- Email: [email protected]
Limiting distribution of weighted processes of residuals. Application to parametric nonlinear autoregressive models
Limiting distribution of weighted processes of residuals. Application to parametnlc nonlinear autoregressive models Jean DIEnOLT, Naamane LAin and Joseph NGATCIIOU WANDJI
J. D. : CNRS & IMAGILMC/SMS, np 53, 380U Crenohle C("llex 09. E-mail: [email protected] N. L. : LSTA, Universite Paris-VI, 4, place Jussieu , 75252 Paris Ct'dt'x 05. E-mail: [email protected] J. N. W. : INRIA Rhone-Alpes, 38330 MontLonnot-Saint-Martin. E-mail: [email protected]
Abstract.
We introduce methods for testing the goodness-of-fit of linear or nonlinear parametric autoregressive models of order 1, under stationarity and ergodicity assumptions. We establish two functional limit theorems for the process of deviations A"(.) between the weighted process of residuals under consideration and its parametric counterpart, under the null hypothesis Ho . We discuss several possible tests based on these results and show that the half-sample method introduced by [10] for parametric distribution function models can be adapted to the present setting.
Loi limite [onctionnelle tl'un processus residuel cumule, Application des tests d'ajustement pour ties modele» autoregressifs parametriques cl'orclre 1
a
Resume.
Nous etabltssons des theoremes fonctionnels limites pour un processus residue! cumule, dans le cas 014 La suite des observations est ergodlque et stationnaire. Nous en deduisons des tests d'adequation de modeles parametriques autoregressifs d'ordre I.
Version fran{:aise abregee Notre but est de mettre au point des methodes statistiques pour discrimincr entre rnodelcs parametriques autoregressifs lineaires ou non lineaires de la forme (0.1 ) Note presentee par Paul 0764-4442197103250535
DEllf:UVEl.s.
Academle des ScienceslElsevier. Paris
535
J, Dieholt, N. Laib and J, Ngatchou Wandji en utilisunt une approche non parametriquc, et en prenant en compte I'estimation de B sous l'hypothese nulle lI u d'uppartcnance de la fonction d'autoregression a un modele parametrique M = {11/.(, j fJ) : fJ E (-)}. Les bruits e, sont des differences de martingales de variance conditionnelle I par rapport aux tribus 9, = a(X 1, e l , ... ,X e,) : " (11,2)
La suite (X,) est sculcrnent supposee stationnaire et ergodique. Nos statistiques de test sont construites scion le processus defini par : • :) A,,(:I
(lI.a)
"
= n -1/2",( L X'+I
' n ) ) I(X , ~ x), - m(X,; B
xER,
' =1
0" dcsignant un estimateur 'It 1/2-convergeant du vrai parametre fJ o, et verifiant l'hypothese (AS) ci-dcssous, Sous 11 0 , ce processus comcide avec
(ll.·1)
iJ,,(:J~) = B,,( :r:) -
1 2 11.- /
L"
(m(X,; B,,) - m(X, ; Bo) )I(X,
s x) ,
I=l
ou (lUi)
11,,(x)
= '1/.-1/2 L"
a(X,)I(X, ~ X)e,+l'
1=1
Nous en dcduisons plusicurs statistiqucs de test possibles: en particulier, un test de type KolmogorovSmirnov base sur lu statistiquc S" = sup rER Iii" (:r:) I, el un test de type Cramer-von Mises base sur
5" =
1:
A~(:r.)1II(F,,( :I:))dF,,(x)
(w(.) dtSsigne une fonction poids et F" est la f.r. empirique de l'echantillon). D'apres Ie Theoreme I ci-dcssous, Ie processus converge en distribution vers iJ(·) defini en (3.2) . Par consequent, sous
ts; (.)
lI u, S" converge en loi vcrs SUPrER
Ill(:r.)1 =
sUPuE(o.l\lll(F-1(u))I, et 5" converge en loi vers
j-~ Il2(:J~)tIJ(F(x)) dF(:r.) = ht no
iJ2 (F- 1 (u ))w(u) du,
s;
Pour en dcduirc une procedure de test, on doit connaitre les lois Iimites de S" et de Pour Sn, on pcut utiliscr par exemplc 191. Un autre test possible est presente ci-dessous. Lorsque les fonctions (:. flu cl ho qui intcrvicnncnt dans l'expression de la loi limite sont inconnues, on les remplace par des estimuteurs uniformcmcnt convergents (voir (3».
