On testing for the mean vector of a
multivariate distribution with
generalized and 2-inverses
Duchesne, Pierre and Francq, Christian
Université de Montréal, Université Lille 3
January 2010
Online at
https://mpra.ub.uni-muenchen.de/19740/
PierreDUCHESNE y
Département de mathématiquesetstatistique
Universitéde Montréal
Christian FRANCQ
EQUIPPE-GREMARS
UniversitéCharlesde Gaulle- Lille3
Deember15, 2009
Abstrat
Generalized Wald's method onstruts testing proedures having hi-squared limiting distributions
from test statistishaving singular normal limiting distributions by use of generalized inverses. In
this artile, the use of f2g-inverses for that problem is investigated, in order to propose new test
statistis with onvenient asymptoti hi-square distributions. Alternatively, Imhof-based test
sta-tistis an also be dened, whih onverge in distribution to weighted sum of hi-square variables;
Theritialvalues ofsuhproedures anbefoundusingImhof's(1961)algorithm. Theasymptoti
distributions of the test statistis under the null and alternative hypotheses are disussed. Under
xed and loal alternatives, the asymptoti powers are ompared theoretially. Simulation studies
arealsoperformedto ompare the exat powers oftheteststatistis innite samples. A data
anal-ysis on thetemperature and preipitation variability inthe European Alps illustrates theproposed
methods.
Key words and phrases: f2g-inverses; generalized Wald's method;generalized inverses; multivariate
analysis; singularnormaldistribution.
Mathematis subjet lassiation odes (2000): primary 62H10; seondary 62F05,62H15.
1.INTRODUCTION
LetT
n
=(T
n1 ;:::;T
np )
>
,n1, be a sequene ofstatistis, and introdue Z
n
()=n
1=2 (T
n ),
=(
1 ;:::;
p )
>
. Thelassial testingproblemonfronts hypothesesH
0
and H
1 :
H 0
:=
0 ;
H 1
:6=
0 :
(1)
Abbreviated title: "Testing for themean vetor ofa multivariatedistribution".
Correspondingauthor: Pierre Duhesne, UniversitédeMontréal;Département demathématiqueset
de statistique; C.P. 6128 Suursale Centre-Ville; Montréal, Québe H3C 3J7; Canada. tel: (514)
343-7267;fax: (514)343-5700;e-mail: duhesnedms.umontreal.a.
y
Thisworkwassupportedbya grantfromtheNaturalSiene andEngineeringResearh Counil
Underthenullhypothesis,itisassumedthatthesequeneofteststatistisonvergestowardanormal
distribution: Z n
:=Z
n (
0 )
d
!N
p (0
p
;)asn!1,where0
p
=(0;:::;0) >
isthep-dimensionalnull
vetor. Theovariane matrix6=0,possiblysingular, is supposedtobeunknown. Based on
sam-pledata,thehypothesistestingproblemistodeide whetheritseemslikelythatthenullhypothesis
to be true. To study limitingalternative distributions, under thexed alternative H
1
:=
1 ,we
supposethatZ
n (
1 )
d
!N
p
(0; ) asn!1,andunderanysequeneof loalalternativehypotheses
H 1n
:
n
=
0
+n
1=2
n
,
n
!, we supposeinstead that Z
n (
0 )
d
! N
p
(;), with 6=0.
The problemdesribed above represents a general framework for manystatistial problems oming
frommultivariate analysis, non-linear regressionmodels,and timeseries analysis.
1.1Leading examplesof the testingproblem
LetX
i
=(X
i1
;:::;X ip
) >
,i=1;:::;n, be a randomsample froma multivariatedistribution, where
E(X i
) = and var(X
i
) = Ef(X
i
)(X i
) >
g = . The sequene of test statistis ould be
simplybasedonthesamplemeans,T
n =
X
n
=n
1 P
n
i=1 X
i
,andtheasymptotinormaldistribution
underthenull hypothesisisfound invoking themultivariate entral limit theorem. Thatframework
goes bak to the seminal work of Wald (1954). In fat, under the normality assumption, that is
X i
N
p
( ;), withthe ovariane matrix unknown but supposed positive denite, Hotelling's
T 2
test statisti represents the lassial test proedure for problem (1) and Wald (1954) simply
developed the large sample analog of Hotelling's method. See, e.g., Srivastava and Khatri (1979),
Muirhead (1982) or Anderson (1984), among others. When thesample is obtained from a singular
multivariatedistribution, Bhimasankaram and Sengupta(1991) proposed a methodology similar to
Hotelling's test statisti. If XN
p
(;)withdet ()=0,it iswell-known thatX belongs to
theolumn spae of rank r,say, of the ovariane matrix with probability one (w.p.1); visually,
the data lies in the r-dimensional ane subspae of R
p
, r < p. Furthermore, a ertain linear
transformation ofX follows a nonsingular normal distribution (see,e.g., Bilodeau and Brenner
(1999,p.62)orEaton (2007)). Thustheinitial problemanbereformulatedina smallerdimension
using a non-singular normal distribution. However, to work with the datainthe transformed sale
maybe seenasadisadvantagefroma pratialpoint ofview,and,more importantly, therank of
mustbeknownapriori,whihan be arestritiveassumption. Wald'smethod hasbeengeneralized
byMoore(1977)to sequenesoftest statistishaving singularnormaldistributions bya naturaluse
ofgeneralized inverses.
Multivariate sampling is just a simple example and the study of other test statistis may result
in asymptoti singular normal distributions. In parametri models, when is the vetor of ell
probabilities in a multinomial model and T
n
represents the vetor of observed relative frequenies,
the ovariane matrix is singular; that example has been studied in detail by Moore (1977). In
non-linear regressionmodels,underertainonditions,theasymptoti distribution oftheregression
parametersinnon-linearregressionmodelsisasingularnormaldistribution,seeRobinson(1972). See
alsoHadiandWells(1990),whogiveseveralexamplesofnon-linearmodelswithsingularinformation
matries. Another example istaken fromtimeseries analysis,where aentral problemis to testfor
residualautoorrelationsisapproximatelyasingularnormal distribution,seeBoxandPiere(1970),
Liand MLeod(1981) andLjung (1986),amongstothers.
1.2Testingproedures
In this paper, we onsiderseveral test proedures whih an be used whether is singular or
non-singular, withoutassuming normality. Morespeially,westudy thegeneral lassoftest statistis:
Q n
(W n
)=Z
>
n W
n Z
n
; (2)
where W
n
is a weight matrix. For the testing problem (1), the null is rejeted for large values of
Q n
(W n
). We disussindetailthreetesting proedures orresponding totheweightingmatries: (i)
W n
=I
p
,where I
p
denotes thepp identitymatrix, (ii)W
n
=
k n
,where
k n
representsa
f2g-inverseof
n
,andnally(iii)W
n
=
n
,where
n
istheMoore-Penroseinverse(orpseudo-inverse)
of
n
. Theestimator
n
isassumedto be stronglyonsistentfor ,thatis
n
!,almost surely.
Note that the strong onsisteny of
n
is assumed to be true under the null hypothesis, under
sequenes of loal alternatives and for xed alternatives. Our framework is general, and the test
statistisan be applied for all the testingproblems desribed above. For example,in multivariate
sampling, natural andidates would be proedures based on sample means for Z
n
, and
n ould
be the sample ovariane matrix S
n
=n
1 P
n
i=1 (X
i X
n )(X
i X
n )
>
. The test statistis inlass
(i) are simply based on the usual Eulidian norm of Z
n
. The test proedures in lass (ii) onsider
to usef2g-inverses for the weight matrix W
n
. The literatureon generalized and pseudo-inverses is
onsiderable,see, e.g.,RaoandMitra(1971)andRao (1973),amongothers. Ontheotherhand,the
important roleof f2g-inverses instatistis,more partiularly inthestudy ofquadrati forms,seems
lesswell doumented. Appliations of f2g-inverses instatistis are desribed in Getson and Hsuan
(1988). Finally,the lass of test statistis (iii) is omposed of thegeneralized Wald's test statistis
introdued in Moore (1977). See also Andrews (1987) and Hadi and Wells (1990). Duhesne and
Franq (2008) investigated diagnosti heking time series models with portmanteau test statistis
relying on generalized inverses and f2g-inverses. In their appliations, Z
n
was based on a vetor
of sample autoorrelations and
n
wasa ertain onsistent estimator of theasymptoti ovariane
matrix of the sample autoorrelations. Here the framework is onsiderably more general, and we
investigate the theoretial and empirial properties of the test statistis Q
n (W
n
) under xed and
loalalternatives.
