XAP x i LUC LUdNG KIEM BTNH DIEM CHUYEN CUA MO HINH HOI QUY TUYEN TINH HAI PHA
Nguyen Thi QtQ'i**
Khoa Todn Email: [email protected] Ngdy nhgn bdi: 1/2/2018
Ngdy PB ddnh gid: 21/3/2018 Ngdy duyit ddng: 23/3/2018 TOM T A T
Lyc lupng kiim dinh diim chuyin cua md hinh hdi quy tuyen tinh hai pha, vdi diem ehuyen da biit, phu thupc vdo dp rap ciia ham md binh. Trong trudng hpp chua biit diim chuyin, dp rap ciia ham md hinh dupe ude lupmg dua vdo diim chuyin udc lupng. Trong bai bao nay, tac gid gidi thieu nhung dilu ki6n dam bao su hdi tu hdu chdc ebdn ciia dd rap udc lupng tdi dp rdp thuc cua ham md binh, khi diem chuyen dupe udc lupng bang phuang phap phdn du dieh ehuyin. Tit do, bai bao dua ra dupe each xap xi lue lupng kiem dinh diim chuyin cho md hinh hdi quy tuyin tinh hai pha khi chua biet diem chuyen thyc.
Tie khda: Do rdp, mo hinh hoi quy hai pha. diim chuyin, li^c lugng kiem dinh.
APPROXIMATE POWER OF CHANGE-POINT TEST FOR TWO-PHASE LINEAR REGRESSION MODEL ABSTRACT
The power of change-point test of two-phase regression model with a known change- point depends on the roughness of model function. In case the change-point is unknown, the roughness of model function is estimated basing on the estimated change-point. In this paper, we propose the conditions for ensuring the convergence of the estimated roughness and the real roughness of the model function, when the change-point is estimated by using the shift residual. From that, we obtain an approximate power of the change-point test of two-phase regression in ease the real change-point is unknown.
Keywords: Roughness, two-phase regression model, change-point, testing power.
1. G l 6 l THI$U
Nhieu qua trinh trong thuc t l tudn theo md hinh hoi quy tuyin tinh hai pha, d do cae tham sd dieu khien md hinh giii nguyen gia tri trong pha ddu, tai mdt thai diem nao dd nd chuyen sang gia tri khae vd giii nguyen trong pha edn Iai. Viec nghien cuu md hinh cd thay ddi trang thai nhu vdy - edn gpi Id md hinh d i i m chuyin - da dupe phat trien hon nira the ky qua vd dat dupe nhung thanh tyu rue rd, dupe ap dung rdng rdi
90 I TRirdNG D ^ H p c HAl PHONG
trong nhieu ITnh vyc khae nhau. Trong kinh te, ngudi ta thdy md hinh diem ehuyen bdi la phii hpp khi nghien ciiu mdi quan he giiia Iai suat (interets rate) ddi vdi thay ddi lai sudt chiet khdu (discount rate) quy dinh bdi FED. Sit dyng md hinh ARCH de nghien ciiu ehuoi thdi gian trong miln tdn sd, ngudi ta da phdt hien ra sy chuyen ddi eua chudi thdi gian ehi sd chiing khoan, cung nhu thi trudng ngoai hdi lien he mat thiet vdi khiing hoang tai chinh d chdu A vd Lien xd. Theo Caussinus H., Lyazrhi F., trong giai doan nghien ciiu, tdng sdn phdm qudc ndi Hoa Ky tudn theo md hinh diem chuyen bdi. Md hinh diem chuyen dupe dp dyng thanh cdng trong nghien ciiu hang khdng vu try, bien ddi khi hau, ehe dp thiiy van, luong mua, du bao, nghien ciiu the thao... Viec nghien cuu md binh diem ehuyen la can thiet va lien tue dupe phdt triin trong nhiing nam gdn ddy. Cu the, xet md hinh