1. Introduction Our aim is to introduce methods for testing the goodness-of-fit of linear or nonlinear parametric uutnrcgrcssion models of order 1 defined by (0.1), when neither the variance function nor the distribution or the noise or the stationary distribution of the observations is known, under stationarity
53()
Limiting distribution of weighted processes
and ergodicity assumptions. The response variables X t + 1 are related to X, by a relation of the form (0.1) for some measurable function 0-(') and the rv 's Ct are Ot-martingale differences with conditional variance 1, with 9t = a (X 1 , Cl , •. . , Xt , cd (t ~ 1). Our procedure is based on a measure of the deviation between a weighted process of residuals and a parametric estimate of the cumulated conditional mean function, under the null hypothesis H o. The process (0.1) has been studied , e.g., by [1] and [2], whereas [8] establishes a functional limit theorem for cumulated regressograms when the parameters are unknown, and [II] studies a testing process similar to (0.3) in the GLM regression setting with iid X,' s, basing its test on maximum likelihood estimation. In this Note, we state functional limit theorem s (Theorems I and 2) under Ho for the processes An(.), and a variant of the latter, when the sequence {X t } is only assumed stationary and ergodic. and On is the conditional mean-square estimator of the K-dimensional parameter 0, We show how the asymptotically linear structure of the sequence, of estimators {O,,} (see assumption (AS) below) allows us to determine the limiting distribution of An (.) under Ho. We show that the half-sample method introduced in [10] for distribution function models can be adapted to the present setting (Theorem 2). The behavior of the process An (-) under the alternative, and the estimation of the functions G, go, and ho, and of the matrix Vo. under H o, will be examined in a forthcoming Note.
2. Notation and assumptions The parameter space e is a subset of R K with nonempty interior. int«(-)) . Our basic assumptions are: (AI) There exists "( > 0 such that SUPt>o E{lctHI 2+'"Y19d < 00. (A2) The common distribution function F(. ) of the Xt'S is continuous and increasing on R (A3) The function a{ ·) is measurable. a{x) > 0 for all x E R, and J~oo laI 2 +I" dF < 00 for some I' > O. (A4) The function m(x; e),x E R , e E e, has partial derivatives with respect to Ol-, k = 1•.. . , g. up to order 2, and: (i) The integrals J~oo !Dm/Dek{y; e)!8=80 !dF (y ) are finite for k = I . .. . .K, (ii) There exists TO > 0 and a finite function Mo(x) ~ 0, x E R. such that the closed ball B(eo, TO) is contained in int(e) •
I
.:'up D2m/DO jD(}j' (x; 0)18=80 8EB(80,ro)
I~
Mo(x ) for all ;,; E R ,
i .i' = I , .... «.
