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Transcript of I -- I › ~boos › library › mimeo.archive › ... · 2019-07-25 · Madan Lal Puri and Pranab...
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A CLASS OF RANK-ORDER TESTS FOR AGENERAL LINEAR HYPOTHESIS
by
Madan Lal Puri and Pranab Kumar Sen
Courant Institute of Mathematical Sciences,New York University and University of North Carolina
Institute of Statistics Mimeo Series No. 551
September 1967
Research sponsored by the Office of Naval Research,Contract Nonr-285(38), and by the National Instituteof Health, Public Health Service, Grant GM-12868-03.Reproduction in whole or in part is permitted forany purpose of the United States Government.
DEPARTMENT OF BIOSTATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
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A CLASS OF RANK-ORDER TESTS FOR A
GENERAL LINEAR HYPOTHESIS*
By Madan Lal Puri and Pranab Kumar Sen
Courant Institute of Mathematical Sciences, New York University
and University of North Carolina at Chapel Hill
~~. For a general multivariate linear hypothesis testing
problem a class of permutationally distribution-free rank-order
tests is proposed and studied. Along with a multivariate cum
"multistatistics generalization of Hajek's (1966) results on
the asymptotic normality of rank-order statistics (for
regression alternatives), the asymptotic distribution theory
of the proposed tests and their asymptotic power properties
are studied. Asymptotic optimality of the proposed tests for
specific alternatives is established and a characterization
of the multivariate multisample location problem [cf. Puri
and Sen (1966)] in terms of the proposed linear hypothesis
is also considered.
* Research sponsored by the Office of Naval Research, ContractNonr-285(38), and by the National Institute of Health,Public Health Service, Grant GM-12868-03. Reproductionin whole or in part is permitted for any purpose of theUnited States Government.
2
ratio test is one of the mostly adopted ones. In this paper,
(1958, Chapter 8) and Rao (1965, Chapter 8)]; the likelihood
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= 1, •.• , Nv
l<v<oo,,i < i < N
H : R = OpXq ,o S\. I'IJ
(1.1 )
_ ( (1) (q))1are unknown parameters and ~Vi - c Vi , ••. , cVi ,i
the assumption of multinorma1ity of F(x) is relaxed and aI'IJ,.
class of Chernoff-Savage-Hajek type of rank-order tests is
proposed and studied. These tests are valid for all continuous
where a' = (al , .•• , a p ) and R = ((~jk))j-l p. k-l q1"\1 s\:, - , ••• " - , ••• ,
against the set of alternatives that ~ is non-null.
A variety of tests for Ho in (1.2), based on the assumed
normality of F(x), is available in tne literature [cf. AndersonI'IJ
1. Introduction. Let ue consider a sequence of stochastic"V'VVVVV'V~
are known non-stochastic vectors. Having observed ~v and
assuming some conditions on ~Vi's (to be stated in Section 2),
we want to test the null hypothesis
matrices ~v = (~Vl'···' ~VN ), 1 ~ v < 00, wherev
- ( (1) x(p))' i - 1~vi - XVi ,.~., vi ' - , ... , Nv are independent stochastic
vectors having continuous cumulative distribution functions
(cdf) FVi(~)' ~ € RP, i = 1, ... , Nv , respectively. It is
assumed that
(1.2)
is studied.
rank-order statistics are defined. Section 3 is concerned
3
for i-I < t i i 1, ... , Nv ;Nv< - , =-Nv
j = 1, ... , p .~v . (t ), J
F(x). In Section 2, along with the preliminary notions, these'"
the last Section, relationship of the multivariate multisample
location problem with the linear hypothesis in (1.1) and (1.2)
with certain permutational invariance structure of the joint
utilized for the construction of permutationally distribution
free tests for Ho in (1.2). In Section 4, by a generalization
of Hajek's (1966) results, the asymptotic joint normality of
the proposed rank-order statistics (for nearby alternatives)
is established. Section 5 deals with the asymptotic power
distribution of Ev (when H in (1.2) holds), and this is then'" 0
properties of the proposed tests for nearby alternatives,
and Section 6 is concerned with the asymptotic optimality of
the proposed tests, granted certain conditions on F(x). In"-
2. Preliminarv notions and fundamental assumDtions. Let Rv(~)~ I'V'V'VV"V'\AI 'V'V\.I~'VVVV'V~ 1
(.) (.) (.)be the rank of Xvr among the Nv observations Xvi , ... , Xv~ ,
vfor i = 1, ... , N ; by virtue of the assumed continuity of F(x)v "-ties may be ignored, in probability, for all j = 1, •.. , p.