ian + a,x, + £, khi 1 <i<k*
{Po+Pl^i+^i khi k'^<i<n
ti-ong dd a < X j < . . . < x „ < 6 , eae sai so {£•,} Id ng^u nhien, Q:Q,aj,;ffQ,y^,A:*chuabilt Neu ttQ = PQ \k ai= /3y thi md hinh (1) dupe gpi la khdng cd chuyen. Ngupc lai, neu it nhat mdt trong hai dang thitc ndy khdng xdy ra, md hinh dupe gpi la cd chuyen va k* dupe gpi la thdi diem ehuyen. Ddi vdi md hinh cd chuyin a^^ f\, va hai dudng thdng
y~ct(^-\- cx^x va y — PQ+ I\X eat nhau tai diem cd hoanh dp x= r thi r dupe gpi la diem chuyen. Trong trudng hop t G \XJ^*,XJ^^^) thi hdm md hinh dupe gpi la gay khiic lien tyc, md hinh dupe gpi la lien tye. Trai lai, md hinh dupe gpi la gian doan. O day, chiing ta ehi xet trudng hpp md hinh lien tue. Dat h = /3i-ai, Md hinh (1) dupe viit lai dudi dang
yi=f(Xi) + £i, irong do
f(x) ^aQ + a^x -I- h(x - T)I(X > r) (2) La ham md binh vd / ( . ) Id hdm chi tieu.
De udc lupng diem chuyen T ciia md hinh (2), da cd nhieu cdng trinh dua ra cac phuong phap khac nhau; tuy nhien cac phuong phap nay thudng dya tren mdt sd gia thiit kha chat che ve phan bd ciia sai sd. Trong trudng hpp khdng cd gia thiet ve phan bd eiia sai sd, dya tren phuang phap dich chuyen tham sd dupe de xuat bdi Liu [5], chimg tdi da dua ra phuang phap de udc lupng diem chuyen, xem [2].
Mat khde, khi diem ehuyen T cua md hinh da biet, chung tdi da ehi ra dupe cdng thiic hien de tinh toan lyc lupng kiem dinh, luc lupng nay phy thudc vdo kich thudc mlu, vao phuong sai sai sd va vao "dp rap" eiia md hinh (xem [1]).
Nhu vdy, vdn de dat ra la: trong tiiidng hpp chua biet diem chuyen eiia md hinh, dya ti-en diim ehuyen dupe udc lupng bang phuong phdp phdn du dich chuyin, lieu dp rap ciia hdm md tdnh udc lupng cd bdi tu ve dp rap cua ham md hinh thyc hay khong? Bai bao nay se tap trung chiing minh vdn d l neu ra Id dimg. Khi dd, viec tinh lyc lupng kiim dinh doi vdi bai
TA.P CHi KHOA H p c , S6 28, thang 5/2018 | 91
toan kiim dinh diim chuyen hoan toan ed the xdp xi dupe thdng qua viec tinh toan dd rdp cua bam md hinh ude lupng vdi diim chuyen ude lupng dd.
2. DO R A P C U A HAM M 6 HINH v A P H A N D U ' D I C H CHUYEN
Khi kiim dinh diim chuyin, ham md hinh f{x) cang gd ghi, cdng lech nhieu so vdi dudng thang thi kha ndng phdt hien ra diem chuyen edng ldn. Khdi niem do rap dupe dua ra de do miic dp gd ghe ciia hdm md hinh.
Dinh nghia 1. Dp rap cua hdm f{x) dua vao thiit ke {xi,...,x„} dupe ky hieu bdi 1 - ^
5 ^ ( / , { x , } „ ) va xac dinh theo cdng t h i r c : 5 ' ^ ( / , { x , } „ ) - - ^ ( / ( ; c , . ) - ( a - l ' 6 x ^ ) (3) trong dd d,b la udc lupng bmh phuong cyc tieu cua he sd chdn vd he sd gde tuang img ciia mo hinh hdi quy tuyen tinh dan vdi tap dii Ueu ( X p / ( x , ) ) , i = \,...,n.