and the integral J~oo Mo(Y) dF(y) is finite. (AS) Under H o• n
(2.1 )
n 1 / 2 (O n- eo)
= 71.- 1 / 2 L c,oo(Xt)Ct+l + 01'(1)
as n
-+ 00,
t=1 00 where tpoO = (tpO,l(·) , ... , c,oO, K(·» T denotes a function such that "tpllll 2+I' ,. di" is 00 finite for some "(" > 0 ell . II is a norm on R K ). Under (AS), n l / 2 (O" - ( 0 ) converges as n -+ 00 to a [{-dimensional normal rv with mean 0 and variance matrix
1-
(2.2)
r0 =
i:
c,ootp~ «r. 537
J. Dlebolt, N. Laib and J. Ngatchou Wandji
=
arg min9E9 E~=l (Xt+l In this Note. we consider the conditional mean-square estimator On m(X,; 0))2. studied by [4]. [5J. [6], [7], and [8] for discrete-time processes. We denote by V'm(x; 0) the gradient of m(x; 0) with respect to (), and V'm(x; 0)19=90 = V'mo(x) , where 00 E int(S) is the true value of the parameter. Under technical assumptions (see [8], B, or [7], [5]-[M]-[N], when 17(') is constant). the sequence of estimators {On} is n 1/ 2-consistent, satisfies (AS), and
va
3. \\lain results We define the weighted residual process (:U)
x ER,
where I(X E C) denotes the indicator function. Under Ho, An(·) takes the form Bn(x) = lJIl ( ; : ) - n- I / 2 (m(X, ; On) - m(X t ; ( 0 )) It X, $ x), where B n(·) is defined in (0.5),
2:;'=1
I. - Under assumptions (Al)-(A5), the processes BnO converge in distribution as to the centered Gaussian process
TIlEOREM
n .....
00
(:1.2)
x ER,
where
(:1.:1 ) with
D(x) =
'V(.)
i~
(1
d(W
0
F),
x E R,
a standard Wiener process,
(a.4) and where ~ = (6, ' .. I ~ 1\ ) T is normal with mean 0 and variance matrix r 0, and where for all x the rv' .\' ~ and D(x) have J( x 1 covariance matrix
E(~ D(x)) = ho(x) = [~
(1
The covariance function E(B(:r.dB(X2)) of the Gaussian process B(.) is
Remarks. - (I) - In the case where On is the conditional mean-square estimator and where conditions IS of 18J hold, (3.5) takes the form ho(x) = VO- 1 J~oo (12 V'mo dF. (2) - In the case where F(·) has a density 1('), iJ(x)
538
= i~ 11/ 2 dW (T
1 2 /
- 90(x)T ( [ :
dW).
Limiting distribution of weighted processes
=
Example. - Testing for no effect. In this case, m(x; e) e for all x and with mean 0 and variance J~(X) a 2 dF, and the covariance of B(.) is
e, ~ denotes a normal
rv
G(x 1\ y) - G(x) F(y) - G(y) F(x) + G(+oo) F(x)F(y).
=a
is constant, this covariance reduces to a 2 [F(x) /\ F(y) - F(x) F(y)]; therefore, B(.) is then a Brownian bridge. When the function is constant, it is possible to modify the process AnO into a process AnO such that under Ho the AnO's converge in distribution to B(·) (see (3.3». We consider a sequence {qn : n ~ I} of integers 1 ~ q" ~ n increasing to 00, such that lim" --+no q,.j n = f'i" and for each n a qn -subsample {Xii' , .. , Xi q n } of the n-sample {Xl, ... , X n}, where the integers 1 ~ i 1 < ... < i qn _< n are independent of the sequence {Xd· Letting Bn = Oq n denote the conditional mean-square estimate of based on this subsample, we define
If a(x)
aO
e
n
(3.7)
An(x)
=
2 n- 1/
L
(Xt +1
-
m(Xt ; Bn ))I(Xt ~ z ),
x E R.
t=l Under Ho, A n(·) reduces to a modification iJ,,(x) of Bn(x) with On replaced by
B".