Thepaperisorganizedasfollows. InSetion 2,wedisusstheasymptotidistributionsofthetest
statistis Q
n (I
d ), Q
n (
k n
) and Q
n (
n
) under null and loal hypotheses. The asymptoti powers
of these test statistisare ompared in Setions3 and 4, underxed and loal alternatives,
respe-tively. In Setion 5, some simulation experiments are onduted. A data analysis is presented in
Setion 6 on the monthly temperature and preipitation variability in the European Alps for the
period 1659-1999. Conluding remarks areoered inSetion 7. An Appendix gives some tehnial
2.ASYMPTOTICDISTRIBUTIONS UNDER THENULL HYPOTHESIS
For a reason that will be transparent in thenext setion, thetest statisti based on Q
n (I
d
) will be
alledImhof-basedtest.
2.1TheImhof-based teststatisti(ase W
n
=I
p )
A simple and natural test proedure leads to the study of the norm of Z
n
, namely jjZ
n jj
2
. The
asymptotidistributionofQ
n (I
p
)underthenull hypothesisH
0
followseasily invokingthe
multivari-ate entral limit theorem and a spetral deomposition of . Consider thespetral deomposition
=P P
> = P p i=1 i v i v > i withP >
P=I
p
,where=diag (
1 ;:::;
p
)andtheolumnsv
1 ;:::;v
p
of the matrix P onstitute an orthonormal basis of R
p
. The weights
1 2 ::: r > 0 and r+1 = ::: = p
= 0 represent the eigenvalues of and thus r = rank(). The hi-squared
distribution with k degrees of freedom and non-entrality parameter is noted
2
k
(). The vetor
1 p
=(1;:::;1) >
orrespondsto ap-dimensionalvetoromposedonly ofones.
Proposition 1. If Z
n =Z n ( 0 ) d !N p
(;) and
n
! inprobability asn!1, then
Z > n Z n d ! r X i=1 i 2 1 ( 2 i )+ > 0 0 := r X i=1 i N 2 i + > 0 0 ; where 0
= diag 0
> r ;1 > p r P >
. The non-entrality parameters
i satisfy = ( 1 ;:::; p ) > = 1=2 P > with 1=2 = diag 1=2 1 ;:::; 1=2 r ;0 > p r
, and the normal random variables N
i are
omponents of the singular normal distribution N
= (N
1
;:::;N p ) > N p ( ;I r p
) with I
r p = diag 1 > r ;0 > p r .
Proof: The proposition is a well known onsequene of the ontinuous mapping theorem and of
standard results on quadrati forms of Gaussian vetors. See, e.g., Rao and Mitra (1971) or Rao
(1973) for distributions of quadrati funtions in normal random variables. More preisely, it is
obtained by noting that N
n := 1=2 P > Z n d ! N = N p ( ;I r p
) and that kZ
n k 2 = kP > Z n k 2 = k 1=2 N n + 0 k 2 . 2
Underthenull hypothesis=0
p
inProposition1 and itfollows that:
Q n
(I p
)=jjZ
n jj 2 d ! r X i=1 i N 2 i ; (3) where N 1
;:::;N r
orrespond to independent N(0;1) random variables. In pratie, it is possible
to evaluate the distribution of the Gaussian quadrati form in (3) by means of the algorithm of
Imhof(1961). Morepreisely,thetest proedure basedon theweight matrix W
n
=I
p
relieson the
following steps: 1)Compute theeigenvalues
^ 1 ;:::; ^ p of n
,whih provides aonsistent estimator
. 2)Evaluatethe(1 )-quantile
( ^ 1 ;:::; ^ p )of P p i=1 ^ i N 2 i
usingImhof'salgorithm,andnally
3) the null hypothesis is rejeted when Q
n (I p ) ( ^ 1 ;:::; ^ p
). An interesting advantage of that
In a time series framework, a similar proedure has been onsidered in Franq, Roy and
Za-koïan(2005)intesting fornullautoorrelations intheresidualsfromautoregressive-moving-average
(ARMA)models. Intheirontext,theyspeiedZ
n
=n
1=2 ^
,with^=(^ (1);:::; (p))^ >
,where(h)^
denotes thelag-h residual autoorrelation, h =1;:::;p. In orderto test null autoorrelations,their
test statisti redued to the so-alled Box-Piere test statisti jjZ
n jj
2
= n
P p
h=1 ^ 2
(h) proposed by
BoxandPiere(1970),whihreliesonthesumofsquaredresidualautoorrelations. Seeforexample
Li(2004),among others. Intheir proedure, the quantiles werefound estimating theeigenvalues of
a onsistent estimator of the asymptoti ovariane matrix of the residual autoorrelations. That
strategy has been adapted in Duhesne and Franq (2008) for diagnosti heking non-linear time
seriesmodels. Notethat the useof the test statisti Q
n (I
p
) intheontext of multivariatesampling
seems to be absent from the literature, probably due to the non-standard limiting distribution of
thattest proedure.
2.2Thetest statistibasedon f2g-inverses (aseW
n
=
k n
)
Ageneralized inverse (g-inverse) ofis amatrix
~
satisfying theondition:
~
=: (4)
Itisalsoalledaf1g-inversesine (4)isusuallytherstofthefour onditionsdeningthe(unique)
Moore-Penrose inverse of(see, e.g.,Getsonand Hsuan(1988)). Ontheother hand,af2g-inverse
of is any matrix
satisfying the seond relation dening theMoore-Penrose inverse of , that
is:
=
: (5)
Whenrequirements(4)and(5)aresatised,theresultingmatrixisalledaf1;2g-inverse orreexive
g-inverse (see, e.g.,Rao (1973,p. 25)).
Note that the matrix = P P
>
is the Moore-Penrose inverse (or pseudo-inverse) of ,
where =diag (
1
1
;:::; 1
r ;0
>
p r
). For k =1;:::;r, dene the matrix
k
= P
k P
>
,where
k
=diag (
1
1
;:::; 1
k ;0
>
p k
):Thematrix
k
is always af2g-inverse,but this isnot ag-inverse
ofwhen k<r.
Empirial versions are easily onstruted. Sine
n
! almost surely, as n ! 1, a natural
estimatorof
k
relies on
k n
,where thespetraldeompositionof
n
isgivenby
n
=P
n
n P
>
n .
Whenall thenonnulleigenvaluesofaredistint,thematrix
k
isuniquelydened. However,
when some eigenvalues display multipliities,
k
is not uniquely dened, beause it depends on
the partiular hoie of the orthonormal basis in the spetral deompositions of . That aveat
is xed using projetions and the Gram-Shmidt orthogonalization proess. Consider an arbitrary
basisB =fu
1 ;:::;u
p gof R
p
. For eaheigenvalue
k
=
k
()ofmultipliitym
k
(), letV =V
k ()
be the assoiatedeigenspae withdim (V)=m
k
(). Single eigenvalues do not pose problem;
onse-quently suppose k >1. Theprojetion on V isdenoted P
V
,whihis uniquely dened. The vetors
P V
(u 1
);:::;P V
(u p
) span V sine any vetor v 2V an be expressed asv =
P p
i=1
i u
i
=P
V (v )=
P p
i=1
i P
V (u
i
). From the vetors P
V (u
1
);:::;P V
(u p
), a basis B
V
= fP
V (u
i 1
);:::;P V
(u i
k
)g of V
is extrated as follows: let i
1
be the smallest index of f1;:::;pg suh that P
V (u
i
`2f2;:::;kg,i `
represents thesmallestindex offi
` 1
+1;:::;pg suhthatP V
(u i
`
) isnot spanned
by P V (u i 1
);:::;P V
(u i
` 1
) . Using theGram-Shmidt proess, the basisB
V
is transformed in an
orthonormal basis of V. This proess allows to dene a unique ommon basis of eigenvetors for
the spetral omposition of and
k
. More preisely, we dene a unique matrix P
B suh that =P B P > B and k =P B k P > B
for all k r. The matrixP
B
will be alledtheB-eigenvetor
matrix of.
A similaronstrution holds for
n with eigenvalues ^ 1 ::: ^ p
. SinetheB-eigenvetor
ma-triesP
B
andP
n;B
ofand
n
areuniquelydened,thef2g-inverses
k = k B and k n = k n;B
arenowuniquelydened by
k B =P B k P > B and k n;B =P n;B k n P > n;B
:An algorithmisgivenin
theAppendixon the onstrution ofthematries
k
B
and
k
n;B
. Inpratie, a tolerane isneeded
to estimate the rank of , themultipliities, and to distinguish null and non null eigenvalues. The
proposedalgorithmdenesafuntion,notedA
B;k;
(),basedonatolerane. Thefollowing
Assump-tionA()isneessaryinorder to speifytheminimumdistane between the dierent eigenvalues of
.
