DISTRIBUTION APPROXIMATIONS FOR CUSUM AND CUSUMSQ STATISTICS

Transcription

1 58 R. Habb: Dstrbuto approxatos STATISTICS IN TRANSITION-ew seres, Deceber Vol., No., pp DISTRIBUTION APPROXIMATIONS FOR CUSUM AND CUSUMSQ STATISTICS Reza Habb ABSTRACT The cuulatve su (cusu) s a portat statstcs testg for a chage pot. Ths paper s cocered wth the dstrbuto approxatos to the cusu statstc uder the ull ad alteratve hypotheses. We also cosder dstrbuto approxatos for the cuulatve su of squares (cususq) test statstcs. Fally, a dscusso secto s gve. Key words: Beta approxated; Chage pot; Cuulatve su; Cuulatve su of squares; Multvarate oral; Respose surface regresso.. Itroducto It s very portat for ecooc polcy to detfy chage pots ecooc ad facal seres. For exaple, Hsu (979) tested the exstece of chages stoc aret data. K et al. () cosdered the proble of ultple chage pot GARCH odels. Hllebrad ad Schabl () studed chage pot detecto volatlty of Japaese foreg exchage terveto uder GARCH odelg. Haluga et al. (9) detected chages the order of tegrato of US ad UK flato. I face, the portfolo s volatlty ay crease as well as rs preu rses (see e.g. Mosch ad Starca, ). Durg the last four decades, dfferet ethods are eployed for detectg chage pots. Page (954) studed chage pot aalyss the cotext of qualty cotrol. Cheroff ad Zacs (964), usg a quas-baysa approach, odeled the chage pots. Hely (97) derved the axu lelhood estato of chage pot. Worsley (988) costructed cofdece tervals for chage pot the expoetal faly dstrbutos. Habb et al. (5) cosdered the chage pot detecto a geeral class of dstrbutos. A excellet referece chage pot probles s Csorgo ad Horvath (997). The dstrbuto theory used for these tests s typcally asyptotc. So dervg the exact dstrbuto of Departet of Statstcs, Cetral Ba of Ira, Ferdows Ave., Postal Code: 59496, Tehra, Ira, e-al: habb56@gal.co.

2 STATISTICS IN TRANSITION-ew seres, Deceber 59 test statstcs s very portat. Oe of the ost useful approach to detect shft eas of observatos s cuulatve su (cusu) whch s descrbed as follows. I ths paper, we study the fte dstrbutos of cusu test statstc. We also cosder the shft varace case. The cusu statstc gve by = M ax s, s a portat statstc testg for a chage pot, at whch s = e, e = x x, x = x,, =,...,, =. Here, x,..., x s a sequece = of depedet oral rado varables whose eas areθ,,...,, where θ =,,..., θ = θ + δ = +,...,, wth a coo ow varaceσ. It s terestg to test fθ are chaged at uow te pot, that s: H : δ = agast H : δ. The large values of M rejects H. The exact ull dstrbuto of M s coplcated. Note that the ull dstrbuto of M does ot deped oθ. The ltg ull dstrbuto of M s gve by σ Sup B() t where B() t s the stadard Browa brdge o (,) ad the supreu s tae over (,). Coffe ad Specer () proposed a cetral ch-squared approxato for the ull dstrbuto of M +, where M = ax s. + Ther approxato ethod wors well. However, ths ote, the exact ull ad alteratve dstrbuto of M s studed. We also cosder the exact dstrbuto of chage pot estator. The chage pot varace s cosdered. These probles are ot cosdered by Coffe ad Specer (). The proble s to fd the quatle c α such that P ( ) H M > c α = α for fte saple szes. We ote that ad P ( M > c ) = P ( s c, for all =,..., ) =, H H α α α

3 6 R. Habb: Dstrbuto approxatos P ( M < x) = P ( s c, for all =,..., ), H = H ad H. H H α As follows, we show that s= ( s,..., s ) T has ultvarate oral dstrbuto uder the ull ad alteratve hypotheses. That s, M s the axu of absolute of a ultvarate oral dstrbuto ad that c α s a two sded equcoordate α percet quatle of ultvarate oral dstrbuto. The percetage pots of M + s deoted by c + α. The c + α s the oe sded equcoordate quatle of ultvarate oral. Gez (99) proposed soe uercal approaches to calculate the cuulatve probabltes ad equ-coordate quatles of a ultvarate oral dstrbuto wth ay ea vector ad covarace atrx. The fucto pvor vtor pacage of R software perfors ths calculatos. To prove the oralty, let x = ( x,..., x ) T be observato vector. Uder the ull hypothess of o chage pot, x has a varates oral wth ea vector θ j ad covarace atrx σ I where I s the detty atrx ad j s vector of s. The vector of devato fro the eae= ( e,..., e ) T equals to Ax where A s the followg parttoed atrx A= I JM j, where J s atrx of s. The has a varate oral dstrbuto wth ea vector ad the covarace atrx σ AA ( I J). Let L= ( L,..., L ) T s the atrx of vectors L such that L (,...,,,...,) T = at whch the uber of s at L s,...,. Oe ca see that s= Le, ad the s s varate oral wth ea vector ad the covarace atrx σ D, where j Dj = (, j),, j =,...,. Uder the alteratve hypothess, s s aga varate oral wth the covarace atrx σ D but the ea vector sα = ( α,..., α ) T, such that T = σ