2. - Suppose that the function a(.) is constant and that f'i, = 1/2. Then. under assumptions (Al)-(A5) and ([8], B, or [7], [S]-[M]-[N]), the processes B..(-) converge in distribution as n -+ 00 to the Gaussian process B(x) = a W(F(x», x E R. THEOREM
4. Possible tests based on these results (1) - Two possible tests are a Kolmogorov-Smirnov type test based on the statistic S" = sup .rER a Cramer-von Mises type test based on the statistic
IA,. (x) I and
5"
jna .4.;,(:1:) w(F,,(:l'»
=
dF,,(:r.),
-00
where w(·) is a weight function and F" is the empirical d.f. of the sample. Theorem I implies that. under Ho, 5 .. converges in distribution to 5 = sUPrER IB(x)l, and 5.. converges in distribution to iJ2(F- 1 (t» w(t ) dt. If a(·) is constant and f'i, = 1/2, then 5" -+ aSuPtE[o,ljIW(t)l, and 5.. -+ a W (t )w(t) dt (for the tabulation of these tests, see [2]). When a(·) is not constant, we can use [9] ~o obtain a tabulation of the test based on 5". (2) - Another po~sibility is to make use of the the Karhunen-Loeve expansion of the Gaussian process ir(·) = (B 0 F- 1 ) ( . ) defined on the unit interval [0,1],
J;
J/
(X)
B*(·)
= L Ay2 z, Ii('), J=1
°
where Al ~ A2 ~ .. , ~ are the eigenvalues of the covariance operator of iJ* on L2[0, form an orthonormal basis of eigenfunctions of this operator, and the rv's
z, = Aj1/2(11
iJ*(t) !;(t) dt)
1], fll h, ...,
areiidnormal N(O,l).
As discussed, e.g., in [12] in the setting of goodness-of-fit tests for parametric models of distribution functions, it is possible to choose a statistical test of the form T;! Z;.i for some moderate
= Ef=l
539
I. Dicbolt, N. la'.b and I. Ngatchou Wandji
.J > \, where Z".j = >..-;\/'1. J::", A"t1J)fjtF(y))dF(y). The rv's Tj converge in distribution under lin to 7.1 = L~= 1 Z}, a chi-square with .J degrees of freedom, Note remise lc 12 mars 1997. ucceptee apres revision Ie 20 mai 1997.
References III An II. Z. and Ch\'l1~ n., 1991. A Kolmogorov-Smirnov type statistic with application to test for nonlinearity in time series. I",. SIal. R"I'.. ~lJ. pp. 2117-307. 121 Ul\'hull J. IIl1d IMi') N., 1995. Nonparamctric tests for correlation or autoregression models under mixing conditions. C. R. A.../Il. Sd. Paris, s~ric I, 320. pp. 1135-1139. PI UldllIll J., I.u\') N. and Nl:lllChuu Wundjl J., 1997. Limiting functional distribution of weighted processes of residuals. Apphcution to j:ool.lness-of-fit tests for parametric models of nonlinear autoregressive. Manuscript. 141 Dullo 1\1., S\'lIoussl It. and Touall A., 1991. Proprietes usyrnptotiqucs presque sOres de l'estimateur des moindres carres d'un modele 1I11torcgressif vectoriel, Ann. lnst. IIl'11ri Poincare, serie B. 27. pp, 1-25. 151 KlIlIIl\o I.. A. and Nelson I'. I., 1978. On conditional least squares estimation for stochastic processes. Ann. SIal.• 6, 3, pp. 629-642. (611.111 T. I... 199~. Asymptotic properties of nonlinear least squares estimates in stochastic regression. Ann. SIal.• 22. pr. 1917-1930, I7ll\1alll:l'lIs 1\1. and \'ao J. 1"., 1996. Sur l'estirnateur des moindres carres d'un modele autoregressif fonctionnel. To appear in C. R. A/'I/I/. Sci. Paris, serie I. 1111 I\lcKl'lIl:lJl' I. W. and Zhan~ 1\1. J., 199~. Identilication of nonlinear time series from first order cumulative characteristics, A"". SIII/., 22. pp. 495-514. (lJI 1'lIl'rhllr~ V. I., 1996. A.vymplotic MI'lhods in the Theory of Gaussian Processes and Fields, Translation of Mathematical Mllfllll!rnphs, 14K. American Mathcmntical Society: Providence. 1101 Sh'lIhells 1\1. A., 1978. On the half-sample method for goodness-of-fit, J. Roy. SIal. Soc., 40, pp. 64-70. 111I SU J. Q. und Wel I.. J.t 1991. A lack-of-til lest for the mean function in a generalized linear model, J. Amer. SIal. !\.I.WI('.. llfl. pp. 420-426. 1121 Shorlll'k (;. and Wellnl'r J., 1996. Elllpiri('(/Ip'lwe.un with application 10 statistics, Wiley: New York.