Assumption A(). LetB =fu
1 ;:::;u
p
g be an arbitrarybasisof R
p
. Thetolerane >0 issuh
that:
C 1
: minf j
i () j ()j: i ()6= j
()g>,
C 2
: For k rank(),the appliation A
B;k;
suhthat A
B;k;
()=
k
B
isontinuous at .
Condition C
1
in Assumption A() ensures that the multipliities are onsistently estimated if is
hosen small enough. It an be seen that C
2
is satised for all but a nite number of basis B.
The following lemma is useful for establishing the asymptoti distribution of Q
n (W
n
) in the ase
W n = k n .
Proposition 2. Suppose that Z
n = Z n ( 0 ) d ! N p
(; ) and
n
! almost surely, as n! 1.
Let B =fu
1 ;:::;u
p
g be a basis of R p
. Under AssumptionA(), if krank (), it follows that:
Z > n k n;B Z n d ! 2 k ( (k) 1 ); where (k) 1 = > P B k P > B
,and P
B
is the B-eigenvetor matrix of .
Proof: Assumption C
2
inA() andthealmost sureonvergene of
n
to give:
k nB =A B;k; ( n )! k B =A B;k;
()a:s:; asn!1:
The ontinuous mapping theorem then entails that Z
> n k n;B Z n d ! Z > k B
Z. The
Ogasawara-Takahashitheoremestablishesthehi-squarelimitingdistribution(see,e.g.,RaoandMitra(1971)or
Rao(1973)): Sine
k
B
isaf2g-inverse,theonditions(
k B ) 3 =( k B ) 2 and > k B k B = > k B
holdtrivially. Finally,
k
B
belongstotheolumnspaeof
k B ,sine k B = k B
, = C
k diag ( 1 1 ;:::; 1 k )C > k
, where P
B
=(C
k C
p k
), with C
k
and C
p k
of
dimen-sionspk and p(p k),respetively. Thenumberofdegrees offreedom isk=rank(
k B )with non-entrality parameter > k B k B k B = > P B k P > B
. Thisonludes theproof. 2
Q n
( k n
) is
k
under thenull. The testsbased on the f2g-inverses arethus dened bythe ritial
regions n
Q n
( k n
)> 2
k;1
o
,where
2
k;
denotes the-quantileof the
2
k
distribution.
2.3Thetest statistibasedon the generalizedinverse(aseW
n
=
n )
Inorderto testthe nullhypothesis,Hotelling's typeproedureisbasedontheweighting W
n
=
1
n
andthus theteststatistiQ
n (
1
n
),providedthat
n
isinvertible. Whenisnon-singular,
n will
be non-singular whenn ishosenlarge enough. Inthe singular ase, theweighting W
n
=
n oers
a natural hoie, sine the Moore-Penrose inverse is uniquely dened. Reall that the generalized
inverse is obtained by inverting the non zero eigenvalues. In pratie, as for the f2g-inverses, a
tolerane>0is requiredfor assessingthe non zeroeigenvalues. Thefollowing propositionlaries
theroleof and givesthe asymptotidistribution of Q
n (
n ).
Proposition 3. Suppose that Z
n
= Z
n (
0 )
d
! N
p
(; ) and
n
! almost surely, as n! 1.
Let the spetral deomposition
n
= P
n
n P
>
n
where
n
= diagf
1 (
n
);:::; p
( n
) g. For any
>0, let n;
=P
n
n;" P
>
n
, where
n;
is the matrix obtained by replaing by zero the elements of
n
whih are less than . If issuiently small, so that:
Prfrank (
n;"
)=rank ()g!1; (6)
as n!1, it follows that:
Z >
n
n;" Z
n d
!
2
r (
2 );
where r=rank () and
2
=
>
.
Using similar arguments as in Proposition 2, Proposition 3 follows. See also the asymptoti
distributiontheoryinMoore(1977,1978) orTyler (1981,Lemma 2.4). Theproposition showsthat,
if is hosen suiently small, under the null hypothesis Q
n (
n
) = Z
>
n
n;" Z
n
+o
P (1)
d
!
2
r .
Beause =
r
,underthe assumptionsA() and (6),for suiently smallwehave
Pr
Q n
( n
)=Q
n (
r n
) !1; (7)
asn!1. Inthissense,thegeneralizedinversestatistianbeonsideredasapartiularf2g-inverse
statisti. Notethatintheasewhere
n
isonlysupposedtobeaf1g-inverse,additionalhypotheses
arerequired inorder to have thestated result (inpartiular mustbein theolumn spae of).
In view of Theorem 9.2.3 of Rao and Mitra (1971, p. 173), the onlusion of Proposition 3 is also
truewhen the estimator
n
is a given symmetrireexive g-inverse of thematrix
n
and when
isa ontinuity point of the appliation whih givesthis partiular symmetrireexive g-inverse.
The test statisti Q
n (
n
) has been onsidered in a time series ontext for testing null
autoor-relations in time series analysis. In fat, Li (1992) investigated the use of Q
n (
1
n
) in non-linear
time series analysis. However, in the time series framework, it seems diult to formulate preise
onditionswhihguaranteetheinvertibilityoftheasymptotiovarianematrix. Forexample,the
non-lineartimeseriesmodelofLi(1992)rulesoutlinearmodelssuhastheARMAmodels,inwhih
singularovarianematrixmayresultinempiriallevelsfarfromthenominallevels. Inordertohave
ateststatistiwell-denedinlinearandnon-linearmodels,DuhesneandFranq(2008)investigated
the use of Q
n (
n
), with
n
the Moore-Penrose inverse of a ertain estimator of the asymptoti
ovariane matrix of theresidual autoorrelations.
3.POWER UNDERFIXED ALTERNATIVES
Wenowexaminethe asymptotipowersunderxedalternatives, adoptingtheapproahof Bahadur
(1960). In this approah, the eieny of a test statisti is measured by its slope, dened as the
rateof onvergene of its p-valueunder axed alternative hypothesis H
1
:=
1
6=
0
. Using the
notation in(3), let
S I
(t)=P
r X i=1 i N 2 i >t ! ; S k
(t)=P
2
k >t
; and S (t)=P
2 rank() >t
be therespetive asymptoti survivalfuntionsof thetest statistisQ
n (I p ), Q n ( k n
) and Q
n (
n )
underthenull hypothesis H
0
. Denote by V(f
i 1
;:::; i
j
g) the linearvetor spae generated bythe
olumns i
1 ;:::;i
j
of P = P
B
. When, with obvious onventions,
i 1 1 < i 1 i j < i j +1 thenV(f i1 ;:::; ij
g) denotes theeigenvetor spaeassoiated to theeigenvalues
i1 ;:::;
ij
of.
Proposition4. LetabasisB andatoleranesuhthatA()and(6)holdtrue,andletkrank().
Underthe alternative H
1
:=
1
6=
0
, the (approximate) Bahadurslopes of the test proedure (2)
withthe weightingsW
n =I p , W n = k n andW n = n
are given by:
I = lim n!1 2 n logS I fQ n ( I p )g= k 1 0 k 2 1 ; (8) k = lim n!1 2 n logS k Q n k n =( 1 0 ) > k ( 1 0 ); (9) = lim n!1 2 n logS Q n n =( 1 0 ) > ( 1 0 ); (10)
respetively, where the onvergene in probability (9)and(10) hold, respetively, when:
1 0 62V(f k+1 ;:::; r
;0g); (11)
1
0
62V(f0g): (12)
The onvergene in probability in formula(8) stands without further restrition.
Note that, even when the variane is known, i.e. when
n
=, the test statisti based on the
f2g-inverse (resp. the generalized inverse) is not onsistent when (11) (resp. (12)) does not hold
true. Indeed, when
1 0 2V(f k+1 ;:::; r
;0g)wehave
k (
1
0
)=0
p
,and itfollowsthat:
Q n k = Z > n ( 0 ) k Z n ( 0 ); = n Z n ( 1
)+n
1=2 ( 1 0 ) o > k n Z n ( 1
)+n
1=2 ( 1 0 ) o ; = Z > n ( 1 ) k Z n ( 1
)6!1 inprobability;
asn! 1. Similarly,when
1
0
2V(f0g) the statisti Q
n =Z > n ( 1 ) Z n ( 1
(10),thepreviousderivationsshowthatoneanset,byontinuity, k
=0when(11)isnotsatised,
and =0 whenthe relation (12) does not hold. Thefollowing orollary presents a omprehensive
omparison ofthe Bahadur slopesofthetest statistisQ
n ( I
p ), Q
n
k n
and Q
n
n
.
