Research Article Testing for Change in Mean of Independent Multivariate Observations with Time Varying Covariance

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1 Joural of Probabiliy ad Saisics Volume, Aricle ID , 7 pages doi:.55// Research Aricle Tesig for Chage i Mea of Idepede Mulivariae Observaios wih Time Varyig Covariace Mohamed Bouahar Isiue of Mahemaics of Lumiy, 63 Aveue de Lumiy, 388 Marseille Cedex 9, Frace Correspodece should be addressed o Mohamed Bouahar, mohammed.bouahar@uiv-amu.fr Received 8 Augus ; Acceped 4 November Academic Edior: Ma Lai Tag Copyrigh q Mohamed Bouahar. This is a ope access aricle disribued uder he Creaive Commos Aribuio Licese, which permis uresriced use, disribuio, ad reproducio i ay medium, provided he origial work is properly cied. We cosider a oparameric CUSUM es for chage i he mea of mulivariae ime series wih ime varyig covariace. We prove ha uder he ull, he es saisic has a Kolmogorov limiig disribuio. The asympoic cosisecy of he es agais a large class of aleraives which coais abrup, smooh ad coiuous chages is esablished. We also perform a simulaio sudy o aalyze he size disorio ad he power of he proposed es.. Iroducio I he saisical lieraure here is a vas amou of works o esig for chage i he mea of uivariae ime series. Se ad Srivasava,, Hawkis 3,Worsley 4, ad James e al. 5 cosidered ess for mea shifs of ormal i.i.d. sequeces. Exesio o depede uivariae ime series has bee sudied by may auhors, see Tag ad MacNeill 6, Aoch e al. 7, Shao ad Zhag 8, ad he refereces herei. Sice he paper of Srivasava ad Worsley 9 here are a few works o esig for chage i he mea of mulivariae ime series. I heir paper hey cosidered he likelihood raio ess for chage i he mulivariae i.i.d. ormal mea. Tess for chage i mea wih depede bu saioary error erms have bee cosidered by Horváh e al.. I a more geeral coex of regressio, Qu ad Perro cosidered a model where chages i he covariace marix of he errors occur a he same ime as chages i he regressio coefficies, ad hece he covariace marix of he errors is a sep-fucio of ime. To our kowledge here are o resuls esig for chage i he mea of mulivariae models whe he covariace marix of he errors is ime varyig wih

2 Joural of Probabiliy ad Saisics ukow form. The mai objecive of his paper is o hadle his problem. More precisely we cosider he d-dimesioal model Y μ Γ ε,,...,,. where ε is a i.i.d. sequece of radom vecors o ecessary ormal wih zero mea ad covariace I d, he ideiy marix. The sequece of marices Γ is deermiisic wih ukow form. The ull ad he aleraive hypoheses are as follows: H : μ μ agais. H : There exis / s such ha μ / μ s. I pracice, some paricular models of. have bee cosidered i may areas. For isace, i he uivariae case d, Sarica ad Grager show ha a appropriae model for he logarihm of he absolue reurs of he S&P5 idex is give by. where μ ad Σ are sep fucios, ha is, μ μ j if j,..., j, j [ λ j ], <λ < <λ m <, j if j,..., j, j [ τ j ], <τ < <τ m <,.3 for some iegers m ad m. They also show ha model. ad.3 gives forecass superior o hose based o a saioary GARCH, model. I he mulivariae case d >, Horváh e al. cosidered he model. where μ is subjec o chage ad Σ Σis cosa; hey applied such model o emperaure daa o provide evidece for he global warmig heory. For fiacial daa, i is well kow ha asses reurs have a ime varyig covariace. Therefore, for example, i porfolio maageme, our es ca be used o idicae if he mea of oe or more asses reurs are subjec o chage. If so, he akig io accou such a chage is very useful i compuig he porfolio risk measures such as he value a Risk VaR or he expeced shorfall ES see Arzer e al. 3 ad Holo 4 for more deails.. The Tes Saisic ad he Assumpios I order o cosruc he es saisic le B τ Γ τ ( Y Y τ,,. where Γ is a square roo of Σ, ha is, Σ Γ Γ, ( Y Y (, Y Y Y Y.

