Kolmogorov-Smirnov Tests for AR Models Based on Autoregression Rank Scores Author(s): Faouzi El Bantli and Marc Hallin Source:

Lecture Notes-Monograph Series, Vol. 37, Selected Proceedings of the Symposium on Inference for Stochastic Processes (2001), pp. 111-124 Published by: Institute of Mathematical Statistics Stable URL: http://www.jstor.org/stable/4356146 Accessed: 28/07/2010 06:06
Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=ims. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.

Institute of Mathematical Statistics is collaborating with JSTOR to digitize, preserve and extend access to Lecture Notes-Monograph Series.

http://www.jstor.org

Ill

Institute LECTURE

of Mathematical ?

Statistics SERIES

NOTES

MONOGRAPH

Kolmogorov-Smirnov Based on Autoregression

Tests

for Rank

AR

Models Scores

Faouzi

El Bantli1

Marc

Hallin2

I.S.R.O. Universit? Libre de Bruxelles Belgium and

Brussels,

E.C.A.R.E.S., I.S.R.O., de Math?matique D?partement Libre de Bruxelles Universit? Brussels, Belgium

Abstract of the Kolmogorov-Smirnov for the patype are constructed rameter of an autoregressive model of order p. These tests are based on autoregression rank scores, and extend to the time-series context a method rank scores and proposed by Jureckov? (1991) for regression models with independent observations. Their asymptotic regression distributions are derived, and they are shown to coincide with those of classical as under null hypotheses statistics, Kolmogorov-Smirnov well as under contiguous alternatives. Local asymptotic efficiencies are A Monte Carlo experiment is carried out to illustrate the investigated. of the proposed tests. performance Keywords: sion rank Autoregressive scores, models, Autoregression test, Local quantiles, asymptotic Autoregresefficiency. Tests

Kolmogorov-Smirnov

Introduction the model of order

Consider

autoregressive Yt = 0iit-i

? -h et, tez (Li)

+ ? ? ? + ?pyt_p

iARip) vector

where ? > 1 is a fixed integer, model), of unknown autoregressive coefficients,

? = (0i,...,0p)'ap-dimensional and {et, t 6 Z} a process

of

1 Research supported by the Fonds d'Encouragement ? la Recherche de l'Universit? Libre de Bruxelles. 2Research supported by an A.R.C, contract of the Communaut? fran?aise de Belgique and the Fonds d'Encouragement ? la Recherche de l'Universit? Libre de Bruxelles.

112

EL BANTLI

AND

HALLIN

independent cumulative

and identically distributed distribution function F and = 0

random probability :=

variables, density

with

unspecified

/ satisfying < oo.

? xdFix)

and

0 <

s)

f x2dFix)

(1.2)

Letting

:= jo

G Rp

zp

-S???*^

0 W G C, \z\ < 11,

(1.3)

we shall sumption)

assume

: {et, of the ARip) density, process by Yn := (Yi,..., Yn)' a {Yt, t G ?}. Denote realization of length ? of some solution of (1.1); we do not require Yn to be since all solutions are asymptotically stationary, stationary (see Hallin and Werker for a detailed Assume discussion on this issue). furthermore (1998) that in case they are not, they safely Yq) also are observed; (Y-p+i,..., be put equal to zero without results. asymptotic affecting Koul and Saleh (1995), to the time-series context Koenker extending Bassett quantit? concept (1978)'s for model (1.1) := (?Sn)(a),?(n)(a)) problem of regression quantiles, as the solution defined the a-autoregression can and

throughout t G Z} then

paper that is the innovation

the

? G ?

as(the usual causality and / the innovation process,

?(n)(a) of the

e?n)(a)

G R,

0(?)(a)

6 RP

minimization

?(n)(a)

:= arg

min

?a (Yt

?0

?',.,?),

(1.4)

where

the minimum Qa(u)

is taken

over all ? = (0o,0)

such

that

Oo ? ? and ? G IP, a G [0,1],

:= \u\ {al[u

> 0] + (1 - a)I[u t = 0, ...,n quantile of the can optimal 1.