Let ~v .(t) be a function defined on (0,1) and taking constant, J
values over intervals [i/Nv ' (i+l)/Nv )' 0 ~ i ~ Nv ' i.e.,
there is some function ~j(t) defined for 0 < t < 1, for which
(2.1)
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definitions lead to asymptotically equivalent results and the
theory follows on almost parallel lines, for simplicity of
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for all
:::l, •.• ,p;for j
j =1, ... , p.
J'l
lim [~V .(t) - ~j(t)J2dt ::: 0V-><D 0 ' J
NV
-> 00 as v -> co ), and
(2.4)( II)
j =:: 1, ... , p ,
k=::l, •.• ,q.
Our proposed test for Ho in (1.2) is based on the stochastic
matrix
Consider now
Define
(2.2)
notations and for intended brevity of discussion, we will
consider only the scores defined by (2.2).) Concerning
~V .'s, we assume that, J
(2.3)(I) ~i~oo ~V,j(t) ::: ~j(t) exists for all 0 ~ t ~ 1 and
constant, for j = 1, ... , p, (where of course
where UV(l) ~ ..• ~ UV(N ) denote an ordered sample of Nvvobservations from the ~ectangular distribution over (0,1).
(We may also definea~j)(a) =:: ~j(a/Nv+l) for a =:: 1, ... , Nv
and j = 1, ... , p. However, as it is known that the two
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(this can always be done by suitably adjusting ~ in (1.1)),
Concerning ~Vils, the following assumptions are made:
k = 1,.,., q;
for all
1- Nv
NV
Sup (2:: [c (k) ]2} < co for all k = 1, ..• , q •vi=l vi
M = ((nz o"))j 0' 1 •""v v,JJ ,J = , ... ,p
NV (k)L cVi = 0 for k = 1, ... , q and 1 < v < co ,
i=l
lim (Max [c (k) ]2j;V [c (k)]2 ] ~ 0V->OO. 1<i <N ' vi 1=1 vi
--v
(2.9) (V)
Let then £v = ((CV,kk'))k,kl=l, ... ,q where
(2.7) (III)
(2.8)(IV)
(2.6)
(2.l0)(VI) Rank of £v = q ~ 1 .
(In fact, by reparameterization in (1.1), £v can always be
made of full rank. Hence, (2.11) is no restriction to our
model.) Let us also define
N
(2 12) 1 (~a(j)(R(j))a(j')(R(jl)• ~V,jjl = Nv-i f;r--V vi --v vi
(2.13)
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(Later on, we shall consider a theorem which establishes that
under certain conditions on F(x), (2.14) holds.) Finally, let.....
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j=l, ... ,p;
j=l, ..• ,p,
Rank of ~v = pq .
Rank of M = P > 1 •.....v -
J 1 2 T2~jj = 0 ~j(t)dt - ~j ,
and its rationality will be made clear in the next section.
denoted by H[j](X) (j = 1, ... , p) and the bivariate (joint)
marginals bYH[j,j'](X'y) for j I j' = 1, ... , p. Further, let
-1 (( ))-1 (( jk j'k'We denote by £v = dv, jk; j I k' = dv ' )) the
reciprocal matrix of £v. We may define formally our proposed
test statistic as
Before that, we would like to introduce the following notations.
Let H(x) be a p-variate cdf whose univariate margina1s are.....