3 Joural of Probabiliy ad Saisics 3 are he empirical covariace ad mea of he sample Y,...,Y, respecively, x is he ieger par of x, adx is he raspose of X. The CUSUM es saisic we will cosider is give by B sup B τ, τ,.3 where X max X i if X (X,...,X d..4 i d Assumpio. The sequece of marices Γ is bouded ad saisfies Γ Γ Σ > as..5 Assumpio. There exiss δ> such ha E ε δ <, where X deoes he Euclidia orm of X. 3. Limiig Disribuio of B uder he Null Theorem 3.. Suppose ha Assumpios ad hold. The, uder H, B L B sup B τ, τ, 3. L deoes he covergece i disribuio ad B τ is a mulivariae Browia Bridge wih idepede compoes. Moreover, he cumulaive disribuio fucio of B is give by ( F B z k exp { k z } d. 3. k To prove Theorem 3. we will esablish firs a fucioal ceral limi heorem for radom sequeces wih ime varyig covariace. Such a heorem is of idepede ieres. Le D D, be he space of radom fucios ha are righ-coiuous ad have lef limis, edowed wih he Skorohod opology. For a give d N, led d D d, be he produc space. The weak covergece of a sequece of radom elemes X i D d o a radom eleme X i D d will be deoed by X X. For wo radom vecors X ad Y, X law Y meas ha X has he same disribuio as Y.

4 4 Joural of Probabiliy ad Saisics Cosider a i.i.d. sequece ε of radom vecors such ha E ε ad var ε I d. Le Γ saisfy.5 ad se W τ Γ τ Γ ε, τ,, 3.3 where Γ is a square roo of Σ, ha is, Σ ΓΓ. May fucioal ceral limi heorems were esablished for covariace saioary radom sequeces, see Bouahar 5 ad he refereces herei. Noe ha he sequece Γ ε we cosider here is o covariace saioary. There are wo sufficie codiios o prove ha W W see Billigsley 6 ad Iglehar 7, amely, i he fiie-dimesioal disribuios of W coverge o he fiie-dimesioal disribuios of W, ii W i is igh for all i d,ifw W,...,W d. Theorem 3.. Assume ha ε is a i.i.d. sequece of radom vecors such ha E ε, var I d ad ha Assumpios ad hold. The W W, 3.4 where W is a sadard mulivariae Browia moio. Proof. Wrie F Γ Γ, F i, j he i, j -h ery of he marix F,ε ε,...,ε d. To prove ha he fiie-dimesioal disribuios of W coverge o hose of W i is sufficie o show ha for all ieger r, for all τ < <τ r, ad for all α i R d, i r, Z r α i W τ i i L Z r α i W τ i. i 3.5 Deoe by Φ Z u E exp iuz he characerisic fucio of Z ad by C a geeric posiive cosa, o ecessarily he same a each occurrece. We have [ ( iu r Φ Z u E exp α k k E exp iu r r k ] τ k F ε α j j k τ k τ k F ε, τ 3.6 r Φ k, u, k

5 Joural of Probabiliy ad Saisics 5 where Φ k, u E exp iu r α j j k τ k τ k F ε. 3.7 Sice ε is a i.i.d. sequece of radom vecors we have ( law ( ε τk,...,ε τk ε,...,ε τk τ k. 3.8 Hece Φ k, u E exp iu r α τ k τ k j j k F τk ε. 3.9 Le I A if he argume A is rue ad oherwise, k τ k τ k, ξ,i r α j j k k F τk iε i, M,k ξ,i, i 3. F, σ ε,...,ε, k he filraio spaed by ε,...,ε. The M,i, F,i, i k, is a zero-mea square-iegrable marigale array wih differeces ξ,i. Observe ha k i ( E ξ,i F,i r α k j F τk if τ k i j k i σ k τ r k τ k α j j k r α j j k as. 3. Now usig Assumpio we obai ha Γ <Kuiformly o for some posiive cosa K, hece Assumpio implies ha for all ε>, k i E (ξ,i I ξ,i >ε F,i k E ( ξ ε δ,i δ F,i i Ck δ/ as, 3.