4 STATISTICS IN TRANSITION-ew seres, Deceber 6 α δ (, ){ ax(, )},,...,. = = I the ext secto, we copute oe-sded ad two-sded cv s c α + ad c α We copare our crtcal pots (cv s) wth the cv s obtaed by Mote Carlo (MC) sulato. We study the power of test. We also cosder dstrbuto approxatos for the cuulatve su of squares (cususq) test statstc for chage pot detecto varace secto.. Cv s ad power of test Tables ad gve oe-sded ad two-sded cv s. Wthout loss of geeralty, we assue thatσ = ad θ =. It s see our approxated cv s are close to true cv s estated by Mote Carlo study. Ths fact shows our approxato s accurate. The absolute errors of our approxato s easured by e = P (ax s q ), α H α α where q α s the true quatle of ax s obtaed by Mote Carlo experet ad PH (ax s x) s coputed by our oral approxato. Table gves the axu ad eda of errors e α, α =.9(.).999 for each saple szes. Table, we coclude that our approxato wors well. Table 4 gves the oral approxated power of test for soe selected saple szes. We let =, δ =,, σ = adθ =. Table. Coparso of oral ad MC two-sded cv s α Noral cv, = MC cv, = Noral cv, = MC cv, = Noral cv, = MC cv, = Noral cv, = MC cv, = Noral cv, = MC cv, =

5 6 R. Habb: Dstrbuto approxatos Table. Coparso of oral ad MC oe-sded cv s α Noral cv, = MC cv, = Noral cv, = MC cv, = Noral cv, = MC cv, = Noral cv, = MC cv, = Noral cv, = MC cv, = Table. Errors of oral approxated probabltes Maxu error Meda error Table 4. Noral approxated power, α =.5 / θ Rear Ploberger ad Kraer (99) used the cusu statstc replacg e = x x, by regresso resduals. Our results ca be exteded to regresso resduals. Cosder the regresso odel Y = Xβ + ε,

6 STATISTICS IN TRANSITION-ew seres, Deceber 6 at whchy, X p, β ad p ε are observato vector, desg atrx, uow coeffcets vector ad resdual vector, respectvely. Suppose that X s of full ra ad defeq = I Pwhere ( T T P = X X X) X The the vector of estated resduals = (,..., ) T s gve by = Qε ad ε =. T Let s = ε for,...,.the s = L. The j cov( s, s ) = σ = L T QL = L T ( I P) L = (, j) LQL T. j j j j j Followg secto, t ca be show that the ull dstrbuto of s, j =,..., are ultvarate oral wth zero ea ad covarace atrx j = ( σ j ). The slar results ca be exteded to alteratve dstrbutos. The above results relates to ow varace case. I ost practcal stuatos, varace wll have to be estated. Ths case s cosdered secto 4.. Chage Varace I the prevous secto, we studed the chage pot the eas of observato. To detect chage pot varace usg the cusu statstc, let y = x. Here, uder H, x,..., x s a sequece of depedet oral observatos such that Ex ( ) = ad var( x ) = σ = ξ j,,..., = +,...,, whereσ ξ. The eas of y are chaged uder H. The cusu test statstc s M = ax ( y y ). σ The ltg ull dstrbuto of M s gve by sup Bt ( ). The Table 5 gves 5% quatle of M. A respose surface regresso for 5% quatle of M s estated as follows. The adjusted R of regresso s 99.. q =

7 64 R. Habb: Dstrbuto approxatos Table 5. 5% quatle of M q q q q Followg Coffe ad Specer (), we propose a ch-squared approxato the for of a χ ( a > ad df > ) for M +. The oet estates of a ad df are df a λ µ = = µ λ ad df µ ad λ are the ea ad varace of where M + uder H. Table 6 gves, λ, a ad df forσ =. Followg Coffe ad Specer (), we let =,,, 4, 6, 8,. Table 7 gves the eda ad axu of absolute errors. Table 6. Values of µ, λ, a ad df µ λ a df Table 7. Coparsos ch-squared wth MoteCarlo probabltes Maxu of error Meda of errors µ