540
J. D. : CNRS & IMAGILMC/SMS, np 53, 380U Crenohle C("llex 09. E-mail: [email protected] N. L. : LSTA, Universite Paris-VI, 4, place Jussieu , 75252 Paris Ct'dt'x 05. E-mail: [email protected] J. N. W. : INRIA Rhone-Alpes, 38330 MontLonnot-Saint-Martin. E-mail: [email protected]
Abstract.
We introduce methods for testing the goodness-of-fit of linear or nonlinear parametric autoregressive models of order 1, under stationarity and ergodicity assumptions. We establish two functional limit theorems for the process of deviations A"(.) between the weighted process of residuals under consideration and its parametric counterpart, under the null hypothesis Ho . We discuss several possible tests based on these results and show that the half-sample method introduced by [10] for parametric distribution function models can be adapted to the present setting.
Loi limite [onctionnelle tl'un processus residuel cumule, Application des tests d'ajustement pour ties modele» autoregressifs parametriques cl'orclre 1
a
Resume.
Nous etabltssons des theoremes fonctionnels limites pour un processus residue! cumule, dans le cas 014 La suite des observations est ergodlque et stationnaire. Nous en deduisons des tests d'adequation de modeles parametriques autoregressifs d'ordre I.
Version fran{:aise abregee Notre but est de mettre au point des methodes statistiques pour discrimincr entre rnodelcs parametriques autoregressifs lineaires ou non lineaires de la forme (0.1 ) Note presentee par Paul 0764-4442197103250535
DEllf:UVEl.s.
Academle des ScienceslElsevier. Paris
535
J, Dieholt, N. Laib and J, Ngatchou Wandji en utilisunt une approche non parametriquc, et en prenant en compte I'estimation de B sous l'hypothese nulle lI u d'uppartcnance de la fonction d'autoregression a un modele parametrique M = {11/.(, j fJ) : fJ E (-)}. Les bruits e, sont des differences de martingales de variance conditionnelle I par rapport aux tribus 9, = a(X 1, e l , ... ,X e,) : " (11,2)
La suite (X,) est sculcrnent supposee stationnaire et ergodique. Nos statistiques de test sont construites scion le processus defini par : • :) A,,(:I
(lI.a)
"
= n -1/2",( L X'+I
' n ) ) I(X , ~ x), - m(X,; B
xER,
' =1
0" dcsignant un estimateur 'It 1/2-convergeant du vrai parametre fJ o, et verifiant l'hypothese (AS) ci-dcssous, Sous 11 0 , ce processus comcide avec
(ll.·1)
iJ,,(:J~) = B,,( :r:) -
1 2 11.- /
L"
(m(X,; B,,) - m(X, ; Bo) )I(X,
s x) ,
I=l
ou (lUi)
11,,(x)
= '1/.-1/2 L"
a(X,)I(X, ~ X)e,+l'
1=1
Nous en dcduisons plusicurs statistiqucs de test possibles: en particulier, un test de type KolmogorovSmirnov base sur lu statistiquc S" = sup rER Iii" (:r:) I, el un test de type Cramer-von Mises base sur
5" =
1:
A~(:r.)1II(F,,( :I:))dF,,(x)
(w(.) dtSsigne une fonction poids et F" est la f.r. empirique de l'echantillon). D'apres Ie Theoreme I ci-dcssous, Ie processus converge en distribution vers iJ(·) defini en (3.2) . Par consequent, sous
ts; (.)
lI u, S" converge en loi vcrs SUPrER
Ill(:r.)1 =
sUPuE(o.l\lll(F-1(u))I, et 5" converge en loi vers
j-~ Il2(:J~)tIJ(F(x)) dF(:r.) = ht no
iJ2 (F- 1 (u ))w(u) du,
s;
Pour en dcduirc une procedure de test, on doit connaitre les lois Iimites de S" et de Pour Sn, on pcut utiliscr par exemplc 191. Un autre test possible est presente ci-dessous. Lorsque les fonctions (:. flu cl ho qui intcrvicnncnt dans l'expression de la loi limite sont inconnues, on les remplace par des estimuteurs uniformcmcnt convergents (voir (3».
