the followng product process alternatve s consdered. Assume that the data generatng mechansm for ths alternatve consst of the followng product process U =X &1 X, (7) where the X 's are ndependent and dentcally dstrbuted random varables wth mean zero and fnte varance. Ths process s a 1-dependent sequence wth zero lag 1 covarance c(1). The statstcs descrbed earler are used to detect f the sequence of U 's form an..d. sequence. The classcal portmanteau statstc s usually defned as the sum of squares of sample correlatons, (see Brockwell and Davs, 1991). Such a test wll have poor power, n partcular the power does not tend to 1 as the sample sze tends to. That s, the process (7) s a partcularly dffcult alternatve to be detected by a portmanteau statstc. Snce the jont bvarate dstrbutons at varous lags for ths alternatve process are not products of the margnals, one should expect a test based on the process (1) to have some power aganst ths type of product alternatve. To ths end, a smulaton study s consdered next. For the purpose of ths study assume that X s a N(0, 1) random varable. The smulaton TABLE V Product Process Rejecton Rates at 0.05 Level Test, at Varous Lags k=1, 2, 3

19 NONPARAMETRIC TEST OF INDEPENDENCE 209 conssts of generatng sequences of observatons from the alternatve (12) and notng the rejecton rates for each of the Crame rvon Mses and the KolmogorovSmrnov functonals dscussed earler. Ths was done for dfferent sample szes wth 2000 Monte Carlo replcates. The estmated crtcal values from Table IV were used and the study was conducted for three dfferent lags. Table V summarzes the results. Snce the product process s a 1-dependent sequence, the rejecton rates usng the lag 2 or 3, processes should be and are The lag 1 process has good power rejectng the null hypothess of ndependence. The Crame rvon Mses statstc does better than the KolmogorovSmrnov statstc. The Crame rvon Mses statstc has power at sample sze 100 and ncreases to power at sample sze 400. The KolmogorovSmrnov statstc has somewhat smaller power, but stll has good power for the product process alternatve. The rejecton rates for the lag 2 and 3 processes are Thus the tests based on these lags also recognzes that the data s consstent wth a 1-dependent sequence. Agan one should notce that a portmanteau TABLE VI Percentage of Rejecton of U t =bu t&1 +$ t

20 210 GHOUDI, KULPERGER, AND RE MILLARD statstc based on sample correlatons has power of about 0.05 for all sample szes Comparson wth Delgado's Test. In ths secton a smulaton study s carred out to compare the power of the tests statstcs presented here to that of Delgado (1996). For sake of comparson wth Delgado (1996), the smulaton s done usng hs two alternatves. Frst a sequence of observatons followng an AR(1) model U t =bvu t&1 + $ t s consdered, then a second sequence where U t =b$ 2 t&1+$ t s used. In both stuatons $ t ; t1 are ndependent N(0, 1) random varables. For each of these studes, 5000 Monte-Carlo replcates are generated and the percentage of tme the ndependence hypothess s rejected s recorded. The cut-off values for all tests were obtaned by smulatons. Table VI provdes the results for the frst model and Table VII for the second model. It can be seen that there s no clear wnner, that s Delgado statstc performs a lttle better for the AR(1) settng. But V, V, V* orv * are more powerful n detectng the nonlnear alternatve consdered n the second study. TABLE VII Percentage of Rejecton of U t =b$ 2 t&1 +$ t

21 NONPARAMETRIC TEST OF INDEPENDENCE PROOFS Ths secton provdes the proofs of the results stated earler n the manuscrpt. Each subsecton s devoted to one proof. Most of the results stated n ths paper nvolve the covarance functon of the processes R A, whch can be easly manpulated usng the followng extenson of the bnomal formula. Proposton 6.1. (Multnomal Formula) let u, v # R A. Then Let A be a nonempty set and \ ()+\ )+ u v ( j # B j # A"B = (u () +v () ). (8) 6.1. Proof of Proposton 2.1. Snce for any { j, +, j (t)=p(u t (), U j t ( j ) )&K () (t () ) K ( j ) (t ( j ) ), 1, j p, the property +, j #0 yelds the ndependence of U and U j. Next f [U ; # B] are ndependent for all B/I p wth B k, then ths s also true for all sets A/I p wth A =k+1, because + A #0 mples that for all t # R p, 0=+ A (t) =K p (t A )+, B{A # A (&1) A"B K p (t B ) =K p (t A )+ K ( j ) (t ( j ) ) j # A =K p (t A )& K ( j ) (t ( j ) ), j # A provng that [U ; # A] are ndependent. j # A"B (&1) A"B, B{A K ( j ) (t ( j ) ) 6.2. Proof of Theorem 2.1. Frst, defne R n, A(t)= 1 - n n =1 k # A [I[= (k) t (k) ]&K (k) (t (k) )], (9) where K (k) denote the kth margnal dstrbuton of = 1. Observe that + A (t) s the expectaton of R n, A(t). The proof of the Theorem proceeds as follows. Frst t wll be shown that the processes R n, A(t), A/I p converge to the lmtng processes R A 's gven n the statement. Next t wll be establshed that sup t R n, A (t)&r n, A(t) converges n probablty to zero.