< 0]} , u G R,

and

xt := (Yi, ..., Yi-p+i)', This a-autoregression G R^1 (ojn),0(n)) linear program alnr+

be

obtained

as

the

component ? R2n+P+1

?(n)= of the

solution ^(n),f+,f")

-f (1 - a)lnr~ = r+-r-

:= min

Yn-Mo-XnZ r*

, 1

(1.5)

. zGf,

eK}.,

0<a<

K-S

TESTS

FOR

AR

MODELS

113

where

for the ?-dimensional ln stands with rows xt, 0 < t < ? ? 1. matrix dual program The corresponding Yna

vector

(1,...,

1) , and

Xn

is the

? ? ?

is

:= max -

l'na

= n(l

a) (1.6)

X^a

= (1 - a)X'nln 0<a<1 and Gutenbrunner (1995), the (1992), to adapted n = ? the solutions (a) later a-autoregression that, for all? rank = 1,..., ?

k aG[0,l]n, Following the idea of Jureckov? context (an, duality

the autoregressive (a), (?i scores. The and ...,a^ formal

by Koul

and Saleh

0 < a < 1 of (1.6) between

are called

(1.5)

and (1.6)

implies

a G [0,1], ? 1 = I [ ? for t G It \Yt = ?o(a) by the 0 < a equality < 1} if Yt>?oia)+*'t-l?{n)ia)

M ?tn (a)

if

yt<?0(a) ?

+ xU?(n)(c) L the of ?(n)(a) sample and such are paths that

while,

+ ?'^?? constraints

(a)

components Clearly, linear, the

determined {?n (a),

in (1.6).

are

continuous,

piecewise

atn (0) = of Koenker of the

of the algorithms modification 1, and ?t" (1) = 0. An obvious efficient and d'Orey computation (1987 and 1994) allows for an n solutions ? (a) and ? (a) over the whole interval [0,1]. property i.e., denoting of autoregression by atn (a, Yn) + ???) from rank scores is their of (1.6), (?,?) autoregressionthe fact that

A crucial invariance,

the solution

???)(a,?? which tween immediately

+ ?1? follows

=???)(a,??),

gRp+1,

(1.7)

berelations Some further algebraic (1.6). rank and the corresponding quantiles autoregression autoregression in Lemma scores are provided 2.1 of Hallin and Jureckov? Quite (1999). no preliminary in order to compute estimation of ? is needed remarkably, with the more rank score statistics. This is in sharp contrast autoregression familiar aligned rank methods (Hallin and Puri, 1994), where ranks are comand Gutenbrunner from estimated see Jureckov? residuals; puted (1991),

114

EL BANTLI

AND

HALLIN

Jureckov? Harel merical and

Gutenbrunner (1992), Puri (1998), or Hallin new

et al. (1993), Hallin et al. (1997a, and Jureckov? for details (1999) tests based on

1977b), and nu-

Kolmogorov-Smirnov of these tests is investigated The asymptotic behaviour in Sec(1.1). tion 3, where we show that the limiting of the test statistics distributions coincide with those of the classical both unstatistics, Kolmogorov-Smirnov der the null hypothesis alternatives. Our results extend as under contiguous model those The the with of Jureckov? local (1991) from asymptotic efficiency of the proposed performance Normal, Laplace and regression of these tests models tests to autoregression is also investigated. on simulated respectively. models. Finally, AR series

applications. In the present paper, version of the traditional

a autoregression rank score statistic are introduced for

is illustrated

Cauchy

innovation

densities,

Limiting Statistics

Distributions Based the density on

of

Kolmogorov-Smirnov Rank Scores

Autoregression

Assume

that

(1.1) ties satisfying Jureckov?, (FI) fix)

remains

unspecified (1.2) and 1999) :

/ of the innovations the family ? within the following

in the autoregressive model of exponentially tailed densifrom Hallin and conditions (borrowed

is positive /'

for all

G R,

and

absolutely X(/)

continuous, := / J

with

a.e.

derivative

and finite

Fisher

information

there exists < oo; moreover, two bounded derivatives, / (F2) is monotonically decreasing > 0, r = r/ > 1, bf /