Trong trudng hop cd vd han diem thiet ke, ta eoi mdi hdm phdn bd F(x) cd gid J a [a,b] chiia it nhdt hai diem la mdt thiit ke suy rdng tren J, dp do xac sudt img vdi ham phdn bd F(x) dupe ky hieu la (dF).
Dinh nghia 2. Dp rdp cua hkmffx) dya vdo thiet ke F(x) cd gid tren J dupe ky hieu bdi S {f,F) vd xdc dinh theo cdng thitc:
S^U',F)= nia Uf(x)-(a+bx)fdFM. W
Bat Zy(x)-\, Z2(x) = x,va <z,z >f- < l , l > f <1,JC>^
<^J>F= <hf>F
<X,f>f
< X,\>f < x,x>
trong do (k, t)f. = \ k(x)i{x)dF(x). Nhiing thiet ki ma ma tran < z , z > ^ kha nghich dugc goi la tiii6t kg tuang thich vai he ham co so {i,:^}. Nhu vay, theo phuong phap binh phuang nho nlidt, cue tiSu o (4) t6n tai va dat dugc tai
( a , i ) ' ' = ( < z , z ' " > ^ ) ' < z , / > ^ . (5)
Ro rang, moi thiet ke rai rae {x,, i = 1,...,«} co it nhat liai di^m phan biet la mgt thi^t kg suy rgng F^ ^ .^^(x)-la ham plian bo m^u cita miu quan sat [x^,...,x„] nen d6 thay rang (3) la trucmg hgp dac biet cua (4).
Trong [I], xet ham mo hinh f(x) va diem chuySn r da biSt, dita vao day quan sat {(^/'J'/J} , voi thoi diem chuyen k*, sii dung phuang phap binh phuong nho nhat, tim dugc RSS, RSSi, RSS2 tuong iing la tong binh phuong phSn du dua vao toan bp quan sat, k*
92 I TRirdNG DAI H p c HAl PHONG
quan sat ddu, n-k* quan sat sau. Kbi do, dudi gid thuylt khdng cd chuyen cua md hinh. Chow [3] da chi ra rdng thdng ke:
^^{RSS~(RSS,+RSS^))I2 ^^^
{RSS^+RSS2)/n-4
cd phdn bo Fisher-Snedecon vdi 2 va n—4 bdc ty do. Vi vdy, mien bac bd miic a tuong iing 1^ ^ > yi^ ( 2 , « — 4 ) . Cdng thiic tmh luc lupng kiim dinh dupe eho bdi djnh ly sau:
Dinh ly 1. ([2, tr. 2999]) Gid sii day thiit k l {x,} tucmg thich vdi he { l , x } , n l u m d hinh cd diem chuyin da hilt thi F xac dinh d (6) cd phan bd Fisher- Snedeeon phi trung tdm vdi 2, n—4 bac tu do vd tham sd phi trung tam \a A. = ^—^— . Khi do, lyc lupng
CT
kiem dinh diem ehuyen dupe tinh toan bdi edng thiie:
^|^J«S(^_2,«-4]>/,(2,«-4)}. (7)
Tuy nhien, trong trudng hpp ham md hinh f{x) chua biet diem chuyen thi viec kiem dmh vd tinh lye lupng kiem dinh theo (7) khdng the thyc hien dupe. Vi vay, van de dat ra la:
de cd the diing xdp xi dupe cdng thiic (7), ta cdn udc lupng diem chuyen eho md hinh, sau do xem xet xem lieu dp rap ciia ham md hinh udc lupng tuang irng cd hdi ty ve dp rap ciia ham md hinh thyc hay khdng. De giai quyet van de nay, ta se xem xet phuong phap udc lupng bang phan du dich chuyen nhu sau.