Corollary 1. Under the assumptions of Proposition 4, the followingomparisonsan be made:
i)The Imhof-based testis always onsistent (i.e. we always have
I >0);
ii)The test based on the f2g-inverse
k n
isonsistent (i.e.
k
>0)if and onlyif (11);
iii)The generalized-inverse based testis onsistent (i.e. >0)if and onlyif (12);
iv) For all k r := rank() we have =
r
k
k 1
1
with
k
=
k 1 i
1
0
2V(f
k g
);
v) When
1
0
2V(f
1 ;:::;
k
g) with k > 1 we have
k
I
, with strit inequality i there
existsk 0
suh that 1<k 0
k,
k 0
< 1
and
1
0
62V( f k
0g
) ;
vi) When
1
0
62V(f
1 ;:::;
k
g) we have
I >
k ;
vii)The BahadurslopeofImhof-basedteststatistiisalwayslargerthantheoneoftheteststatisti
based on
1
, that is I
1
, withequality i
1
0
2V(f
1 g).
The most notieable result of this orollary is that, ontrary to the other test proedures, the
Imhof-based test oers a stritly positive Bahadur slope for all
1
0 6= 0
p
. From that point of
view,Q
n (I
p
) represents theonly omnibustest statisti with non trivialpowerunder all alternative
hypotheses,andisinthespiritoftheso-alledportmanteauteststatistisinthetimeseriesliterature.
However, foranalternative hypothesisinthenon-zeroeigenspae of,theslopeoftheImhof-based
test issmallerthan that ofthetest based onthegeneralized inverse
n
. Notealso that, interm of
theBahadurslope,the f2g-inversetest statisti basedon
k n
dominatesthetest statistibased on
k
0
n
when k>k 0
.
Figure1 displays theBahadur slopesof thedierent testswhen =diag (1;1;1=2;1=2;0;0 ) and
when
1
0
isaunitvetorwithdiretiondintheplaneontainingu
1
=1=
p
5(1;1;1;1;0;1) >
and
u 2
=(0;0;0;0;1;0) >
. Thelengthofthevetorgoingfromtheoriginto theurveP
k
inthediretion
dgivestheBahadurslopeoftheteststatistiQ
n (
k n
). Inthisexample,theBahadurslopeofQ
n (I
6 )
isalways one, sine
1
=1and the length of
1
0
is normalized to one. Sineu
2
2V(f0g), only
theImhof-basedtestispowerfulforalternativesinthediretionofu 2
andtheslopesoftheothertests
anel for that alternative hypothesis. This gure thus illustrates the points i)-iv) of Corollary 1.
Figure2illustratesotherpoints. Inpartiular,inthisgure,
3
= beausethediretiondofthe
alternative belongs toV(f
4 g
) and
3
I
beause dbelongsto V(f
1 ;
2 ;
3 g).
Proof of Proposition 4: Under the null hypothesis,n
1=2 Z
n
=n
1=2 Z
n (
0 )
p
!0 and underthe
alternative H
1
:=
1
we have n
1=2 Z
n
=n
1=2 fZ
n (
1
)+n
1=2 (
1
0 )g
p
!
1
0
. A large
deviation resultyields:
logP r X
i
N 2
i >x
!
x
2 1
Figure 1. The Bahadur slopes k
of Q
n (
k n
) and
I
of Q
n (I
p
) when =
diag (1;1;1=2;1=2;0;0 ), for alternatives inthediretion 1
0
=d=a
1 u
1
+a
2 u
2 ,
jjdjj=1, where u
1
=1=
p
5( 1;1;1;1;0;1) >
and u
2
=(0;0;0;0;1;0) >
. The Bahadur
slope
k
orresponds to thelength of the vetor going from the origin to theurve
P k
inthe diretion d. The slope ofthe Imhof-basedtestdesribesairleP beause
itisonstant.
−1.0
−0.5
0.0
0.5
1.0
−1.0
−0.5
0.0
0.5
1.0
u1
u
2
P
1
P
2
P
3
P
4
P
−
P
1
−
P
2
−
P
3
−
P
4
−
P
d
asx!1, where N
1 ;:::;N
r
areindependent N(0;1) random variables(see Zolotarev (1961)). The
Bahadurslope ofthe rst testis thus given by:
I
= lim
n!1 2
n logS
I kZ
n k
2
= lim
n!1 jjZ
n jj
2
n 1
= jj
1
0 jj
2
1
:
Now notethat
lim n!1
Q n
k n
= lim
n!1 n(
1
0 )
>
k (
1
0
)=1 inprobability
undertheondition(11), andthatQ
n
n
!1inprobabilityundertheondition(12). Usingthe
large deviation result logP(
2
k
>x) x=2 as x ! 1, under the ondition (11) (resp. (12)) the
Bahadurslope
k
(resp. ) is thenobtainedbytheargumentsused toompute
I
. 2
Proof of Corollary 1: Points i) iii) are diret onsequenes of Proposition 4. To show iv),
onsider the spetral deomposition =P P
>
where P
>
Figure 2. The Bahadur slopes k
of Q
n (
k n
) and
I
of Q
n (I
p
) when =
diag (1;1;1=2;1=2;0;0 ), for alternatives inthediretion 1
0
=d=a
1 u
1
+a
2 u
2 ,
jjdjj=1,where u
1
2V(f
1
g) and u
2
=1=
p
3(1;1;1;0;0;0) >
2V(f
1 ;
2 ;
3
g). The
Bahadurslope
k
orrespondstothelengthofthevetorfromtheorigintotheurve
P k
inthe diretion d. The slope ofthe Imhof-basedtestdesribesairleP beause
it isonstant.
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
u1
u
2
P
1
P
2
P
3
P
4
P
−
P
1
−
P
2
−
P
3
−
P
4
−
P
d
notethatfor 1k <k 0
r the dierene
k
isnon-negative:
k
=(
1
0 )
> Pdiag
0 >
k ;
1
k+1 ;:::;
1
k 0
;0 >
p r
P >
( 1
0
)0;
withequality ifandonly if
1
0
2V(f
k+1 ;:::;
k 0g
). Points v) vii) ome from
k
I
=(
1
0 )
>
Pdiag 0;
1
2
1
1
;:::; 1
k
1
1
;
1
1
;:::; 1
1
P >
( 1
0
):
Thisshows theresult. 2
Other omparisons between the test statistis an be performed under loal alternatives. In the
4.POWER UNDERLOCAL ALTERNATIVES
ConsiderasequeneofloalalternativesoftheformH
1n
:
n
=
0 +n
1=2
n
,where
n
!6=0
p .
Thefollowing propositiongivestheAsymptoti LoalPower (ALP)of thetest proedures.
Proposition 5. When Z
n (
0 )
d
! N
p
(; ), and with the notations and assumptions of
Proposi-tions1-3,the ALP ofthe testproedure (2)withthe weightingW
n
=I
p
, W
n
=
k n
andW
n
=
n
under the loal alternatives H
1n
are given by:
ALP I
() = Pr
r X
i=1
i
2
1 (
2
i
)+
>
0
0 >
(
1 ;:::;
r )
!
; (13)
ALP k
() = Pr
2
k (
(k)
1
)> 2
k;1
; (14)
ALP () = Pr
2
r (
2 )>
2
r;1
: (15)
Theproof of Proposition 5 represents adiretonsequene of Propositions 1-3and thereforeit is
omitted. Thefollowing orollaryompares theALP ofthedierent test proedures.
Corollary 2. Under the assumptionof Proposition 5:
i)The Imhof-based testis always loally asymptotially powerful (i.e. ALP
I
()> 8);
ii)The test based on the f2g-inverse
k n
isloally asymptotially powerful (i.e. ALP
k
()>)
if andonlyif 62V(f
k+1 ;:::;
r ;0g);
iii) The generalized-inverse based test is loally asymptotially powerful (i.e. ALP () > ) if
andonlyif 62V(f0g);
iv) When2V(f
1 ;:::;
k
g) withk <r then ALP k
()>ALP ();
v) When2V(f
1
g) we have ALP
1
()>ALP
I ();
vi) When2V(f
1
g) we have ALP
1
()ALP
2
()ALP
r 1
()ALP ().
Thisorollary shows that, asfor the Bahadur slopes, the rankingof the loalasymptoti powers
ofthe dierenttestsdependsontheposition ofthealternativewithrespettotheeigenspaes of.
However, ompared to Bahadur'sapproah,Pitman's approah highlights therelative merits of the
testproedureswithadierentviewpoint. Inpartiular,intermofALP,theperformaneofthetest
statistisbased ofthe f2g-inverse does not neessarilyinreases with k (ompare iv) inCorollary 1
withvi) inCorollary 2).