(2.l4)(VII)
(where ® refers to the Kronecker-product of two matrices).
By (2.9) and (2.14), we obtain that
(2.18)
(2.16)
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1, •.. , P ,j f. jl
r (H) = lim'" v->oo
and
Permutation distribution of Sv and the rationalitv of ~v'"V'V'VVVV'VV'V 'V'V"VVVV'VVVV """ """'" """""~ """
Finally, let
It follows from (2.9), (2.11) and (2.11) that under the
assumption (2.23) (VIII) k(H) is of rank p and is finite,
~(H) will also be of rank pq, and ~(H) is finite, provided
lim £v exists. However, SUPvllr~(H) II will be finite, by (2.9).v->oo '"
Under Ho in (1.2), ~v is composed of Nv independent and identically
distributed random vectors. Hence, the joint distribution
of ~v remains invariant under the Nvl permutation of the Nv
vectors ~Vl"'" ~VN among themselves when (1.2) holds. Wev
consider now the basic rank-permutation principle, which is
essentially due to Chatterjee and Sen (1966) and discussed also
by Puri and Sen (1966). We define a p X Nv rank matrix ~v
3.
(2 . 20 ) Tl j j I (H)
(2.21)
(2.22)
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R(l) . R(l)vI VNv
(3.1) ~v = = (~Vl' ..• , R ) ,",-VNv. .R(P) R(P)
vI VNv
where ~Vi(1 ) R(P)), for i 1, ..• , Nv · Each= (RVi , ••. , = rowvi
of ~V is a permutation of the numbers 1, ••• , Nv ' there being
thus in all (Nv!)P possible realizations of ~v. Let us
rearrange the columns of ~v in such a way that the first row
has elements 1, ••. , Nv in the natural order, and denote the
*corresponding matrix by ~v. ~v is said to be permutationally
equivalent to a matrix ~~l), if it is possible to obtain
~~l) from ~v only by permutations of the columns of the latter.
* *Thus, corresponding to each ~v' there will be a set ~(~v) of
Nv! possible realizations of ~v' such that any member of this
*set will be permutationally equivalent to ~v. Now, the probability
distribution of Bv over the (Nv!)P possible realizations will
depend on F(~), even when Ho in (1.2) holds (unless F(~) has
coordinate-wise independent marginals). Thus, rank-statistics,
like Qv in (2.6), are, in general, not distribution-free
*under H in (1.2). However, given a particular set ~(Rv)o "'-(of Nv! realizations), the conditional distribution of Bv
*over the Nv! over the Nv! permutations of the columns of Bv
would be uniform under Ho in (1.2), i.e.,
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1, ... , P ;
~(H)'"
j 1= j'
limv->co
and
Permutation distribution of Sv and the rationalitv of ~v."V'V'VVV'V'VVV~ IV" t'\I'V" t'\I'V"~ IV"
Finally, let
It follows from (2.9), (2.11) and (2.11) that under the
assumption (2.23) (VIII) k(H) is of rank p and is finite,
~(H) will also be of rank pq, and ~(H) is finite, provided
lim £v exists. However, sUPvlll~(H) II will be finite, by (2.9).v->oo '" .
Under Ho in (1.2), ~v is composed of Nv independent and identically
distributed random vectors. Hence, the joint distribution
of ~v remains invariant under the Nvl permutation of the Nv
vectors ~Vl' •.• ' ~VN among themselves when (1.2) holds. Wev
consider now the basic rank-permutation principle, which is
essentially due to Chatterjee and Sen (1966) and discussed also
by Puri and Sen (1966). We define a p X Nv rank matrix Bv
(2 . 20 ) Tl j j , (H)
(2.21)
3.