Kf > 0 such that, and /", respectively; to 0 as a: ?y

[ J fix)dx J \fix) for all \x\ > Kf, f has

?oo

and,

for some

b =

x-?-oo We will focus on the

-y(*>=Um-'?e('-f(*)) x-KX) 6|?|G of testing 0(!) null

= 1 b\x\r hypotheses of the form

problem

%n : ?? = 0, against alternatives ?" Such (see tests Garel : ?? f 0,

:= (??,...,??-?)unspecified,

0(i)

:= (0?, ...,0?-?)

unspecified. identification process

play a crucial role, and Hallin 1999).

for instance,

in the order

K-S

TESTS

FOR

AR

MODELS

115

Write

the

AR{p) Yn

model = Xn0

(1.1)

as + Xn;20p with t = + ??? (2.1) 1 < t < n, and e? :=

+ en = Xn;10(l) is the (hence,

where Xn;2 (e?,...,

Xn := :=

? Xn;i:Xn;2

(? ? ?) matrix Xn^t = Yt-P,

rows 1,...

x'^, ,n),

(?-?+?,.?.,^^)' e?)'. Denote

by Pn := Xn;l(Xn;lXn;l) W1 onto the Xn;l linear space spanned by the

the

matrix projecting (random) of Xn;i. columns Define Xn;2 := PXn;2 and Xn;2

:= Xn;2 - Xn;2 ~ [Xn\2 = [I-Pn](Xn;2-Xn;2ln),

? ^nfl)

*n

where n-p ^-n-^X^^n-1^ t=l and D\ let := n"1 X?;2 = ?~X (Xn;2 ??;21?)' [ln Pn] (Xn;2 *n;2ln) ? of t=-p+l Yt=:Y and X?n.2 := n"1 ?X?;2it, t=l

(x?2)'

solution the autocovariances of the stationary Denoting by 7fc(0), A; = 1,... of empirical autocovariances under ARip) dependence (1.1), the consistency in probability, as ? -> oo, to that D2 converges implies I D2:=7oW-(7p-lW,---,7xW) V 7,-20) which, in view of classical Yule-Walker p-l ?2 = 7o(0) 5>7i(*) 1=1 depend = s}, nor on the equations, ... under ?? t? W J to 70 W 71W 71W ToW ??? ??? ??-2(T)?1

reduces

a simple innovation Let rank

scale

factor /.

that

does

not

on 0,

shape

of the

?n

density (a) =

scores

(?(1?)(a),...,??? under ??, computed Yn =

0 < a < (a)J, i.e., corresponding Xn;10(l) + ?n

1, be

the

autoregression submodel (2.2)

to the

116

EL BANTLI

AND

HALLIN

n of course ? does not depend (though (a), as a statistic, the process : 0 < a < 1} defined n, consider by {Tnia)

on 0(i))?

For each

Tnia)

:= rCll2D~l

???;2^?)(a), t=i

0 < a < 1.

(2.3)

has trajectories in the space of continuous functions process C[o,i] h> c(a), with the Borei s-field a G [0,1] (as usual, is equipped C[0>i] with the uniform C associated metric ||ci ? c2|| := maxo<a<i lci(a) ~c2(?:)|). a Following sistently functionals Section refer V.3.2 of H?jek original and Sid?k to the (1967) define on we con[for convenience, the continuous (C[o,i],C)

This

edition],

h+ (c(?)) The one-

:=

max 0<a<l

c(a)

and

?? (c(?))

:=

max 0<a<l

|c(a)|.

and two-sided

statistics

rank score-based autoregression we are considering in the problem of testing

Kolmogorov-Smirnov Wq are

K+

:= ?+ (T?(.))

= n-^D-1

max ?^lt-l

J^w(o)

(2.4)

and K? := ?* (Tn(?)) The = ?"1/2!)-1 max xn\2tt?tn U<Q<1 S t=l the asymptotic (a) , (2.5) of

respectively. K+ and ifj^ Theorem

following under the null Assume that

results hypothesis (Fl)-(FS)

provide ?%.

distributions

2.1

are satisfied.