1. Introduction Our aim is to introduce methods for testing the goodness-of-fit of linear or nonlinear parametric uutnrcgrcssion models of order 1 defined by (0.1), when neither the variance function nor the distribution or the noise or the stationary distribution of the observations is known, under stationarity
53()
Limiting distribution of weighted processes
and ergodicity assumptions. The response variables X t + 1 are related to X, by a relation of the form (0.1) for some measurable function 0-(') and the rv 's Ct are Ot-martingale differences with conditional variance 1, with 9t = a (X 1 , Cl , •. . , Xt , cd (t ~ 1). Our procedure is based on a measure of the deviation between a weighted process of residuals and a parametric estimate of the cumulated conditional mean function, under the null hypothesis H o. The process (0.1) has been studied , e.g., by [1] and [2], whereas [8] establishes a functional limit theorem for cumulated regressograms when the parameters are unknown, and [II] studies a testing process similar to (0.3) in the GLM regression setting with iid X,' s, basing its test on maximum likelihood estimation. In this Note, we state functional limit theorem s (Theorems I and 2) under Ho for the processes An(.), and a variant of the latter, when the sequence {X t } is only assumed stationary and ergodic. and On is the conditional mean-square estimator of the K-dimensional parameter 0, We show how the asymptotically linear structure of the sequence, of estimators {O,,} (see assumption (AS) below) allows us to determine the limiting distribution of An (.) under Ho. We show that the half-sample method introduced in [10] for distribution function models can be adapted to the present setting (Theorem 2). The behavior of the process An (-) under the alternative, and the estimation of the functions G, go, and ho, and of the matrix Vo. under H o, will be examined in a forthcoming Note.
2. Notation and assumptions The parameter space e is a subset of R K with nonempty interior. int«(-)) . Our basic assumptions are: (AI) There exists "( > 0 such that SUPt>o E{lctHI 2+'"Y19d < 00. (A2) The common distribution function F(. ) of the Xt'S is continuous and increasing on R (A3) The function a{ ·) is measurable. a{x) > 0 for all x E R, and J~oo laI 2 +I" dF < 00 for some I' > O. (A4) The function m(x; e),x E R , e E e, has partial derivatives with respect to Ol-, k = 1•.. . , g. up to order 2, and: (i) The integrals J~oo !Dm/Dek{y; e)!8=80 !dF (y ) are finite for k = I . .. . .K, (ii) There exists TO > 0 and a finite function Mo(x) ~ 0, x E R. such that the closed ball B(eo, TO) is contained in int(e) •
I
.:'up D2m/DO jD(}j' (x; 0)18=80 8EB(80,ro)
I~
Mo(x ) for all ;,; E R ,
i .i' = I , .... «.
and the integral J~oo Mo(Y) dF(y) is finite. (AS) Under H o• n
(2.1 )
n 1 / 2 (O n- eo)
= 71.- 1 / 2 L c,oo(Xt)Ct+l + 01'(1)
as n
-+ 00,
t=1 00 where tpoO = (tpO,l(·) , ... , c,oO, K(·» T denotes a function such that "tpllll 2+I' ,. di" is 00 finite for some "(" > 0 ell . II is a norm on R K ). Under (AS), n l / 2 (O" - ( 0 ) converges as n -+ 00 to a [{-dimensional normal rv with mean 0 and variance matrix
1-
(2.2)
r0 =
i:
c,ootp~ «r. 537
J. Dlebolt, N. Laib and J. Ngatchou Wandji
=
arg min9E9 E~=l (Xt+l In this Note. we consider the conditional mean-square estimator On m(X,; 0))2. studied by [4]. [5J. [6], [7], and [8] for discrete-time processes. We denote by V'm(x; 0) the gradient of m(x; 0) with respect to (), and V'm(x; 0)19=90 = V'mo(x) , where 00 E int(S) is the true value of the parameter. Under technical assumptions (see [8], B, or [7], [5]-[M]-[N], when 17(') is constant). the sequence of estimators {On} is n 1/ 2-consistent, satisfies (AS), and
va
3. \\lain results We define the weighted residual process (:U)
x ER,
where I(X E C) denotes the indicator function. Under Ho, An(·) takes the form Bn(x) = lJIl ( ; : ) - n- I / 2 (m(X, ; On) - m(X t ; ( 0 )) It X, $ x), where B n(·) is defined in (0.5),
2:;'=1
I. - Under assumptions (Al)-(A5), the processes BnO converge in distribution as to the centered Gaussian process
TIlEOREM
n .....