8 STATISTICS IN TRANSITION-ew seres, Deceber 65 Rear Whe σ s uow t s replaced by ts estate uder the ull hypothess,.e., y ad the test s proposed by X T = ax D, D =, =,...,. X Ths statstc s the cuulatve su of square (cususq) proposed by Icla ad Tao (994). The ltg ull dstrbuto of T s gve by sup Bt ( ). For oderate saple szes, Saso et al. (4) estated a respose surface regresso for 5% quatle of T as follows.5 = q Rear Whe Ex ( ) = µ (ow), we let y = ( x µ ). Whe µ s uow, we ca use M aga, sce eas of x are chaged. We ca also use M wth lettg y = ( x x). Call ths statstc by ˆM. The crtcal values (cv) are gve Table 8 for α =.5. Paraeter µ s chose by coputer. As we expect, the crtcal values of ˆM does ot deped o µ. I Table 9, we copare the power of tests for M ad ˆM cases, σ =, ξ = ad =. It sees test procedure, based o ˆM wors uch better. We are worg o ull ad alteratve dstrbutos of ˆM. For other possbltes, see dscusso secto. Table 8. Crtcal values µ cv of M cv of ˆM

9 66 R. Habb: Dstrbuto approxatos Table 9. Power of tests M ad ˆM Power of M Power of ˆM Dscusso Followg referee coets, ths secto s added to paper to preset a lst of future research topcs. We are worg o the ad they wll be copleted future. 4.. Chage ea: uow varace Coffe ad Specer also developed tests for the uow varace case. I ost practcal stuatos, varace wll have to be estated ad uless saple sze s very large the estate caot safely be treated as a ow value. Ths s why, we cosdered ths part. I guess, I ca approxate the dstrbuto of test statstc by ultvarate t dstrbuto. 4.. Chage ea ad varace Soetes whe varace chages so does the ea. For exaple, face theory says that f a portfolo s volatlty creases the rs preu wll rse ad chage the retur. The cusu test statstcs ad ther dstrbutos are terestg. 4.. Test procedures based o regresso resduals Slar to Rear, t s terestg to detect chage pot varace usg regresso resduals. We are worg o ths topc.

10 STATISTICS IN TRANSITION-ew seres, Deceber Dstrbuto of ˆM The ull ad alteratve dstrbuto of ˆM s terestg. Acowledget I tha the referee for hs valuable coets o y auscrpt. These coets helped e to prove the paper. REFERENCES CHERNOFF, H. ad ZACKS, S. (964). Estatg the curret ea of a oral dstrbuto whch s subjected to chages te. A. Math. Statst. 5, CONNIFFE, D. ad SPENCER, J. E. (). Approxatg the dstrbuto of the axu partal su of oral devates. Joural of Statstcal Plag ad Iferece, 88, 9 7. CSORGO, M., ad HORVATH, L., (997). Lt Theores Chage Pot Aalyss, Wley. UK. GENZ, A. (99). Nuercal coputato of ultvarate oral probabltes. Joural of Coputatoal ad Graphcal Statstcs,, 4 5 HABIBI, R., SADOOGHI-ALVANDI, S. M. ad NEMATOLLAHI, A. R. (5). Chage pot detecto a geeral class of dstrbutos. Coucatos Statstcs, Theory ad Methods 4, HALUNGA, A. G., OSBORN, D. R. ad M. SENSIER (9). Chages the order of tegrato of US ad UK fato. Ecoocs Letters,. HILLEBRAND, E. ad SCHNABL, G. (). The effects of Japaese foreg exchage terveto: GARCH estato ad chage pot detecto. Dscusso Paper No.6. Japa Ba for Iteratoal Cooperato (JBIC). HINKLEY, D., (97). Iferece about the chage-pot a sequece of rado varables. Boetra 57, 7. HSU, D.A., (979). Detectg shfts of paraeter gaa sequeces, wth applcatos to stoc prce ad ar traffc flow aalyss. J. Aer. Statst. Assoc. 74, 4.

11 68 R. Habb: Dstrbuto approxatos INCLAN, C. ad TIAO, G. C. (994). Use of cuulatve sus of squares for retrospectve detecto of chages of varaces. J. Aer. Statst. Assoc. 89, 9 9. KIM S., S. CHO ad S. LEE (), O the cusu test for paraeter chages GARCH(,) odels, Coucatos Statstcs, Theory ad Methods 9, MIKOSCH, T. ad STARICA, C. (). Chage of structure facal te seres, log rage depedece ad the GARCH odeld. Revew of Ecoocs ad Statstcs, forthcog. PAGE, E. S. (954). Cotuous specto schees. Boetra 4, 5. PLOBERGER, W. ad KRAMER, W. (99). The cusu test wth ols resduals. Ecooetrca 6, SANSO, A., ARAGO, V. ad CARRION, J. L. (4). Testg for chages the ucodtoal varace of facal te seres, Uversty of Barceloa Worg Paper. WORSLEY, K. J. (986). Cofdece regos ad test for a chage pot a sequece of expoetal faly rado varables. Boetra 7, 9 4.