Then, *

under

Wq,

= lim v { n-?oo P(ff+ n <x) ' <?*.>-{; and

?? 1-

.o2. exp(-2ar)

? J ?? > 0

v (2-6) '

?;<0 l-2f)(-l)*-1exp(-2ibV) ?0 The proof is based 2.1 Define on the following lemma. x>0 (2.7)

Lemma

the scores 0 na - na Rn]t -1 Rn? < Rntt 1 (2.8)

a\ia)

:=

?^ 0

<na<Rn;t < na ,

K-S

TESTS

FOR

AR

MODELS

117

where

Rn-t denotes are satisfied, let

the rank ofet

among

e?,...

,e?.

Assuming

that

(F1)-(F2)

?t(a):=7[et>F-1(a)], Then,

1 < ? < ?,

O < a < 1.

-1/2 sup ? 0<a<1 and

(2.9) t=l

sup 0<a<1

? -1/2

S [(*??* ?=1

*?;2)

4?? (a)

???2,??]

?,

(2.10)

as ? ?>? oo. bution Proof

Moreover,

in (C[o,i],C) of Lemma

: O < a < 1} converges the process {??(a) : 0 < a < 1}. to ?/ie Brownian bridge {Zia) 2.1. The

in distri-

of application convergence (2.9) is a direct The approximation on results Theorem 3.2 of Hallin and Jureckov? (1999). the rank score process then in H?jek (1965) also Jureckov? 1999) given (see : 0 < a < 1} follows from of {Tn(a) The asymptotic behaviour yield (2.10). the fact that

0 < a < ?D-'n-y^X^a^ia), to the Brownian converges weakly V.3.5 of H?jek and Sid?k (1967). Proof we have, Theorem for positive 2.1. From ? and lj bridge {Z(a)} : see Theorem 1 in Section D V.3.3.a of H?jek and Sid?k (1967),

Theorem

y and

all 0 < a < 1, = 1 - exp(-2zy)

? [Zia) and

< ?{1 - a) + ay]

? [-x(l

-a)-ay<

Zia)

< x(l

a) + ay)

= 1-2

?(-l)*+1 k=x follow

exp(-2A;2:n/).

The

asymptotic ? = y. Thus, ability the

distributions

(2.6)

and

(2.7)

directly

from

letting D

one-sided a whenever

test K+

based

on K+ than

rejects the

Hq critical

at (asymptotic) value ?? flogen

prob,

level

is larger

118

EL BANTLI

AND

HALLIN

while K?

the exceeds

two-sided the

based

on K^ ?(a)

a-quantile

level a whenever rejects ?? at probability := Q"~1(a) of the distribution function

Qix):=2J?i-l)k-lexvi-2k2x2). fc=l Note that both Q(x) has been tabulated by Smirnov (1948). are entirely as they do not depend neither distribution-free, on / nor on the nuisance 0(X). We now turn to a study of the power of the tests based on K+ (the case of K? follows The following lines, and is left to the reader). along similar result provides local alternatives of the form asymptotic powers against critical values Un : 0p = := ?~1/2t, 0p 2.2 Assume t G R, that (F1)-(F2) ?(1) = (0lf ...,0?_?) G R^1 Let 1/21 ??(T(?)\t) where P^n is computed :=P$n iC>(4loea) (0(i),0?). Then, for any 0 < a < 1, unspecified. This function

Theorem

are satisfied.

under f, r)

?? :=

= Bia, Urn - v n->oo pn(0(i)? > r) := ? where ma* (z(u) stand +

1 ?2/i(Fr1(u))X-1/2(/)r} for the standardized > (?loga) , (2.11)

versions of f and F, respectively. ~ 0 (the notation means that the ratio Furthermore, for r ?> ? (t) ?(r) tends to one as t ?? 0; F, as usual, stands for the standard normal ?(T)/C(T) distribution function), /i ? 2s/?"1/2(/)a

and F\

Bia,f,r)-a~

G-?

log

a)

<f>iu,f)il>ia,u)du\

t, (2.12)

with ^/):=-J/(F-i(u)) :=2F il (--loga)

and

1/2 Vl? (2u-l)(u(l-ti))-

?{a,?)