Figure 3 displays the ALP's for several diretions of the loal alternative, and for the same
matrix as that used in Figures 1 and 2. From this gure, writing ALP
k
ALP
k
(), the
following relations are satised:
ALP 1
>ALP 2
>ALP I
>ALP 3
>ALP ; when /(1;0;0;0;0;0)
> ;
ALP >ALP
3
'ALP
I
>ALP 2
>ALP 1
; when /(1;1;1;1;0;0)
> ;
ALP 2
>ALP I
>ALP 3
>ALP >ALP 1
; when /(1;1;0;0;0;0)
> ;
ALP 3
>ALP >ALP I
>ALP 2
=ALP
1
; when /(0;0;1;0;0;0)
> :
Tosummarize,intermofALP,thetestbasedon
k
Figure 3. Asymptoti loal powers of the tests when = diag (1;1;1=2;1=2;0;0 ),
for alternativesindierent diretions.
0
1
2
3
4
0.0
0.2
0.4
0.6
0.8
1.0
∆ = δ
u
1
,
u
1
=
(
1
,
0
,
0
,
0
,
0
,
0
)
T
δ
ALP(
∆
)
k=1
k=2
k=3
Moore−Penrose
Imhof
0
1
2
3
4
0.0
0.2
0.4
0.6
0.8
1.0
∆ = δ
u
2
,
u
2
=
(
1
,
1
,
1
,
1
,
0
,
0
)
T
δ
ALP(
∆
)
0
1
2
3
4
0.0
0.2
0.4
0.6
0.8
1.0
∆ = δ
u
3
,
u
3
=
(
1
,
1
,
0
,
0
,
0
,
0
)
T
δ
ALP(
∆
)
0
1
2
3
4
0.0
0.2
0.4
0.6
0.8
1.0
∆ = δ
u
4
,
u
4
=
(
0
,
0
,
1
,
0
,
0
,
0
)
T
δ
ALP(
∆
)
The Imhof-based test statisti and the one relying on the generalized inverse oer power for more
alternatives, but itappearsthat none testis dominatedbyanotherone.
Proof of Corollary 2: To show i) we note that k
k
2
=
>
= 0 i 2 V(f0g), and that
k 0
k 2
=0 i 2V(f
1 ;:::;
r
g). Thepointsii) andiii) are obtained similarly. To showiv), rst
notethat
2
(k)
=
> P
B diag
0 >
k ;
1
;:::; 1
r ;0
>
p r
P >
B
withequalityi2V(f k+1
;:::; r
g
). Thus
2
=
1
>0when2V(f
1 ;:::;
k
g). Nowiv) is
impliedbythefatthatP
2
k
()> 2
k;1
stritly dereaseswithk forall >0and all2(0;1)
(seeTheorem 2 inGhosh (1973)).
In order to show v) and vi), we adapt arguments alled upon by Ghosh (1973). Let X
1
be a
2
k
()-distributed random variable, and let X
2
be a random variable independent of X
1
. Consider
thetestingproblemH
0
:=0againstH
0
:>0basedon theobservations (X 1
;X 2
). Assumethat
the distribution of X
2
is the same under the null and alternative hypotheses. Using the
Neyman-Pearson lemma, the most powerful test statisti of H
0
: = 0 against H
0
: =
1
, with
1 > 0,
rejetsthenullhypothesisifthelikelihoodratio islarge. A straightforward buttediousomputation
showsthat thislikelihood ratiois givenby:
L(X 1
;X 2
; 1
)
L(X 1
;X 2
;0)
= (k=2)exp (
1 =2)
1 X
i=0 (
1 X
1 )
i =f4
i
i! (i+k=2)g:
NotethatthisratioisaninreasingfuntionofX
1
. Consequentlytheritialregionoftheuniformly
mostpowerful(UMP)testisgivenbyfX
1
2
k;1
g. Wenowapplythisresultwhen 2V(f
1 g),
setting =
2
1
, X
1
=(N
1
+)
2
and X
2 =
P r
i=2
i =
1 N
2
i
,with thenotations of Proposition 1 and
(N 1
;:::;N r
) >
N
r (0;I
r
). Noting alsothat
(1)
1
=
2
1
when 2V(f
1
g), we thenobtain:
ALP 1
() = P
(N 1
+)
2 >
2
1;1
;
> P (
(N 1
+)
2 +
r X
i=2
i =
1 N
2
i >
( 1
;:::; r
)
1
)
=ALP
I ();
whihshows v). Setting X
1
=(N
1
+)
2 +
P k
i=2 N
2
i
and X
2
=N
2
k+1
,thesame argument entails:
ALP k
() = P
(
(N 1
+)
2 +
k X
i=2 N
2
i >
2
k;1
)
;
> P (
(N 1
+)
2 +
k+1 X
i=2 N
2
i >
2
k+1;1
)
=ALP
k +1 ();
andpoint vi) follows. 2
Inthenext setion, furtheromparisons andundertaken using MonteCarlo experiments.
5.SIMULATIONEXPERIMENTS
Inthe previous setions, we have presentedthe asymptoti nulldistributions ofthree lasses oftest
statistis, and we have given some asymptoti properties under xed and loal alternatives. It is
natural to inquire about their nite sample properties, in partiular their exat levels and powers.
Furthermore, the theoretial results obtained inSetions 3 and 4 need to beompleted empirially.
The power omparisons between the weighting W
n
=
n
and W
n
= I
p
seem also of partiular
interest. To partially answer these onsiderations, some MonteCarlo experiments were onduted.
ThemainomputerodefortheexperimentsdesribedbelowhasbeenwrittenusingtheRlanguage,
5.1 Desriptionof thesimulation experiments
In order to ompare the test statistis, we onsidered multivariate sampling from the multivariate
normal distribution, for several hoies of the ovariane matrix . The test statistis inluded in
our simulation experiments used the weighting W
n = I p , W n = k n
, k r, r = rank(), and
W n = n , where n = S n
represents the sample ovariane matrix. For eah random sample
of size n = 100, we examined the empirial frequenies of rejetion of the null hypothesis H
0 :
= 0 when the latter was true by using test statistis with three nominal levels (1, 5 and 10%).
Multivariate sampling appears partiularly onvenient to study the power of the test proedures,
given the analytial results demonstrated in the previous setions. Several xed alternatives have
been inluded inthe study,whihhave been hosenbyexaminingthespetraldeomposition ofthe
ovariane matrix and their assoiatedeigenspaes.
Table 1. Mean vetors
(k)
ij
in multivariate sampling from the normal distribution
N 3 ( (k) ij ; (k) ij
),i=I;II,j =1;2, k=1;:::;4.
Experiment i=I
(1) I;1 =( 3 10 ;0;0)
> , (2) I;1 =( 3 10 ; 3 10 ;0) > , (3) I;1 =( 3 10 ; 3 10 ; 1 10 ) > , (4) I;1 =( 3 10 ;0; 1 10 ) > , (1) I;2 =( 4 5 ;0;0)
> , (2) I;2 =( 3 10 ; 3 10 ;0) > , (3) I;2 =( 3 10 ; 3 10 ; 3 10 ) > , (4) I;2 =( 4 5 ;0; 3 10 ) > ;
Experiment i=II
(1) II;1 =( 3 10 ;0;0)
> , (2) II;1 =(0; 2 25 ; 2 5 ) > , (3) II;1 =( 1 10 ; 1 5 ; 1 50 ) > , (4) II;1 =( 1 10 ; 3 10 ; 1 5 ) > , (1) II;2 =( 1 10 ; 1 10 ; 1 5 ) > , (2) II;2 =( 1 10 ; 1 10 ;0) > , (3) II;2 =( 1 10 ; 1 10 ; 1 10 ) > , (4) II;2 =( 1 10 ; 1 10 ; 1 10 ) > ;
Table 2. Mean vetors
(k)
ij
in multivariate sampling from the normal distribution
N 6 ( (k) ij ; (k) ij
),i=III;IV,j=1;2;3, k=1;:::;4.