Gia sir chiing ta cdn udc lupng thdi diem chuyen eiia md hinh (I) dua tren n quan sat:
(Xj,yj),i = l,..., n . De thyc hien viec udc lupng, ta chia day quan sat thanh hai nhdm: nhdm thu nhdt chiia k quan sat ddu (Xj,yj), i = X,...,k, nhdm thit hai chita n—k quan sat cdn lai (Xj,yj),i = k + l,...,n, gja sit d^/^, d^^ va J5Q^, /3if^ ldn lugt la udc lupng binh phuong cue tieu cho he sd chan, he sd gdc cua md hinh tuyen tinh dya tren hai nhdm quan sat tren. Theo [5], cae phdn du dich chuydn dupe xdy dyng:
yyi ~i^Qk + ^\k^i) khi k + \<i< n.
Nhu vdy, khi tinh phdn du cho nhdm quan sat dau (pha ddu), chung ta dung tham sd ude lupng cua nhdm quan sat sau (pha sau) va ngupc lai. U'u diem eiia phdn du dich chuyen la ehung gdn vdi phdn du thdng thudng dudi gid thuyet khdng cd chuyen ciia md hinh (khi ( a Q , a i ) = (^0'A))'"^^"^S^^'^^S duoc phdng dai len dudi ddi thuylt (aQ,ay)^ (pQ,pi).
Viec udc lupng diem chuyen dupe the hien trong dinh ly sau.
Dinh ly 2 ( [ 2 , ti". 37]). Gid su thda man cac dilu kien sau day:
i) Thiit k l Xi ti-ai diu tren doan [(3,6] = [0,1]: x^^i/n, i ^\,...,n.
TAP CHi KHOA H p c , So 28, thang 5/2018 | 93
ii) Ham md hinh ed thi viit dudi dang
f„{x) = tto + ccxX-\-h{x-Xk*)I{x > Xf^*X h^O.
iii) Tdn tai rQG(0,l/2) sao eho ATQ < fc* < « - A Q , ti-ong dd ko=[«ro] + l vdi [a] la phdn nguyen ciia sd thye a.
iv) x^*—>r khi « ^ o o .
v) Cdc sai sd {Sj} Id eae biln ngdu nhien ddc lap, cd ky vpng khdng, £(£,• ) = cr, > 0 vdi/ = 1,...,A:*, E(£f)=o-2>0vmi = k*-\-l,...,n, erf, cr| ehua biit.
Khidd, l i m - ^ = r hdu chdc chdn (h.c.c.), trong dd k„ = argm^c ^e^/^- n ^ « ' n ko<k<n-kQ /=i Han nua, vdi i = \,2:
lim«.r = « ; , lim/Lr =aQ-hT, lim^ftr = a^ + hz (h.c.c.).
Cae gia thiit d Dinh ly tr6n la khd tdng quat va dl dap iing dupe ti-ong nhung dieu kien thyc te. Gia thiet ii) ddm bao rdng md binh la gay khue lien tyc. Theo gia thiet iii), ta chi can xet thdi diim chuyin tii ATQ din K - A:Q , tiie la diem chuyen khdng gdn qua hai ddu miit. Gia thiit nay dam bao sy hdi ty cua cdc tham so udc lupng. Theo [2], chpn AQ sao eho
^0 = CQH + 0(1),CQ G (0,0.5). Gia thiit ii) cd ngMa rdng x^ Id diem chuyen cua md hinh cd n quan sat. Tit iii), rd rang rdng ATQ ^ oo khi vd chi khi n —> oo. Gia thiet v) rdt tdng quat, d dd cac phuang sai d hai pha (T^ va cr2 ndi chung khdc nhau. Ket ludn d Dinh ly 2 khdng dinh diem chuyen udc lupng dupe k^ / n se hpi tu hdu chac chan, Ioai hpi ty rat manh ciia Iy thuyet xdc sudt, den diem ehuyen thue r.