-1/2

-1.

Proof. Sections appearing

This

VI.4.5

proof, as well as the proof of Theorem and VII.2.3 of H?jek and Sid?k (1967); there here are to be taken as ct = n_1/2Xn;2)t (hence,

3.1, heavily the various

relies constants

on

dt = n~ll2rYt-p,

c = 0),

pi = 1,

K-S

TESTS

FOR

AR

MODELS

119

and 62 = ?im Z^rV-1 (Yt.p ? t=l Sid?k (1967) ?)2 = r2I(/)4 under the

Theorem cess

VI.3.2

in H?jek

and

entails

that,

Hn,

pro-

Tn(a)-f(F-\a))DnI-V2(f)T converges in distribution and to the Brownian bridge = K+ Since {Zia)}. that in turn implies (2.6)

maxo<a<iTn(a) Pn(0(i)

?T) converges,

D2 = s2 + op(l), as ? ?? oo, to

equation

amaxi(z(u)

/(F-1(u))a/X-1/2(/)r)

> (-\\oga)'

= As for for any a; (2.11) follows from noting that ajf(F~l{u)) fi{F{l(u)). 1 the linear approximation of Theorem consequence (2.12), it is an immediate in Section D of H?jek and Sid?k (1967). VI.4.5

Local onK+ ?

Asymptotic

Efficiency

of

the

Test

Based

Pitman to the spect

asymptotic

relative

efficiencies

of the tests multiplier

based

corresponding Lagrange to the autoregression rank score test proposed cannot be computed as easily as in the usual case of asymptotically (1999) normal or chi-square as ratios test statistics, for which AREs are obtained of noncentrality the analytical form of asymptotic parameters. Actually, for given 0 and t does not allow for any simple and AREs powers, result, typically classical dent tion denote will case on a, /, 0 and r. This problem depend already of Kolmogorov-Smirnov tests for linear models ARE see H?jek and Sid?k (1967), results can be obtained Section from VII.2.3. the linear

Gaussian

respect or with reprocedures, by Hallin and Jurckov?

on K+

with

in the appears with indepen-

observations; local However, (2.12) of the

by e(a,/,0,r) to the locally Gaussian instance, optimal from Section VII.2.3 of H?jek and Sid?k e(a,/):=lime(a,/,0,r) does mate not on 0 and r anymore,

approximaas t -> 0. More precisely, power asymptotic Bia,f,r) the ARE of the test based on K+ with respect, for : the following shows that (1967), test result, inspired the limit

(3.1) and thus allows for local or approxi-

depend

comparisons

of asymptotic

performance.

120

EL BANTLI

AND

HALLIN

Theorem ?* with

3.1 respect

Under

the local asymptotic relative (F1)-(F2), to the locally optimal Gaussian test is 1

efficiency

of

i(a,f)=4na2(-loga)l where denotes

(f)exp(zl) the (1 ? a)-quantile

\ j

f{?,f)ip(a,u)du distribution.

(3-2)

za

of the normal

Moreover,

lim eia) a-?) Proof. D under instance, we have ^exp(?\x\), For The proof

X_1(/)

[ / [Jo

<?(u, /)sgn(2u

1) dui

(3.3)

readily

follows

from Theorem

VII.2.3

of H?jek

and Sid?k

(1967).

the

Laplace

(double-exponential)

density

fix)

0(n, One can easily -i / so that (3.2) </>(u,/)sgn(2u becomes = check that

/)

= sgn(2u

1),

0 < u < 1.

l)dti

= 1 - 2F

(-(-loga)1/2)

?a,

e(a)

47r(-loga)exp(z?_Q)

[a

- 2F

(-(-loga)1/2)]

. still involves

In general, the local asymptotic however, actual innovation density /. If we apply to (3.2), we obtain the uniform (in /) upper the := supe(a,/) fer fact that lim ???oo it follows that lim 4pa2(-loga)exp(z?_a) Q->0 hence lim eia) a??() can easily check =0.844. and?(O.l) One by computations ? 1. 2p?2???(a:2)(1 = 4pa2(?

efficiency e(a, /) the Cauchy-Schwarz bound

inequality

e(a)

loga)exp(z2_a)

/ Jo

rpia,u)2

du.