(2.22)
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R(l) . R(l)vI VNv
(3.1) ~v = = .(~Vl' .•• , R ) ,",VNv. .R(P) R(P)
vI VNv
where ~Vi(1 ) R(P)), for i 1, .•• , Nv ' Each= (RVi , ... , = rowvi
of ~v is a permutation of the numbers 1, ... , Nv' there being
thus in all (Nv!)P possible realizations of ~v. Let us
rearrange the .columns of ~v in such a way that the first row
has elements 1, ... , Nv in the natural order, and denote the
*corresponding matrix by ~v. ~v is said to be permutationally
equivalent to a matrix ~~l), if it is possible to obtain
~~l) from ~v only by permutations of the columns of the latter.
* *Thus, corresponding to each ~v' there will be a set ~(~v) of
Nv ! possible realizations of ~v' such that any member of this
*set will be permutationally equivalent to ~v. Now, the probability
distribution of ~v over the (Nv!)P possible realizations will
depend on F(~), even when Ho in (1.2) holds (unless F(~) has
coordinate-wise independent marginals). Thus, rank-statistics,
like ~v in (2.6), are, in general, not distribution-free
*under H in (1.2). However, given a particular set ~(Rv)o '"
(of Nvl realizations), the conditional distribution of ~v
*over theNv ! over the Nv! permutations of the columns of Bv
would be uniform under Ho in (1.2), i.e.,
9
may work with a positive-semidefinite quadratic form in the pq
elements of S , where the discriminant of the quadratic form""y
E( S I~} == opxq .""y y ""
whatever be F(~). Let us denote by~y the permutational
(conditional) probability measure generated by the conditional
law in (3.2). Then, by routine computations (along the lines
of Puri and Sen (1966)), we obtain that
where d ok" 'k I I S are defined by (2.15).y,J ,JSince ~y is a stochastic matrix, for actual test con-
struction it is more convenient to use a real valued function
E( Sy ,jk Sy, j 'k' I~} = dy , jkj j 'k I for j, j I = 1, , p,
k,k' = 1, , q,
of ~y as a test-statistics. By an adaption of the same arguments
as in Chatterjee and Sen (1966) and Puri and Sen (1966), we
is the inverse of the matrix Qy' which has the elements
dY,jkjj'k'
in (3.2). This leads to the test statistic Ly '
defined by (2.17). Ly will be stochastically large if ~y is
stochastically different from O. For small values of y"" *(i.e., Ny)' the conditional distribution of Ly ' given ~(Ry)'
can be obtained with the aid of (3.2), and a conditionally
distribution-free test for Ho in (1.2) based on Ly can be
constructed. This, however, requires the evaluation of the
Ny! realizations of ~y (under ~), while ~y is ~-invariant.
The task becomes prohibitively laborious and for large values
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Proof: The computation of the means and covariances of
we consider the following.
theorem is a consequence of the first part and the results in
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The proof that M - Ad L> Opxp as v -> (X)"",v ~'"V "'"
Proof:
of v, we are forced to consider the following large sample
THEOREM 3.2. Under~, the joint distribution of the
elements of ~v converges, in probability, to ~ multivariate
normal distribution with null means and covariances given by
approach in which we approximate the true permutation distri-
the elements of ~v follows directly from (3.3) and (3.4).
Since £v is positive definite, in probability (by the preceding
theorem), the asymptotic multinormality of the permutation
brevity, the details are omitted. The second part of the
as v -> co .
follows more or less on the same line as in the proof of
Theorem 4.2 of Puri and Sen (1966), and hence, for intended
bution of ~v (under ~v) by that of a chi-square distribution
with pq degrees of freedom (d.f.). As a basis of this study,
(3.4) .
(2.9) through (2.23). Q.E.D.
NvTHEOREM 3.1. Let Hv(~) = (l/Nv ) f=l FVi(~)' and let
.f6v = lI.(H) IH == H be defined ~ in (2.21). Then, under the
assumptions (I)vto (V) of Section 2, [~v _ ~] ~> £px p ~
V -> co. Thus, under the assumptions (VI) and (VII), ~v
is positive definite, in probability, and £v - ~(Hv) JL> opq~pq"'" .,
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By virtue of the preceding theorem, we arrive at the
if ~V
distribution of S follows readily by an appeal to the multi""v
> 2 , reject H i (1 2)v n • ;- Apq,a 0
Hence we have the following large sample test procedure:
< X~q,a ' accept Ho '
variate extension of the well-known Wald-Wolfowity-Noether
Hoeffding-Motoo-Hajek permutational central limit theorems
[cf. H~jek (1961, p. 522)]. Hence, the theorem.
following theorem through a few simple steps.