3. SU" HQI TU CUA DO R A P UOfC LU'gfNG
Gia su ddi vdi md hinh (I), ehiing ta tim dupe udc lupng cho thdi diem chuyen la ^„ va udc lupng tuang iing cho tham sd d pha ddu va pha sau Idn lupt Id d.r , d.j_ va
A)A ' ^lic ' * ^ ^°' ^^^ ^° ^"^ ^^^ luong dupe cd dang:
fAx)
= i . , " .(9)
Ta se chiing minh vdi n du ldn, dp rap S (f„, {xj„) se xap xi dp rap 5 ( / , x ) , vd do dd, neu S (/„, {x^}^) la ldn, ta cd thi tm tijdng nhiing ket luan thdng kg da dua ra. Trdi
94 TRirdNG DAI HpC HAI PHONG
lai, n l u iS'^(/„, {x,}„) tuong ddi nhd, cac kit ludn v l gia tii eua eae tham sd
^ „ , «0A ' ^ik ' ^oic ' A i cd dp tin tudng thdp. Cy thi, ta ed dinh ly sau:
Dinh Iy 3 . Gia su cac gid thiit d Dinh ly 2 dupe thda man. Khi dd
\im S^(f„,{Xi}„) = S^{f,U) (h.c.c) ti-ong do f(x) xdc dinh theo (2), / „ theo (9) va
«—xc
(7(xJ Id ham phdn bd d i u tren [0,1].
De chiing minh dinh ly 3, ta cdn hai bd de sau.
Bo de 1. Gia sit tbda man eac dieu kien sau:
i)Day thiit k l F„ (x) hgi ty yiu din thiit k l F(x) • F„^>F.
^0 SnO^^-> S(x} 1^ nhung ham do dupe, bi chan deu tren [0,1]; tdn tai iVf > 0 sao cho: |g(x)|, \g„(x)\<M V X G J , V n .
iii) {dF){ES) — Q, trong do (dF) la dp do xae suat iing vdi ham phdn bd F(x), E^={t^J:3{t^]czR,g^{t^)^g(t)}.
Khidd s'^(g^,F^)^S^{g,F).
Chung minh. Tir ehd hdm z^ ( x ) , / = 1,2, la lien tye, hi ehan, va i ^ => F, suy ra
<^t'^j >F„"^ JQ^ii^)^}i^Wn(^) -^ Jo^ii^)^j{x)dF(x) (l, j - 1 , 2 )
Hem nua, cae ma tran < z , z > ^ , < z , z >p kha nghieh, nen d e t l < z , z > ^ j — > d e t l < z , z > ^ U^ 0 , tii dd, suy ra mdi day cdc phdn tir ciia ma trdn I < z, z ->j7 I hdi ty den phdn tii tuong iing ciia ma tran < z, z ^ ^ .
Rd rang, vdi i = \,2, cac ham z^{x)g^(x), z^{x)g(x) la do dupe, bi chan;
E^.^g cz Eg, (dF)(E^^.^g) < (dF)(Eg) - 0 (tii dilu kien iii)). Theo Dinh ly 5.5 trong [4], ta eiing cd
< ^hSn >F„ = J o 2 , ( x ) g „ ( x ) r f f ; ( x ) -^ ^^Zi(x)g(x}dF(x) . Suy ra
( f l „ , 4 f = ( < Z , Z ^ > F J <2,gn>F„-^{<^y >F) <2,g>j,^(a,b)'^. (10)
Mat khac, S^(g„,E^) = ^ g „ ( x ) -(d^ + ^ x ) ) dF„(x), khai triin ve phdi thdnh tdng, su dung (10) va lap ludn tuang ty nhu tren ta dupe
T^P CHi KHOA HpC, So 28, thang 5/2018 | 95
S'{g„.F„)->ll(g(.x)-(a-<-bx)) dF(x)
Nhanthdy, vephaila 5 ^ ( g , F ) = min | ( / ( x ) - ( a + f a ) ) rff (x) nentacodiSuphai
(<i,S)eK^ J
chung minh.