6 6 Joural of Probabiliy ad Saisics where E ( ε δ C K Γ r α j ε δ j k δ, 3.3 cosequely see Hall ad Heyde 8, Theorem 3. M,k L Zk, 3.4 where Z k is a ormal radom variable wih zero mea ad variace σ k. Therefore Φ k, u E ( exp ium,k exp ( σk u as, 3.5 which ogeher wih 3.6 implies ha Φ Z u exp ( r σ k u Φ Z u as, 3.6 k he las equaliy holds sice, wih τ, r r τ k τ k α j k j k α i α j mi ( τ i,τ j. i, j r 3.7 For i d, fixed, i order o obai he ighess of W i followig iequaliy Billigsley 6, Theorem 5.6 : i suffices o show he ( W i E τ W i τ γ i W τ W i τ γ F τ F τ α, 3.8 for some γ>, α>, where F is a odecreasig coiuous fucio o, ad <τ < τ<τ <. We have ( W i E τ W i τ i W τ W i τ T T, 3.9

7 Joural of Probabiliy ad Saisics 7 where T E T E τ τ j τ τ j d ( j F i, j ε, d 3. ( j F i, j ε. Now observe ha T cov d ( j F i, j ε,s j, j d ( j F s i, j ε s τ τ j d ( ( F i, j 3. C τ τ for some cosa C>. Likewise T C τ τ. Sice τ τ τ τ τ τ /, he iequaliy 3.8 holds wih γ α, F C/. I order o prove Theorem 3. we eed also he followig lemma. Lemma 3.3. Assume ha Y is give by., where ε is a i.i.d sequece of radom vecors such ha E ε, var I d ad ha Γ saisfies.5. The uder he ull H, he empirical covariace of Y saisfies a.s., 3. where a.s. deoes he almos sure covergece. Proof. Le W Γ ε, F σ ε,...,ε ad for i, j fixed, i d, j d, e W i W j E W i W j F. The e is a marigale differece sequece wih respec o F. Sice e W i W j d k Γ i, k Γ j, k ad he marix Γ is bouded, i follows ha ( E e δ / ( W i C E W j δ / ( W i E δ /E ( W j δ / 3.3 C,

8 8 Joural of Probabiliy ad Saisics sice by usig Assumpios ad we ge ( W i E δ ( d ( Γ E i, k ε k δ / δ δ k 3.4 C. Therefore, Theorem 5 of Chow 9 implies ha e o almos surely 3.5 or W i W j ( Γ Γ ( i, j o almos surely, 3.6 where Γ Γ i, j deoes he i, j -h ery of he marix Γ Γ. Hece W W a.s Lemma of Lai ad Wei, page 57, implies ha wih probabiliy oe Γ i, k ε k ( o Γ i, k O o O i d, 3.8 or ( Γ i, k ε k o O almos surely, 3.9 which implies ha W ( d k Γ,k ε k,..., Γ d, k ε k a.s.. 3.3

9 Joural of Probabiliy ad Saisics 9 Noe ha Y μ W, hece combiig 3.7 ad 3.3 we obai Y Y YY μ W μ W μμ W W YY a.s Proof of Theorem 3.. Uder he ull H we have Y μ Γ ε, hus recallig 3.3 we ca wrie B τ Γ τ ( Y Y Γ Γ τ [ (Y Γ μ ( ] Y μ ( Γ Γ W τ τ W. 3.3 Therefore he resul 3. holds by applyig Theorem 3., Lemma 3.3, ad he coiuous mappig heorem. 4. Cosisecy of B We assume ha uder he aleraive H followig. he meas μ are bouded ad saisfy he Assumpio H. There exiss a fucio U from, io R d such ha τ,, τ μ U τ as. 4. Assumpio H. There exiss τ, such ha U τ U τ τ U /. 4. Assumpio H3. There exiss μ such ha ( μ μ ( μ μ μ as, 4.3 where μ / μ.