Using

the

F(?))2

= 1,

= 1,

that

e(0.01)

= 0.864,

e(0.05)

= 0.852,

K-S

TESTS

FOR

AR

MODELS

121

4 The

Simulation null hypothesis

Results 0 = 0 has been Yt = 0.5?-? considered 0.2Yt_2 in the model

AR(3)

+ 0Yt_3

+ et.

(4.1)

More

precisely,

= 2000

generated ? = 0.2,

normal, two Kolmogorov-Smirnov K^), respectively), a quadratic form relative reported

by (4.1) with 0 = 0.3, and standard Laplace, and Qc;

of the AR(3) ? = 100 series of length replications initial values YL2 = Y-\ = *o = 0, 0 = 0, 0 = 0.1, 0 = 0.4, and three innovation densities (standard and standard tests to the described Gaussian and have been subjected to the logistic) in this paper on K+ and (based test (based on Lagrange multiplier For each of these tests, 1999). levels a = 5% and a = 1%) are

see

Garel

Hallin

rejection frequencies in Table 1.

(at probability

of the 0 = 0 column of Table 1 reveals that the three tests Inspection considered all are rather thus biased. The bias seems less conservative, for the test based on K+ than for the K? and Qc ones. Since severe, though, the same one-sided test based on K+ is also significantly more powerful, under all alternatives considered here, than the two-sided the two-sided Under procedure might on K+ is slightly more powerful than the Gaussian even under Gaussian test, Lagrange multiplier innovations, the local asymptotic of the latter. Under larger values of despite optimality of K+ over its competitors is clear under the three densities 0, this advantage considered, but Though ertheless indicates sided, based on particularly more extensive marked simulations under densities?as Laplace should be undertaken, expected. Table 1 nevKolmogorov-Smirnov 0 = 0.1, the test based one based on K?, well be non admissible.

asymptotically

that Kolmogorov-Smirnov the onetechniques, especially nonlinear test statistics, can be expected to beat locally tests based on linear statistics. optimal

The authors thank Acknowledgement. referee for their sawa, and an anonymous and helpful comments.

Professor Editor, careful reading of the

the

Ishwar

Ba-

manuscript

References ?., and M. Hallin [1] Garel, Journal of the American [2] Gutenbrunner, regression Rank-based (1999). Statistical Association AR 94, order identification. . scores and

1357-1371 rank

C., and J. Jureckov? Annals

quantiles.

(1992). Regression of Statistics 20, 305-330.

122

EL BANTLI

AND

HALLIN

[3] Gutenbrunner, Tests of linear Nonparametric [4] H?jek, regression J.

J. Jureckov?, based hypotheses C., Statistics

R. Koenker, on regression 2, 307-331. of the

and rank

S. Portnoy (1993). scores. Journal of

Extension (1965). In alternatives.

Kolmogorov-Smirnov

test

to

the

(1813) and J. Neyman, [5] H?jek, J., and New York. [6] H?jek, edition). [7] Hallin, J.,

: Proceedings

Bernoulli Bayes Laplace (1763), (1713), Research Seminar Le Cam an International (L. of eds), 45-60. Berlin, Springer-Verlag. (1967). Theory of Rank Tests. Academic Press,

Z. Sid?k

Z. Sid?k, and P.K. Sen (1999). Academic Press, New York.

Theory

of Rank

Tests

(2nd

and J. Picek, J. Jureckov?, J. Kalvov?, with applications tests in AR models, Nonparametric Environmetrics 651-660. 8, M.,

T.

Zahaf

to climatic

(1997). data.