B6 a l 2. Gia sir ham phan bo mau F (x) cua mau {jc,.}„ hgi tu y6u den F(x) - la mgt thiSt ke tren [0,1], ham F(x) lien tuc tai L> e (0,1), cac ham m6 hinh h{x), /l„ (x) cho boi:
, , , \''on+ai„x,(l<x<u„ {af, + aiX,0<x<u [ i Q „ + i i „ x , Li„ < x < l [bo+l\x, u<x<l,
sao cho: lim tj^ = t; va
M->co
v d i ! = 0 , 1 : l i m a , „ = a , - , limfe|„=fe;.
fl—Mo fl—X»
Khido, limS^(h„,{x,}„) = S^(h,F).
M—Ko
Chimg minh. Ro rang, ede bam /2„(jc), /2(x) la cac ham do dupe, hi ehan deu tren [0,1], lim h^(x) = h{x), \/x^o, x G [ 0 , 1 ] , vdy Efj <^{u} . Theo gid thiet, diem gian doan
n—wo
duy nhdt cd thi ciia h(x) la f;, tit dd (dF)(E^) < (dF){D} = 0 . Ap dyng Bd d l 1, ta duoc
lim S \ K , { X , } „ ) = lim S \ h , , F{ _,^^) = S \ h , F ) .
Chirng minh Binh ly 3. Trudc hit ta thdy / ^ => t / , cac ham f„ix) vd f(x) do dupe, bi ehan deu h.e.e.,/fxj lien tye. Dat f„ — ^„ / « , theo Dinh ly 2, ta cd cac gidi ban h.e.e. sau:
lim f„ = r , lim a.r = «,•,
l i m y ^ r =aQ-hT = PQ, l i m A r =a^ + h = /3^
/i->oo "'^n n^x ^'^n
Vi U(x) Hen tue tren (0,1), ap dyngBd d e 2 , nhdn dupe dieu phdi chiing minh.
Nhan xet. Vdi ket qua eiia Dinh ly 3, ta cd the xap xi lyc lupng ciia bai toan kiem dinh ti-ong tiirdng hpp chua biet diem chuyin thue. D I lam dupe dilu ndy, trudc hit ta udc lupng diem chuyen vd ham md Mnh theo Dinh ly 2, sau dd x ^ xi lyc luong kiim dinh bdi cdng thiie.
tai diem chuyen udc luong.
P\F{ ^:—'-—, 2, n - 4) > / ^ (2,« - 4) I vdi dd rap ciia ham udc lupng dupe tinh toan
96 TRircfNG DAI H p c HAI PHONG
4. KET LUAN
Vdi mdt sd gia thiit d l ddng thye hien trong nhitng dieu kien cua thyc te, chung ta nhdn dupe sy hdi tu hdu ehde chdn eiia dp rap cua ham md hinh udc lupng khi dung phuong phap phdn du djch ehuyin tdi dp rdp eua md hinh thuc. Tir dd, khi n du ldn, dp rap ude lupng dupe cho ta thdng tin hiiu ieh: Neu nd ldn, ta cd co sd de tin tudng cac ket ludn dua ra, neu nd nhd, eac kit ludn thu dupe cd dp tin tudng thdp, ngheo nan.
TAI LIEU THAM KHAO
1. To Van Ban, Nguyen Thi Quyen (2014). The power of change-point test for two-phase regression. Applied mathematics, 5, pp. 2994-3000.
2. To Van Ban, Nguyen Thi Quyen( 2016). Estimating a change-point in two-phase regression model based on the shift of parameter estimates. Theoretical Mathematics and Applications, 6(4), pp. 33-52.
3. Chow G. C. (1960). Tests of equality between sets of coefficients in two linear regressions. Econometriea. 28(3), pp. 591-605.
4. Billingsley P. (1968). Convergence of probability measures. John Wiley&Sons, New York.
5. Liu Z., Qian L.(2009). Changepoint estimation in a segmented linear regression via empirical likelihood. Communications in Statistics-Simulation and Computation. 39, pp.
85-100.
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