00
(:1.2)
x ER,
where
(:1.:1 ) with
D(x) =
'V(.)
i~
(1
d(W
0
F),
x E R,
a standard Wiener process,
(a.4) and where ~ = (6, ' .. I ~ 1\ ) T is normal with mean 0 and variance matrix r 0, and where for all x the rv' .\' ~ and D(x) have J( x 1 covariance matrix
E(~ D(x)) = ho(x) = [~
(1
The covariance function E(B(:r.dB(X2)) of the Gaussian process B(.) is
Remarks. - (I) - In the case where On is the conditional mean-square estimator and where conditions IS of 18J hold, (3.5) takes the form ho(x) = VO- 1 J~oo (12 V'mo dF. (2) - In the case where F(·) has a density 1('), iJ(x)
538
= i~ 11/ 2 dW (T
1 2 /
- 90(x)T ( [ :
dW).
Limiting distribution of weighted processes
=
Example. - Testing for no effect. In this case, m(x; e) e for all x and with mean 0 and variance J~(X) a 2 dF, and the covariance of B(.) is
e, ~ denotes a normal
rv
G(x 1\ y) - G(x) F(y) - G(y) F(x) + G(+oo) F(x)F(y).
=a
is constant, this covariance reduces to a 2 [F(x) /\ F(y) - F(x) F(y)]; therefore, B(.) is then a Brownian bridge. When the function is constant, it is possible to modify the process AnO into a process AnO such that under Ho the AnO's converge in distribution to B(·) (see (3.3». We consider a sequence {qn : n ~ I} of integers 1 ~ q" ~ n increasing to 00, such that lim" --+no q,.j n = f'i" and for each n a qn -subsample {Xii' , .. , Xi q n } of the n-sample {Xl, ... , X n}, where the integers 1 ~ i 1 < ... < i qn _< n are independent of the sequence {Xd· Letting Bn = Oq n denote the conditional mean-square estimate of based on this subsample, we define
If a(x)
aO
e
n
(3.7)
An(x)
=
2 n- 1/
L
(Xt +1
-
m(Xt ; Bn ))I(Xt ~ z ),
x E R.
t=l Under Ho, A n(·) reduces to a modification iJ,,(x) of Bn(x) with On replaced by
B".
2. - Suppose that the function a(.) is constant and that f'i, = 1/2. Then. under assumptions (Al)-(A5) and ([8], B, or [7], [S]-[M]-[N]), the processes B..(-) converge in distribution as n -+ 00 to the Gaussian process B(x) = a W(F(x», x E R. THEOREM
4. Possible tests based on these results (1) - Two possible tests are a Kolmogorov-Smirnov type test based on the statistic S" = sup .rER a Cramer-von Mises type test based on the statistic
IA,. (x) I and
5"
jna .4.;,(:1:) w(F,,(:l'»
=
dF,,(:r.),
-00
where w(·) is a weight function and F" is the empirical d.f. of the sample. Theorem I implies that. under Ho, 5 .. converges in distribution to 5 = sUPrER IB(x)l, and 5.. converges in distribution to iJ2(F- 1 (t» w(t ) dt. If a(·) is constant and f'i, = 1/2, then 5" -+ aSuPtE[o,ljIW(t)l, and 5.. -+ a W (t )w(t) dt (for the tabulation of these tests, see [2]). When a(·) is not constant, we can use [9] ~o obtain a tabulation of the test based on 5". (2) - Another po~sibility is to make use of the the Karhunen-Loeve expansion of the Gaussian process ir(·) = (B 0 F- 1 ) ( . ) defined on the unit interval [0,1],
J;
J/
(X)
B*(·)
= L Ay2 z, Ii('), J=1
°
where Al ~ A2 ~ .. , ~ are the eigenvalues of the covariance operator of iJ* on L2[0, form an orthonormal basis of eigenfunctions of this operator, and the rv's
z, = Aj1/2(11
iJ*(t) !;(t) dt)
1], fll h, ...,
areiidnormal N(O,l).