2 2where P(Xt ~ Xt,a} = a(O < a < 1), the level of significance.
4.~~ 2£ ~v· We introduce the folloWing
notations. Let
and let HV[j](X) be the marginal of the jth variate of Hv(~),
THEOREM 3.3. Under the conditions (I) to (VIII) of
Section 2, ~v' defined by (2.17), has ~ permutation distribution
(under~) asymptotically converging, in probability, to a
chi-square distribution~ pq degrees of freedom.
(4.1)
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for j, j' = 1, .•• , P and k, k' = 1, •.• , q; the corresponding
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• ~j(HV[j]~X))~j(HV[j](Y))dFvr[j](X)dFvs[jJ(Y)
+ rr FVi [ ,](X)[l-FVi[j](Y)]J,J -00 <X <Y<oo J
Ajj(i ) =jl"f FVi[j](X)[l-Fvi [ '](Y);rs J -co <X <Y<OO J
. Ajjl(i;rs) ,
, I
. ~ j ( Hv [ j ] (x ) ) ~ j I (Hv [ j I ] ( y) ) dFvr [ j ] dFV s [ j I ] (y) ,
pqxpq matrix is denoted by ~v. Throughout this section it will
for j = 1, ... , p. Also, let FVi[j](x) and FVi[j,j'](x,y)
be the univariate (jth) and bivariate ((j,j' )th) marginals
corresponding to FVi(~)' for j I j' = 1, .•• , p. We define
for j I jl = 1, ... , P and i, r, s = 1, ..• , Nv; the subscript
V in (4.2) and (4.3) is understood. Let then
for i, r, s = 1, ... , Nv and j = 1, .•. , p;
(4.2)
(4.4)
rOO .00
(4.3) Ajjl(i;rs) = J-ro
)_00 [FVi[j,j,](x,Y)-Fvi[j](X)Fvi[jl](Y)]
as v -> 00, and hence
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-> 0
rna x [A.. I ( i . ) - 'Tl j . , (F)]} = 0 ,l~j,jl<p JJ ,rs J
( maxl<i<N--v
~v is positive definite and finite.
(Jv, jkj j I k' - Cv , kk I 'Tl j j , (F) -> 0 as v -> 00,
( maxl<i,r,s<N- -v
limv->oo
By virtue of (2.8) and (2.9), maxl<i<N--v
limv->oo
Proof:
(4.8)
Before presenting the main theorem of this section, we consider
having a continuous density function f(x), x € RP , and (ii)"" ""
be assumed that (i) F(x) in (1.1) is absolutely continuous""
the following lemma.
LEMMA 4.1. Under (1.1), (2.8) and (2.9),
2: - A(F)® C -> opq)(pq as v -> 00. Hence under (2.11)""V "" ""v "" -and (2.23), ~v is positive definite in ~ limit as v -> 00.
for all j,jl = 1, ••. , Pj k,k' = 1, ... , q. This completes
where 'Tl •• ,(F) is defined by (2.20). From (4.4), (4.7), (2.8)JJ
and (2.9), we obtain after some simplifications that
the first part of the proof. The second part follows from
From (4.2), (4.3), (4.6) and some routine computations, it
follows that
(4.5)
(4.6)
(4·7)
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results on limits of sequences. Q.E.D.
Let us now introduce the following notations. Let
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for all j,j' = l, ... ,p; k,k' = l, ... ,q,
COV{ZV,jk' ZV,j'k'} = crV,jkjj'kl ,
NV(4.11 ) ZV,jk = f=l ZVi (jk) for j = 1, ... , Pj
k = 1, ... , q.