22 212 GHOUDI, KULPERGER, AND RE MILLARD For the asymptotc behavour of R n, A, note that R n, A(t)= (&1) A"B _ 1 - n n =1_ j # B j # A"B K ( j ) (t ( j ) ) I[= ( j ) t ( j ) ]& K ( j ) (t ( j ) ) &. j # B For every fxed B, the summand n the above expresson s an emprcal process obtaned from a sequence of ndependent and dentcally dstrbuted random vectors and s therefore tght. Snce there s only a fnte number of B, the sequence of processes (R n, A) s therefore tght. The convergence of the fnte dmensonal dstrbutons to Gaussan lmt s also easy to establsh. To complete the proof one just need to verfy the expresson of the covarance functon. For, let A, B/I p and let t, s # R p usng representaton (9) one gets Cov(R n, A(s), R n, B(t))={0 [K(mn(t ( j ), s ( j ) ))&K(t ( j ) ) K(s ( j ) )] Fnally, observe that j # A f A{B f A=B. R n, A (t)&r n, A(t) = }, B{< _ 1 - n n (&1) B =1 j # A"B k # B [K (k) n (t(k) )&K(t (k) )] [I[= ( j ) t ( j ) ]&K ( j ) (t ( j ) )] }, B{< k # B K (k) n (t(k) )&K (k) (t (k) ) R n, A"B(t A"B ), whch goes to zero n probablty by the GlvenkoCantell lemma and the fact that R n, A"B s tght by the above arguments Proof of Theorem 2.2. Once agan, defne R n, A(t)= 1 - n n =1 k # A [I[= (k) t (k) ]&K(t (k) )]. The proof proceeds exactly lke the one of Theorem 2.1. Frst the asymptotc behavour of R n, A s establshed, then t s shown that R n, A(x) and R n, A are asymptotcally equvalent.

23 NONPARAMETRIC TEST OF INDEPENDENCE 213 For the frst step, let [.] denote the nteger part, set n& p[np] 1 r n (t)= - n [I[= ( j ) p[np]+h t( j ) ]&K(t ( j ) )] j # A h=1 and observe that t s unformly bounded by p- n. Next p R n, A(t)&r n (t)= h=1 p = 1 - n h=1 _ 1 - n [np]&1 =0 j # A (&1) A"B [np]&1 =0 _ [I[= ( j ) p+ht ( j ) ]&K(t ( j ) ) & j # A"B K(t ( j ) ) I[= ( j ) p+h t( j ) ]& K(t ( j ) ) &. j # B j # B In the above representaton for every fxed h and B, the above sum over s a p dmensonal emprcal process of a sequence of..d random vectors and s therefore tght. Snce there s only a fnte number of B's and h's, the sequence (R n, A) s therefore tght. To complete the proof one must consder the fnte dmensonal dstrbuton of R n, A. Frst, note that for all A{B # A p, Cov(R n, A(t), R n, B(s))=0 and Cov(R n, A(t), R n, A(s))= j # A E[I[= (j ) 1 t(j ) ]&K(t (j ) )] _[I[= (j ) 1 s(j ) ]&K(s (j ) )] = [[(mn(t (j ), s (j ) ))&K(t (j ) ) K(s (j ) )]. j # A Moreover, for any fxed t 1,..., t k the central lmt theorem for p dependent sequence (Bllngsley, 1968) apples and yelds the desred Gaussan lmt. For the second step, note that the same argument as that gven n the prevous proof yelds R n, A (t)&r n, A(t), B{< k # B K n,1 (t (k) )&K(t (k) ) _ R n, A"B(t A"B ) + p whch goes to zero n probablty by the GlvenkoCantell lemma and snce R n, A s tght by the frst step. - n