Experiment i=III
(1) III;1 =( 3 10
;0;0;0;0;0) > , (2) III;1 =( 3 10 ; 3 10
;0;0;0;0) > , (3) III;1 =( 3 10 ; 3 10 ; 3 10
;0;0;0) > , (4) III;1 =( 3 10 ; 3 10 ; 1 10 ; 1 10 ;0;0)
> , (1) III;2 =( 4 5
;0;0;0;0;0) > , (2) III;2 =( 4 5 ; 4 5
;0;0;0;0) > , (3) III;2 =( 4 5 ; 4 5 ; 2 5
;0;0;0) > , (4) III;2 =( 3 5 ; 3 5 ; 3 10 ; 3 10 ;0;0)
> , (1) III;3 =( 4 5
;0;0;0;0;0) > , (2) III;3 =( 2 5 ; 4 5
;0;0;0;0) >
,
(3)
III;3
=(0;0; 3
5
;0;0;0) > , (4) III;3 =( 3 5 ; 3 5 ; 3 10 ; 3 10 ;0;0)
> ;
Experiment i=IV
(1) IV;1 =(0; 3 10 ; 1 5 ; 3 10 ;0; 1 5 ) > , (2) IV;1 =(0; 1 10 ; 1 20 ; 3 100 ; 1 2 ; 1 20 ) > , (3) IV;1
=(0:37; 0:24; 0:18; 0:12 ; 0: 06; 0:0 3)
>
,
(4)
IV;1
=(0:22;0:18;0:14;0:0 9;0:0 5;0:0 2) >
,
(1)
IV;2
=(0:18; 0:17;0:00;0 :35 ; 0: 18; 0:1 8) >
,
(2)
IV;2
=( 0:19; 0:34;0:00;0:1 4;0:1 9;0:1 9) >
,
(3)
IV;2
=( 0:19;0:20; 0:23;0: 20; 0:1 9;0:2 1) >
,
(4)
IV;2
=( 0:20;0:08; 0:22;0: 08 ; 0:1 9;0:0 8) >
,
(1)
IV;3
=(0:00;0:00;0:00;0: 00; 0:3 5;0:3 5) >
,
(2)
IV;3
=(0;0; 1
2
;0;0;0) > , (3) IV;3 =(0; 1 5 ;0; 1 10 ;0;0)
>
,
(4)
IV;3
Table 3. Relative frequeny of rejetion of H 0
:=0 (inperentage) for the test
basedonthe statistisQ
n (W
n
)dened by(2),usingW
n
=I
3
,W
n
=
k n
,k =1;2,
andW
n
=
n
,withZ
n =
p
n X
n
and
n
=S
n
,wherethemeanvetorsaregivenin
Table1.
(p;r)=(3;2) =
I;1
=diag(1;1;0)
=0 =
(1)
I;1
=
(2)
I;1
=
(3)
I;1
=
(4)
I;1
1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10%
Q n
(I 3
) 0.9 4.8 10.6 55.0 77.0 85.0 89.7 97.1 98.4 92.4 97.7 99.4 61.5 82.3 88.6
Q n
(S n
) 1.3 5.8 11.0 57.4 77.0 85.0 90.3 96.9 98.7 90.3 96.9 98.7 57.4 77.0 85.0
Q n
(S 1 n
) 0.4 2.9 7.3 27.6 45.3 54.0 50.5 65.3 71.5 50.5 65.3 71.5 27.6 45.3 54.0
Q n
(S 2 n
) 1.3 5.8 11.0 57.4 77.0 85.0 90.3 96.9 98.7 90.3 96.9 98.7 57.4 77.0 85.0
=
I;2
=diag(10;1;0)
=0 =
(1)
I;2
=
(2)
I;2
=
(3)
I;2
=
(4)
I;2
1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10%
Q n
(I 3
) 1.2 5.6 11.3 50.0 71.5 81.0 8.3 24.4 39.4 12.3 35.5 57.3 56.3 79.4 86.3
Q n
(S n
) 1.3 5.8 11.0 39.7 61.6 73.4 59.9 82.6 89.0 59.9 82.6 89.0 39.7 61.6 73.4
Q n
(S 1 n
) 0.9 5.6 11.0 50.1 71.1 80.9 6.6 17.4 26.5 6.6 17.4 26.5 50.1 71.1 80.9
Q n
(S 2 n
) 1.3 5.8 11.0 39.7 61.6 73.4 59.9 82.6 89.0 59.9 82.6 89.0 39.7 61.6 73.4
For our investigations, we onsidered multivariate normal distributions of dimensions p = 3;6,
where was singular and non-singular. The denitions of the ovariane matries are given in
Tables 3-6. In Tables 3 and 5, the ovariane matries are exatlysingular and they are diagonal.
We investigated situationswhere thenon null eigenvalues areboth equal to one,and whenthey are
dierent. Intheasep=6,weinvestigatedfour uniteigenvalues,multipliitiesofdimensiontwoand
a situation where the non null eigenvalues are distint. In Tables 4 and 6 the ovariane matries
areof theform =I
p
AA >
,where is areal valueand A isa pr matrix; theyare preisely
dened intheTables. It iseasily shown thatthese matrieshave at leastp r uniteigenvalues. In
timeseries,theasymptotiovariane matriesofresidualautoovarianesexhibitsimilarforms,see
Li (2004, Chapter 2), among others. The ovariane matries in Tables 4 and 6 are non-singular,
but theratio of the largest to the smallest eigenvalue is large;onsequently they areapproximately
singular. The values of the non null mean vetors under the alternative hypotheses are given in
Tables 1and 2. For eah ase, 1000 independent realizations havebeen generated.
5.2 Disussion ofthe MonteCarloresults
In Tables 3-6, the results for the level study orrespond to the olumn = 0. For the nominal
level = 5%, the empirial size over the 1000 independent repliations should belong to the
in-terval [3:6%;6:4%℄ withprobability 95% (at the nominal levels = 1% and 10%, theintervals are
[0:4%;1:6%℄and[8:1%;11:9%℄, respetively). Whentherelative rejetionfrequeniesareoutsidethe
95%signianelimits,theyaredisplayedinboldintheTables. Whentherelativerejetion
Table 4. Relative frequeny of rejetion of H 0
: = 0 (in perentage) for the
test based on the statistis Q
n (W
n
) dened by (2), using W
n
= I
3
, W
n
=
k n
,
k =1;2;3,and W
n
=
n
,withZ
n =
p
n X
n
and
n
=S
n
,where the meanvetors
are given inTable 1.
(p;r)=(3;3) =
II;1 =I
3
0:79x 3
x > 3
,x 3
=(1; 1
2 ;
1
10 )
>
=0 =
(1)
II;1
=
(2)
II;1
=
(3)
II;1
=
(4)
II;1
1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10%
Q n
(I 3
) 1.0 4.8 10.7 62.0 99.9 100.0 86.5 94.9 97.5 26.9 48.8 62.2 79.5 92.3 96.0
Q n
(S n
) 1.2 5.4 11.3 100.0 100.0 100.0 83.5 93.0 96.1 24.9 44.2 57.9 98.3 99.8 99.9
Q n
(S 1 n
) 0.4 2.9 7.3 3.7 12.8 21.8 47.9 62.5 68.4 14.1 29.3 40.1 41.6 58.3 67.0
Q n
(S 2 n
) 1.3 5.8 11.0 8.1 21.8 32.1 86.3 95.2 97.9 27.3 50.3 63.3 80.4 91.8 95.6
Q n
(S 3 n
) 1.2 5.4 11.3 100.0 100.0 100.0 83.5 93.0 96.1 24.9 44.2 57.9 98.3 99.8 99.9
=
II;2 =I
3
0:333XX >
,X=(
1 ;
2 );
1
=1
3 ;
2
=(1; 1;0) >
=0 =
(1)
II;2
=
(2)
II;2
=
(3)
II;2
=
(4)
II;2
1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10%
Q n
(I 3
) 1.3 5.8 10.3 45.8 67.7 77.5 6.3 26.9 46.2 6.1 40.3 84.9 15.3 41.4 60.1
Q n
(S n
) 1.2 5.4 11.3 31.6 54.3 66.6 29.4 50.6 65.3 100.0 100.0 100.0 100.0 100.0 100.0
Q n
(S 1 n
) 1.1 5.1 10.9 45.8 68.7 78.8 1.2 6.2 10.9 1.0 5.2 10.8 4.9 14.5 22.4
Q n
(S 2 n
) 1.3 5.8 11.0 36.8 60.0 71.2 34.4 58.2 70.5 1.4 5.9 11.0 37.9 63.0 75.8
Q n
(S 3 n
) 1.2 5.4 11.3 31.6 54.3 66.6 29.4 50.6 65.3 100.0 100.0 100.0 100.0 100.0 100.0
We disuss the results presented in Table 3. In general, the errors of the rst kind of the test
statistisarewellontrolled, exeptfor Q
n (S
1 n
) at the5%and10%nominal levels when=
I;1 .