10 Joural of Probabiliy ad Saisics Theorem 4.. Suppose ha Assumpios ad hold. If Y is give by. ad he meas μ saisfy he Assumpios H, H, ad H3, he he es based o B is cosise agais H, ha is, B P, 4.4 where P deoes he covergece i probabiliy. Proof. We have B τ B τ B τ, 4.5 where B τ τ Γ ( W W, W Γ ε, W W, B τ τ Γ ( μ μ. 4.6 Sraighforward compuaio leads o a.s.. μ 4.7 Therefore B τ L Γ ΓB τ, 4.8 where Γ is a square roo of Σ,hais,Σ Γ Γ,ad B τ a.s. Γ U τ. 4.9 Hece B τ P, 4. which implies ha B P. 4.

11 Joural of Probabiliy ad Saisics 4.. Cosisecy of B agais Abrup Chage Wihou loss of geeraliy we assume ha uder he aleraive hypohesis H here is a sigle break dae, ha is, Y is give by. where μ μ μ if τ if τ for some τ, ad μ / μ. 4. Corollary 4.. Suppose ha Assumpios ad hold. If Y is give by. ad he meas μ saisfy 4., he he es based o B is cosise agais H. Proof. I is easy o show ha are saisfied wih 4.3 τ τ ( μ μ if τ τ U τ τ τ ( μ μ if τ>τ, ( (. μ μ μ μ μ Noe ha 4. is saisfied for all <τ <τ sice μ / μ. Remark 4.3. The resul of Corollary 4. remais valid if uder he aleraive hypohesis here are muliple breaks i he mea. 4.. Cosisecy of B agais Smooh Chage I his subsecio we assume ha he break i he mea does o happe suddely bu he rasiio from oe value o aoher is coiuous wih slow variaio. A well-kow dyamic is he smooh hreshold model see Teräsvira, i which he mea μ is ime varyig as follows μ μ ( ( μ μ F,τ,γ,, μ / μ, 4.4 where F x, τ,γ is a he smooh rasiio fucio assumed o be coiuous from, io,, μ ad μ are he values of he mea i he wo exreme regimes, ha is, whe F ad F. The slope parameer γ idicaes how rapid he rasiio bewee wo exreme regimes is. The parameer τ is he locaio parameer. Two choices for he fucio F are frequely evoked, he logisic fucio give by F L ( x, τ,γ [ exp ( γ x τ ], 4.5 ad he expoeial oe ( F e x, τ,γ ( exp γ x τ. 4.6

12 Joural of Probabiliy ad Saisics For example, for he logisic fucio wih γ>, he exreme regimes are obaied as follows: i if x adγ large he F adhusμ μ, ii if x adγ large he F adhusμ μ. This meas ha a he begiig of he sample μ is close o μ ad he moves owards μ ad becomes close o i a he ed of he sample. Corollary 4.4. Suppose ha Assumpios ad hold. If Y is give by. ad he meas μ saisfy 4.4, he he es based o B is cosise agais H. Proof. The assumpios 4. ad 4.3 are saisfied wih U τ ( μ μ T τ, 4.7 where T τ τ ( ( μ μ μ μ μ F ( x, τ,γ dx τ F ( x, τ,γ dx, F ( x, τ,γ ( dx F ( x, τ,γ dx. 4.8 Sice μ μ /, o prove 4.,isuffices o show ha here exiss τ such T τ /. Assume ha T τ for all τ, he dt τ dτ F ( τ, τ,γ F ( x, τ,γ dx τ,, 4.9 which implies ha F τ, τ,γ F x, τ,γ dx C for all τ, or μ μ ( μ μ C μ, 4. ad his coradics he aleraive hypohesis H Cosisecy of B agais Coiuous Chage I his subsecio we will examie he behaviour B uder he aleraive where he mea μ varies a each ime, ad hece ca ake a ifiie umber of values. As a example we cosider a polyomial evoluio for μ : μ ( ( ( P,...,P d, P j x p j k α j,k x k, j d. 4. Corollary 4.5. Suppose ha Assumpios ad hold. If Y is give by. ad he meas μ saisfy 4., he he es based o B is cosise agais H.