24 214 GHOUDI, KULPERGER, AND RE MILLARD 6.4. Proof of Corollary 2.2. Once more, redefne R n, A(t)= 1 - n n =1 k # A [I[e (k) t (k) ]&K(t (k) )]. To prove ths corollary one needs to show that R n, A converges to the specfed lmt and that the processes R n, A and R n, A are asymptotcally equvalent, that s, sup t R n, A (t)&r n, A(t) converges n probablty to zero as n goes to nfnty. The convergence of R n, A to the specfed lmt s a consequence of Theorem 2.3 and the representaton R n, A(t)= (&1) A"B ; * n, p (t B ) # A"B K(t () ). For the asymptotc equvalence of R n, A and R n, A, the same argument as n the proof of Theorem 2.1 yelds sup t R n, A (t)&r n, A(t), B{< k # B = 1 n B 2, B{< K n, 1(t (k) )&K(t (k) ) R n, A"B(t A"B ) + p k # B ; * n,1 (t (k) ) R n, A"B(t A"B ) + p - n whch goes to zero n probablty, snce by Theorem 2.3, ; * n, p s tght and by the above argument R n, A"B s tght Proof of Proposton 2.2. Frst t shall be shown that the hypotheses of Theorem 2.3 are satsfed. Wth the regresson settng, one easly verfes that Condton (R1) s verfed whenever E(&Z&) s fnte. Condton (R3) holds f U admts a contnuous bounded densty k. To show the frst part of (R2) wrte q, n, j, b g (s, t) & 1 - n [(np]&1 =0 _ (g(x p+q+ j)) _I[= ( j ) p+q t( j ) +sr(x p+q+ j )] k{ j &E {(g(x j)) I[= ( j ) 1 t( j ) +sr(x j )] I[= (k) p+q t(k) ] k{ j - n I[= (k) 1 t(k) ] =&,

25 NONPARAMETRIC TEST OF INDEPENDENCE 215 and observe that n, j, b g (s- n, t)& n, j, b g (0, t) p&1 q=0 That s, (3) wll follow f q, n, j, b g (s- n, t)& q, n, j, b g (0, t). sup q, n, j, b g (s- n, t)& q, n, j, b g (0, t) t # C goes to zero n probablty for each q=0,..., p&1. Snce g(y, Z)=g(Z) s ndependent of U and because R1 and R3 are satsfed the above follows from Lemma 7.2 of Ghoud and Re mllard (1998b) whenever E(&Z& 2 )s fnte. The second part of (R2) s qute easy f a n and b n are the least square estmates of a and b. To complete the proof of ths proposton t suffces to show that the representaton of R A gven n Corollary 2.2 reduces to the R A 's defned n Theorem 2.2. Usng the defnton of ; p one obtans R A(t)= = & (&1) A"B ; p(t B ) (&1) A"B ; p (t B ) p =R A (t)+ _ # A"[j] =R A (t). p (&1) A"B j=1 j=1 # A"B # A"B K(t () ) K(t () ) + j (t B, IH) [A+B$E(Z)]k(t ( j ) ) K(t () ) "[j] # A"B (&1) A &1& B K(t () ) 6.6. Proof of Proposton 3.3. Note that D A (s)&w(s A )=, B{A (&1) A"B W(s B ) > j # A"B s ( j ), whch s K A -measurable. The rest of the proof s acheved n two steps. In the frst Step, one must prove that the process D A (s) s orthogonal to W(t C ), for any t #[0,1] p and for any C/A, C{A. In the second Step one shows that D A and D A have the same covarance functons.

26 216 GHOUDI, KULPERGER, AND RE MILLARD Step 1. Let C be a subset of [1,..., p]. E[D A (s) W(t C )]= (&1) A"B # A"B s () _ (s () 7 t () ) s () # B & C # B"C # C"B t () = A 0 /A & C A 1 /A"C (&1) A & A 0 & A 1 # A"A 0 s () _ s () 7 t () t () # A 0 # C"A 0 = A 0 /A & C _ t () # C"A 0 (&1) C"A 0 A 1 /A"C # A"A 0 s () (&1) A"C & A 1 s () 7 t () # A 0 (s ={ () 7 t () &s () t () ) t (), # A # C"A 0, A/C otherwse. It follows that D A (s)=w(s A )&E[W(s A ) K A ]. Step 2. Ths step s dedcated to the computaton of the covarance between D A and D B. Straghtforward computatons show that Cov[D A (s), D B (t)]= B 0 /B (&1) B"B 0 Cov[D A (s), W(t B0 )]> # B"B0 t (). (10) From Step 1, ths s equal to zero unless A/B, Invertng the roles of A and B n the above mples that Cov[D A (s), D B (t)]=0 f A{B. For A=B, Equaton (10) reduces to Cov[D A (s), D A (t)]=cov[d A (s), W(t A )] = (s () 7 t () &s () t () )=C A (s, t). # A 6.7. Proof of Lemma 4.1. Frst notce that usng Donsker's nvarance prncple (Donsker, 1952, or Bllngsley, 1968), R 2 n, A(t) dk p (t) converges n dstrbuton to R 2 A (t) dk p(t)=! A. Next, usng the fact that f K s

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