For the alternatives =
(1)
I;1
;
(2)
I;1
, the mean vetors belong to the eigenspae generated by the
eigenvalue
1
= 1; it appears preferable to speify k = 2 than k = 1 in that situation, whih
maybeexplained bythe multipliityof thatunit eigenvalue. Interestingly,Q
n (S
2 n
) delivershigher
power than Q
n (S
1 n
) under
(1)
I;1
, even if that alternative belongs to the vetor spae generated by
v 1
=(1;0;0) >
;thisisexplainedbythefatthatinnitesamples
(2)
I;1
doesnotbelongexatlytothe
vetor spae of the rst olumn of P
n
in the spetraldeomposition of S
n
. Under the alternatives
=
(3)
I;1
;
(4)
I;1
,the empirial powers ofQ
n (I
3
) andQ
n (S
2 n
) areverysimilar when=
I;1
,witha
slight advantage for the weighting W
n
=I
3
. These alternatives do not lie in a spei eigenspae,
and Q
n (I
3
) oers high power. When =
I;2
, all theeigenvalues aredierent and the ovariane
matrix is singular. Sine
(1)
I;2
2 V(f10g), Q
n (S
1 n
) is very powerful, but the weighting W
n
= I
3
delivers a similar power. The dierenes in powers between Q
n (S
1 n
) and Q
n (S
2 n
) = Q
n (S
n ) is
signiant. For
(2)
I;2
2V(f10;1g), the weighting W
n
=I
3
giveslowpower, andto usea generalized
inverse provides thebestempirialpower. When=
(3)
I;2
,that alternative doesnot lieina spei
eigenspae; onsequently all teststatistisoer some power, butthegeneralized inverse appearsthe
mostpowerful. Thealternative=
(4)
I;2
2V(f10;0g); themost powerfultest statistisareQ
n (S
1 n
)
Table 5. Relative frequeny of rejetion of H 0
: = 0 (in perentage) for the
test based on the statistis Q
n (W
n
) dened by (2), using W
n
= I
6
, W
n
=
k n
,
k = 1;2;3;4, and W
n
=
n
, with Z
n =
p
n X
n
and
n
= S
n
, where the mean
vetorsaregiveninTable2.
(p;r)=(6;4) =
III;1
=diag(1;1;1;1;0;0)
=0 =
(1)
III;1
=
(2)
III;1
=
(3)
III;1
=
(4)
III;1
1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10%
Q n
(I 6
) 0.7 4.1 9.3 38.9 64.5 76.9 81.9 94.4 97.1 96.6 99.3 99.9 87.2 96.4 98.7
Q n
(S n
) 1.1 5.0 11.3 42.9 67.8 79.6 85.4 94.8 97.2 97.2 99.4 99.8 88.1 96.9 98.7
Q n
(S 1 n
) 0.3 3.6 7.7 10.3 22.6 31.8 23.7 37.4 46.8 34.1 48.2 55.6 25.8 40.4 49.1
Q n
(S 2 n
) 0.3 3.7 7.9 17.9 36.2 46.9 43.9 61.7 70.5 59.0 73.3 79.6 45.9 64.1 73.4
Q n
(S 3 n
) 0.5 4.2 7.8 29.3 51.2 63.1 62.9 79.4 86.8 80.8 89.4 93.1 67.7 82.7 88.3
Q n
(S 4 n
) 1.1 5.0 11.3 42.9 67.8 79.6 85.4 94.8 97.2 97.2 99.4 99.8 88.1 96.9 98.7
=
III;2
=diag(10;10;1;1;0;0)
=0 =
(1)
III;2
=
(2)
III;2
=
(3)
III;2
=
(4)
III;2
1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10%
Q n
(I 6
) 0.7 4.6 9.6 34.6 60.2 72.0 73.1 90.5 95.0 82.7 95.4 98.1 53.6 79.3 89.4
Q n
(S n
) 1.1 5.0 11.3 27.7 49.9 64.4 64.9 83.9 91.1 98.2 99.6 99.8 94.8 98.6 99.6
Q n
(S 1 n
) 0.2 2.8 7.8 16.8 33.2 44.1 41.5 59.2 66.9 41.6 59.6 67.5 21.9 41.7 51.2
Q n
(S 2 n
) 0.9 5.1 10.7 36.0 61.5 73.3 73.6 90.3 94.8 73.6 90.2 94.8 42.2 66.1 75.9
Q n
(S 3 n
) 0.6 5.2 9.8 30.8 54.2 67.0 68.3 87.5 92.4 86.2 93.6 97.2 72.7 86.3 91.1
Q n
(S 4 n
) 1.1 5.0 11.3 27.7 49.9 64.4 64.9 83.9 91.1 98.2 99.6 99.8 94.8 98.6 99.6
=
III;3
=diag(8;4;2;1;0;0)
=0 =
(1)
III;3
=
(2)
III;3
=
(3)
III;3
=
(4)
III;3
1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10%
Q n
(I 6
) 0.8 4.5 10.5 56.1 77.5 84.9 75.7 93.9 97.2 27.0 62.7 78.9 82.0 96.0 99.0
Q n
(S n
) 1.1 5.0 11.3 38.4 62.8 74.5 85.0 95.0 97.5 83.0 92.5 96.9 96.9 99.5 99.9
Q n
(S 1 n
) 0.9 5.1 9.9 57.7 79.0 87.7 14.4 29.7 40.9 1.1 4.8 10.6 31.4 54.5 67.2
Q n
(S 2 n
) 0.8 5.2 10.6 48.4 70.1 80.2 89.3 97.3 98.8 1.8 7.3 14.8 76.2 90.5 94.9
Q n
(S 3 n
) 0.5 5.2 10.9 42.6 66.5 75.8 88.3 96.8 98.0 84.5 94.6 97.2 86.8 94.8 97.6
Q n
(S 4 n
) 1.1 5.0 11.3 38.4 62.8 74.5 85.0 95.0 97.5 83.0 92.5 96.9 96.9 99.5 99.9
From the results presented in Table 4, the empirial sizes were rather satisfatory, exept for
Q n
(S 1 n
) at the 5% and 10% nominal levels, when =
II;1
whih underrejeted and displayed
rejetion rates outside the 99% signiane limits. When =
II;1
, the spetral deomposition
gives
1
=
2
= 1 and
3
= 4:610
3
. A basis for the eigenspae is omposed of the vetors
fv 1
= (0:0; 0:2;1:0)
> ;v
2
= (0:5; 0:9; 0:2)
> ;v
3
= (0:9;0:4;0:1) >
g. Clearly
(1)
II;1
and v
1 are
orthogonal and as a result the empirial powers of Q
n (S
k n
), k 2 were low. The best empirial
powers have been observed for the f2g-inverse with k = 3. The weighting W
n
= I
3
oered less
power. The alternative
(2)
II;1
belongs to V(v
1
). The best empirial powers have been observed by
Q n
(S 2 n
) and Q
n (S
3 n
). The weighting W
n
=I
3
oered highpower. The vetor
(3)
2V(v
1 ;v
Table 6. Relative frequeny of rejetion of H 0
: = 0 (in perentage) for the
test based on the statistis Q
n (W
n
) dened by (2), using W
n = I 6 , W n = k n ,
k =1;2;3;4;5;6, and W n
=
n
, with Z
n = p n X n and n =S n
, where themean
vetorsaregiveninTable2.