13 Joural of Probabiliy ad Saisics 3 Proof. The assumpios H H3 are saisfied wih U τ μ ( p k α,k k p ( i p j i, j k l ( τ k τ,..., l k α i,kα j,l p d α d,k k ( pi k k ( τ k τ, ( pj α i,k α j,k. k k k 4. Noe ha 4. is saisfied for all < τ k, k p i such ha α i,k /. <, provided ha here exis i, i d ad 5. Fiie Sample Performace All models are drive from a i.i.d. sequeces ε ε,...,ε d, where each ε j, j d, has a 3 disribuio, a Sude disribuio wih 3 degrees of freedom, ad ε i ad ε j are idepede for all i / j. Simulaios were performed usig he sofware R. We carry ou a experime of samples for seve models ad we use hree differe sample sizes, 3,, ad 5. The empirical sizes ad powers are calculaed a he omial levels α %, 5%, ad %, i boh cases. 5.. Sudy of he Size I order o evaluae he size disorio of he es saisic B we cosider wo bivariae models Y μ Γ ε wih he followig. Model cosa covariace. ( ( μ, Γ. 5. Model ime varyig covariace. ( ( si ω μ, Γ, cos ω ω π From Table, we observe ha for small sample size 3 he es saisic B has a severe size disorio. Bu as he sample size icreases, he disorio decreases. The empirical size becomes closer o bu always lower ha he omial level. The disorio i he osaioary Model ime varyig covariace is a somewha greaer ha he oe i he saioary Model cosa covariace. However he es seems o be coservaive i boh cases.

14 4 Joural of Probabiliy ad Saisics Table : Empirical sizes %. α 3 5 %..3.4 Model 5%..9 4 % %...3 Model 5% % Sudy of he Power I order o see he power of he es saisic B we cosider five bivariae models Y μ Γ ε wih he followig Abrup Chage i he Mea Model 3 cosa covariace. μ [ ] if if [ ], ( Γ. 5.3 Model 4. I his model he mea ad he covariace are subjec o a abrup chage a he same ime: μ [ ] if if [ ] Γ [ ] if if [ ]. 5.4 Model 5. The mea is subjec o a abrup chage ad he covariace is ime varyig see Figure : μ [ ] if if [ ], ( si ω Γ, ω π cos ω

15 Joural of Probabiliy ad Saisics 5 The hree kids of chage i he mea.8.6 mu Abrup chage Smooh logisic chage Polyomial chage Time Figure : The hree kids of chage i he mea Smooh Chage i he Mea Model 6. We cosider a logisic smooh rasiio for he mea ad a ime varyig covariace see Figure : μ μ ( μ μ ( exp 3 / /,, ( ( ( 5.6 si ω μ, μ, Γ, ω π4 cos ω Coiuous Chage i he Mea Model 7. I his model he mea is a polyomial of order wo ad he covariace marix is also ime varyig as i he precedig Models 5 ad 6 see Figure : μ ( ( ( si ω, cos ω ω π From Table, we observe ha for small sample size 3, he es saisic B has a low power. However, for he five models, he power becomes good as he sample size