As discussed, e.g., in [12] in the setting of goodness-of-fit tests for parametric models of distribution functions, it is possible to choose a statistical test of the form T;! Z;.i for some moderate
= Ef=l
539
I. Dicbolt, N. la'.b and I. Ngatchou Wandji
.J > \, where Z".j = >..-;\/'1. J::", A"t1J)fjtF(y))dF(y). The rv's Tj converge in distribution under lin to 7.1 = L~= 1 Z}, a chi-square with .J degrees of freedom, Note remise lc 12 mars 1997. ucceptee apres revision Ie 20 mai 1997.
References III An II. Z. and Ch\'l1~ n., 1991. A Kolmogorov-Smirnov type statistic with application to test for nonlinearity in time series. I",. SIal. R"I'.. ~lJ. pp. 2117-307. 121 Ul\'hull J. IIl1d IMi') N., 1995. Nonparamctric tests for correlation or autoregression models under mixing conditions. C. R. A.../Il. Sd. Paris, s~ric I, 320. pp. 1135-1139. PI UldllIll J., I.u\') N. and Nl:lllChuu Wundjl J., 1997. Limiting functional distribution of weighted processes of residuals. Apphcution to j:ool.lness-of-fit tests for parametric models of nonlinear autoregressive. Manuscript. 141 Dullo 1\1., S\'lIoussl It. and Touall A., 1991. Proprietes usyrnptotiqucs presque sOres de l'estimateur des moindres carres d'un modele 1I11torcgressif vectoriel, Ann. lnst. IIl'11ri Poincare, serie B. 27. pp, 1-25. 151 KlIlIIl\o I.. A. and Nelson I'. I., 1978. On conditional least squares estimation for stochastic processes. Ann. SIal.• 6, 3, pp. 629-642. (611.111 T. I... 199~. Asymptotic properties of nonlinear least squares estimates in stochastic regression. Ann. SIal.• 22. pr. 1917-1930, I7ll\1alll:l'lIs 1\1. and \'ao J. 1"., 1996. Sur l'estirnateur des moindres carres d'un modele autoregressif fonctionnel. To appear in C. R. A/'I/I/. Sci. Paris, serie I. 1111 I\lcKl'lIl:lJl' I. W. and Zhan~ 1\1. J., 199~. Identilication of nonlinear time series from first order cumulative characteristics, A"". SIII/., 22. pp. 495-514. (lJI 1'lIl'rhllr~ V. I., 1996. A.vymplotic MI'lhods in the Theory of Gaussian Processes and Fields, Translation of Mathematical Mllfllll!rnphs, 14K. American Mathcmntical Society: Providence. 1101 Sh'lIhells 1\1. A., 1978. On the half-sample method for goodness-of-fit, J. Roy. SIal. Soc., 40, pp. 64-70. 111I SU J. Q. und Wel I.. J.t 1991. A lack-of-til lest for the mean function in a generalized linear model, J. Amer. SIal. !\.I.WI('.. llfl. pp. 420-426. 1121 Shorlll'k (;. and Wellnl'r J., 1996. Elllpiri('(/Ip'lwe.un with application 10 statistics, Wiley: New York.
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