(p;r)=(6;6) =
IV;1 =I 6 9 20 x 6 x > 6 ;x 6 =(1; 4 5 ; 3 5 ; 2 5 ; 1 5 ; 1 10 ) >
=0 =
(1) IV;1 = (2) IV;1 = (3) IV;1 = (4) IV;1
1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10%
Q n
(I 6
) 0.9 4.7 8.7 94.3 98.9 99.6 95.5 99.1 99.7 91.9 98.5 99.2 49.4 98.2 100.0
Q n
(S n
) 1.6 7.0 12.0 97.0 99.5 99.7 95.8 99.2 99.5 92.6 98.3 99.2 100.0 100.0 100.0
Q n
(S 1 n
) 0.3 3.2 6.7 26.0 40.4 48.6 24.6 38.2 46.7 23.8 39.2 47.3 0.4 3.3 6.7
Q n
(S 2 n
) 0.2 2.7 6.6 45.7 63.0 71.2 45.7 61.6 69.6 46.6 63.7 71.7 0.2 2.5 6.6
Q n
(S 3 n
) 0.5 3.5 7.0 65.6 79.5 84.9 65.2 77.5 83.2 64.2 79.8 85.7 0.5 3.4 6.9
Q n
(S 4 n
) 0.7 4.1 8.8 83.4 92.1 94.8 82.6 91.9 94.4 82.4 92.2 95.2 0.7 3.9 8.3
Q n
(S 5 n
) 1.5 6.1 11.4 95.5 99.2 99.6 96.8 99.4 99.6 93.7 98.6 99.6 1.7 6.1 11.7
Q n
(S 6 n
) 1.6 7.0 12.0 97.0 99.5 99.7 95.8 99.2 99.5 92.6 98.3 99.2 100.0 100.0 100.0
= IV;2 =I 6 0:16XX >
;X=(
1 ; 2 ; 3 ; 4 ); 1 =1 6 ; 2 =( 1 2 ; 1 2 ; 1 2 ; 1 2 ; 1 2 ; 1 2 ) > ; 3 =( 1 2
;0;0;0;0; 1 2 ) > ; 4
=(0;0; 1
2 ;0;0;0)
>
=0 =
(1) IV;2 = (2) IV;2 = (3) IV;2 = (4) IV;2
1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10%
Q n
(I 6
) 0.8 4.7 9.1 94.5 99.0 99.5 94.3 98.9 99.5 96.7 99.5 99.7 74.3 90.9 96.8
Q n
(S n
) 1.6 7.0 12.0 93.1 98.4 99.3 93.7 98.4 99.4 98.8 99.8 99.8 100.0 100.0 100.0
Q n
(S 1 n
) 0.2 3.2 7.0 34.3 50.4 58.0 30.1 45.8 54.5 4.5 13.9 21.7 2.1 7.3 14.1
Q n
(S 2 n
) 0.2 3.3 7.7 56.4 72.8 79.3 53.6 71.1 77.8 14.5 28.2 39.2 4.9 14.1 22.8
Q n
(S 3 n
) 0.4 3.3 7.3 76.3 88.1 92.2 74.3 85.8 90.8 30.6 46.6 56.0 13.4 26.6 35.6
Q n
(S 4 n
) 0.3 3.8 8.4 89.8 95.8 98.4 88.8 96.3 97.9 58.9 72.1 78.1 30.8 48.5 58.9
Q n
(S 5 n
) 1.5 6.1 11.4 94.5 99.1 99.6 94.7 98.7 99.4 99.3 99.8 99.8 76.1 88.6 93.2
Q n
(S 6 n
) 1.6 7.0 12.0 93.1 98.4 99.3 93.7 98.4 99.4 98.8 99.8 99.8 100.0 100.0 100.0
= IV;3 =I 6 0:49XX >
;X=(
1 ; 2 ; 3 ; 4 ; 5 ); 1 = 2
=(1;0;0;0;0;0) >
;
3
=(0;1;0;0;0;0) >
; 4
=(0;0;0;1;0;0) >
; 5
=(0;0;0;0;1;1) >
=0 =
(1) IV;3 = (2) IV;3 = (3) IV;3 = (4) IV;3
1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10% 1% 5% 10%
Q n
(I 6
) 0.4 4.5 9.2 97.3 99.4 99.8 97.9 99.6 99.8 23.2 49.9 65.1 45.7 71.1 81.3
Q n
(S n
) 1.6 7.0 12.0 92.7 98.1 99.2 94.2 98.6 99.3 45.3 66.1 76.0 56.4 74.9 84.1
Q n
(S 1 n
) 0.2 2.5 7.2 63.0 73.7 77.4 57.4 69.9 76.0 0.3 3.8 9.0 11.6 25.7 35.7
Q n
(S 2 n
) 0.7 4.6 9.7 96.9 99.0 99.7 97.7 99.6 99.7 1.8 6.9 13.4 19.4 38.7 51.4
Q n
(S 3 n
) 0.6 4.8 9.3 96.6 99.1 99.6 96.6 99.3 99.7 20.7 39.0 48.5 36.2 57.6 69.0
Q n
(S 4 n
) 1.1 5.0 11.1 95.6 99.1 99.6 96.3 99.2 99.6 49.2 70.7 79.7 61.3 79.8 87.0
Q n
(S 5 n
) 1.2 4.6 9.9 94.2 98.6 99.5 95.5 98.5 99.3 45.5 67.0 77.3 57.9 76.0 84.5
Q n
(S 6 n
) 1.6 7.0 12.0 92.7 98.1 99.2 94.2 98.6 99.3 45.3 66.1 76.0 56.4 74.9 84.1
the best power has been observed for the f2g-inverse with k = 2. The alternative =
(4)
When = II;2
,
1
= 1:000;
2
= 0:334;
3
= 10
3
. The rst alternative
II;2
belongs to the
eigenspaeassoiated to
1
;it appears thatQ
n (S
1 n
) isvery powerful. The seondalternative
(2)
II;2
belongs to theeigenspae assoiated to
2
,and Q
n (S
2 n
) dominates whereas, asexpeted, Q
n (S
1 n
)
hasnopower. ThethirdalternativeliesinV(f
3
g);thetest statistisbasedonthef2g-inverseswith
k = 1;2 oered no power and the best power has been attained at k = 3. The alternative
(4)
II;2
does not lie in a spei eigenspae. Low power has been observed for k = 1 and k = 2, and the
best empirial power has been observed when k = 3. Note that when =
(2)
II;2
;
(3)
II;2
;
(4)
II;2 , the
weighting W
n
=I
3
displayed lowpowerandit appearedpreferable to useageneralized inverse or a
f2g-inverse.
In Table 5,multivariate sampling from a singular normal distribution of dimension six and rank
fourwasperformed. Ingeneraltherejetionratesweresatisfatoryunderthenull,exeptforQ
n (S
1 n
)
when =
III;1
;
III;2
, whih underrejets slightly. In thease =
III;1
, slight underrejetion
ourred at the10%nominallevelfor Q
n (S
k n
),k =1;2;3. When=
III;1
,theempirialpowers
of Q
n (I
6
) and Q
n (S
n
) = Q
n (S
4 n
) were similar for the alternatives inluded in the study. All the
alternativesbelongtotheeigenspaeV(f1g). Underthesealternativesthebestpowerswereattained
by Q
n (S
n
) = Q
n (S
4 n
). When =
III;2
, the best power was observed for k = 2 when the
alternative was
(1)
III;2
, whih belongs to V(f10g). The best f2g-inverse was the one with k = 2
underthe alternative =
(2)
III;2
, whih isalso in theeigenspae V(f10g). The dierenes withthe
weighting W
n
=I
6
were rather small. When =
(3)
III;2
;
(4)
III;2
,the generalized inverse oered the
highestpower. When =
III;3
,the best power wasreahed bythe test statisti Q
n (S
k n
) based
on a f2g-inverse with k = 1, whih an be explained beause
(1)
III;3
2 V(f8g) and the dimension
of that eigenspae is one. When =
(2)
III;3
, the best power is observed with Q
n (S
2 n
). The test
statistiQ
n (S
1 n
) oeredlowpowerunderthatpartiularalternative. Ontheotherhand,thepower
dierenesbetweenQ
n (S
k n
),k2wererathersmall. Thealternative=
(3)
III;3
belongstoV(f2g).
Consequently, Q
n (S
k n
),k2,haveno power. Thebestempirialpowerisobtainedwhenk =3for
thef2g-inverse,but the generalized inverse exhibitsalso highpower. Theweighting W
n
=I
6 oers
some power, but that proedure was signiantly less powerful than the generalized inverse. The
alternative=
(4)
III;3
belongstotheeigenspaegeneratedbythenonnulleigenvalues. Consequently
thebest poweris attainedwiththe f2g-inverse withk=4.
Finally,weanalyzetheresultsinTable6. Ingeneraltherejetionrates werereasonableunderthe
null. When=
IV;1
;
IV;2
,someunderrejetion hasbeenobservedfor Q
n (S
k n
),k =1;2;3,whih
seemedmore pronouned at the 10%nominal level. Overrejetion ourred for Q
n (S
n
)=Q
n (S
6 n
)
at the5%and 10% nominallevels. Some underrejetion hasbeen observed for Q
n (S
1 n
) when
sam-pling from a normaldistribution with ovariane matrix =
IV;3
. The eigenvalues of =
IV;1
are
1
= ::: =
5
= 1 and
6
= 5:510
3
. The alternatives =
(1)
IV;1
;
(2)
IV;1
;
(3)
IV;1
are all
in the orthogonal omplement of the eigenspae assoiated to
6
. For all these alternatives, the
empirial power inrease with k, and the best powers are attained by Q
n (S
k n
), k = 5;6. In
gen-eral the dierenes between the weighting W
n
= I
6
and W
n
= S
n
were rather small. When
=
IV;2
,
1
=
2
= 1;
3
= 0:97;
4
= 0:94;
5
= 0:75 and
6
= 5:310
3