16 6 Joural of Probabiliy ad Saisics Table : Empirical powers %. α 3 5 % Model 3 5% % % Model 4 5% % % Model 5 5% % % Model 6 5% % % Model 7 5% % icreases. The powers i osaioary models are always smaller ha hose of saioary models. This is o surprisig sice, from Table, he es saisic B is more coservaive i osaioary models. We observe also ha he power is almos he same i abrup ad logisic smooh chages compare Models 5 ad 6. However, for he polyomial chage Model 7 he power is lower ha hose of Models 5 ad 6. To explai his uderperformace we ca see, i Figure, ha i he polyomial chage, he ime iervals where he mea says ear he exreme values ad are very shor compared o hose i abrup ad smooh chages. We have simulaed oher coiuous chages, liear ad cubic polyomial, rigoomeric, ad may oher fucios. Like i Model 7, chages are hardly deeced for small values of, ad he es B has a good performace oly i large samples. Ackowledgme The auhor would like o hak he aoymous referees for heir cosrucive commes. Refereces A. Se ad M. S. Srivasava, O ess for deecig chage i mea, The Aals of Saisics, vol. 3, pp. 98 8, 975. A. Se ad M. S. Srivasava, Some oe-sided ess for chage i level, Techomerics, vol. 7, pp. 6 64, D. M. Hawkis, Tesig a sequece of observaios for a shif i locaio, Joural of he America Saisical Associaio, vol. 7, o. 357, pp. 8 86, K. J. Worsley, O he likelihood raio es for a shif i locaio of ormal populaios, Joural of he America Saisical Associaio, vol. 74, o. 366, pp , B. James, K. L. James, ad D. Siegmud, Tess for a chage-poi, Biomerika, vol. 74, o., pp. 7 83, S. M. Tag ad I. B. MacNeill, The effec of serial correlaio o ess for parameer chage a ukow ime, The Aals of Saisics, vol., o., pp , J. Aoch, M. Hušková, ad Z. Prášková, Effec of depedece o saisics for deermiaio of chage, Joural of Saisical Plaig ad Iferece, vol. 6, o., pp. 9 3, 997.

17 Joural of Probabiliy ad Saisics 7 8 X. Shao ad X. Zhag, Tesig for chage pois i ime series, Joural of he America Saisical Associaio, vol. 5, o. 49, pp. 8 4,. 9 M. S. Srivasava ad K. J. Worsley, Likelihood raio ess for a chage i he mulivariae ormal mea, Joural of he America Saisical Associaio, vol. 8, o. 393, pp. 99 4, 986. L. Horváh, P. Kokoszka, ad J. Seiebach, Tesig for chages i mulivariae depede observaios wih a applicaio o emperaure chages, Joural of Mulivariae Aalysis, vol. 68, o., pp. 96 9, 999. Z. Qu ad P. Perro, Esimaig ad esig srucural chages i mulivariae regressios, Ecoomerica, vol. 75, o., pp , 7. C. Sarica ad C. W. J. Grager, Nosaioariies i sock reurs, The Review of Ecoomics ad Saisics, vol. 87, o. 3, pp. 53 5, 5. 3 P. Arzer, F. Delbae, J.-M. Eber, ad D. Heah, Cohere measures of risk, Mahemaical Fiace, vol. 9, o. 3, pp. 3 8, A. G. Holo, Value-a-Risk: Theory ad Pracice, Academic Press, 3. 5 M. Bouahar, Ideificaio of persise cycles i o-gaussia log-memory ime series, Joural of Time Series Aalysis, vol. 9, o. 4, pp , 8. 6 P. Billigsley, Covergece of Probabiliy Measures, Wiley, New York, NY, USA, D. L. Iglehar, Weak covergece of probabiliy measures o produc spaces wih applicaio o sums of radom vecors, Techical Repor, Saford Uiversiy, Deparme of Saisics, Saford, Calif, USA, P. Hall ad C. C. Heyde, Marigale Limi Theory ad Is Applicaio, Academic Press, New York, NY, USA, Y. S. Chow, Local covergece of marigales ad he law of large umbers, Aals of Mahemaical Saisics, vol. 36, o., pp , 965. T. L. Lai ad C. Z. Wei, Leas squares esimaes i sochasic regressio models wih applicaios o ideificaio ad corol of dyamic sysems, The Aals of Saisics, vol., o., pp , 98. T. Teräsvira, Specificaio, esimaio, ad evaluaio of smooh rasiio auoregressive models, Joural of America Saisical Associaio, vol. 89, pp. 8 8, 994.

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