J. Picek, and T. Zahaf (1998). Nonparametric M., J. Jureckov?, [8] Hallin, time series based on autoreof two autoregressive tests of independence and Inference rank scores. Journal 75, Planning of Statistical gression 319-330. [9] Hallin, models M., and J. Jureckov? based on autoregression Optimal (1999). rank scores. tests Annals for autoregression 27, of Statistics

1385-1414. [10] rank tests for linear models Hallin, M., and M.L. Puri (1994). Aligned errors. Journal with autocorrelated 50, 175Analysis of Multivariate 237. Hallin, M., and B.J.M. Werker models and (1998). : from adaptive Ghosh, Optimal Gaussian tests. Ed.), testing for semi-

[11]

parametric autoregressive rank scores ers to regression parametrics, New York. [12] Harel, M., rank score processes. [13] Jureckov?, gression and Time

Series

(S.

multipliLagrange NonIn Asymptotics, M. Dekker, 295-358.

and related quantiles Autoregression (1998). random coefficient for generalized autoregressive processes and Inference Journal 68, 271-294. Planning of Statistical and M.L. Puri of Kolmogorov-Smirnov In Transactions of the 11th Statistical Ed.), 41-49. Decision Academia, Tests type based on re-

J. (1991). rank scores. Theory, Visek,

Information cesses (J.A.

on Conference Prague Proand Random Functions, Prague.

K-S

TESTS

FOR

AR

MODELS

123

[14]

J. (1999). Jureckov?, Regression natives. Bernoulli 5, 1-18. Koenker, R., and G. Bassett 46, 33-50. Koenker, R., and V. dOrey Journal

rank

scores

against

heavy-tailed

alter-

[15]

(1978).

Regression

quantiles.

Econometrica

[16]

quantiles. regression 383-393. 36, [17] Koenker,

(1987). of the

AS Algorithm Royal Statistical

229

: Computing Series Society

on Remark AS R92. A remark R., and V. dOrey (1994). and regression dual regression AS 229 : Computing quantiles Algorithm Series C 43, 410rank scores. Journal of the Royal Statistical Society 414. Koul, H.K., and A.K.Md.E. rank scores processes. related Smirnov, pirical Saleh Autoregression (1995). Annals 23, of Statistics the quantiles 670-689. and

[18]

[19]

N.V. Table for estimating (1948). distributions. Anna/5 of Mathematical Marc Hallin

of fit of emgoodness Statistics 19, 279-281. El Bantli

Faouzi

mhallin@ulb.ac.be Institut de Statistique

elbantl@ulb.ac.be et de Recherche de la Plaine de Bruxelles Brussels CP210 Op?rationnelle

Campus Libre Universit? B 1050

Belgium

124

EL BANTLI

AND

HALLIN

? = 0.3 ? = 0.2 ?=0 $ = 0.1 a = 5% a = 1% a = 5% a = 1% a = 5% a = 1% a = 5% a = 1% Normal /C+ Laplace Logistic Normal ? * Laplace Logistic Normal Laplace Logistic 0.035 0.038 0.043 0.041 0.036 0.050 0.035 0.031 0.046 0.006 0.009 0.008 0.003 0.007 0.005 0.003 0.003 0.006 0.002 0.374 0.525 0.377 0.248 0.462 0.263 0.335 0.324 0.221 0.155 0.226 0.176 0.131 0.174 0.138 0.144 0.112 0.102 0.578 0.854 0.675 0.403 0.661 0.477 0.483 0.442 0.402 0.011 0.238 0.452 0.287 0.153 0.325 0-185 0.224 0.211 0.147 0.011 0.708 0.873 0.769 0.604 0.785 0.654 0.723 0.733 0.683 0.011 0.428 0.663 0.514 0.324 0.554 0.382 0.366 0.521 0.383 0.011

: 5% a = 1% 0.908 0.982 0.934 0.845 0.955 0.883 0.854 0.731 0.868 .010 0.753 0.915 0.797 0.634 0.843 0.714 0.747 0.742 0.783 0.011

Qc

standard errors

? = 0 for ? = 100, 1. Rejection of the null hypothesis frequencies innovation densities and Logistic standard /, reNormal, Laplace, = 5% and a = 1%. The number and for various values of T, a spectively, for each errors are provided standard of replications is ? = 2000; maximal Table under column.