14
Straight-forward but somewhat lengthy computations yield that
where cr v 'k 'lk,IS are defined by (4.4)., J , J
In this context, we bring now the findings of Hajek
(1966), who considered the special case of p = q = 1 and
for j = 1, •.. , Pj k = 1, ... , q and i = 1, ... , Nv • Finally
let
for j = 1, ... , p, i,i' = 1, ... , Nv . Also let
(4.10 )
(where u(x) is equal to 1 or 0 according as x is positive or not),
(2.11), (2.23) (with H =F), (4.8) and some well-known
(4.12)
that
as Y -> co
o ,2lim E[[Sy 'k-E(Sy 'k)] - Zy 'k} /Var(Zy 'k)v->oo ' J , J , J , J
details, we obtain that
for all j = 1, ••. , p; k = 1, ••• , q. We shall utilize (4.13)
to establish a comparatively stronger result. With this end
in view, we first consider the following lemma.
15
discussed the asymptotic normality of the corresponding rank
order statistic (which is just anyone of the SY,jk's). Thus
referring to his elegant paper and thereby avoiding the
By virtue of (4.12), (4.13) and lemma 4.2, it follows
LEMMA 4.2. Let ~y = (a yl , ••. , avt ) and £y = (byl '···' byt )
t > 1, be two sequences of stochastic vectors, such that
v(aYj - b ,)/V(a ,) -> 0 as y -> 00 for all j = 1, ..• , t.YJ YJ --
Then
(4.13)
making use of the conditions stated in the lemma.
for all j, £ = 1, .•• , t.
The proof follows by expressing cov(ayj,ay£) as
COV(byj,by£) + Cov(byj,ay£-by £) + cov(aYj-bYj,by£)
+ cov(aYj-bVj,ay£-by£)' applying Cauchy-Schwarz inequality to
the second, third and fourth terms on the right hand side and
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f or all j, j I = 1, ..• , p; k, k I = 1, •.. , q.
16
Thus it follows from (4.13) and Lemmas 4.2 and 4.3 that
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p q
Zv = 2:: 2:: £jk ZV,jk 'j=l k=l
N1 ~~ t;q (k) (k) (j)= N L-. JJ'k( c Vi -c Vi )B j (r i) (XVi ) ,V r=l j= k=· '
where
An immediate consequence of Lemma 4.2 is the following,
for i = 1, ... , Nv . By definition, gi{~Vi) are independent
random variables. Further
view, we define
by showing that any arbitrary linear combination of ZV,jk'S
has asymptotically a normal distribution. With this end in
LEMMA 4.3. If ~V has asymptotically a multinormal distri
bution with~ vector ~V and dispersion matrix kv and if for
each j(= 1, ... , t) v(aVj - bVj)/V(a Vj ) --> 0 as v --> 00, then
bv ~ also asymptotically the~ multinormal law.
for proving the asymptotic normality of ~v' it is sufficient
to show that (ZV,jk' j = 1, .. " p; k = 1, ... , q) has asymptoti
cally a multinormal distribution. This will be accomplished
where Jjk's are real and finite and not all of them are zero.
From (4.10), (4.11) and (4.15), we obtain that
Nv(4.16) Zv = f=l gi(~Vi) ,
(4.17)
(4.15)
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is finite and positive by (4.5). Proceeding then precisely
on the same line as in Htjek (1966, Section 5 specifically and
using a linear combination such as (4.15) with his Yv's),
it can be shown that under (4.5), the random variables
(gi(~Vi)) satisfy the Lindeberg condition for the applicability
of the central limit theorem. Hence, we arrive at the
following.
THEOREM 4.1. Under the assumptions (I) - (V) of Section 2
and (4.5), the elements of ~v - E~v has asymptotically a
multivariate normal distribution with zero means and dispersion
matrix L: v •""
Now, under the model (1.1) and granted the assumptions
made in Section 2, it follows from routine computations that
for all j = 1, ... , p; k = 1, ... , q; and by Lemma 4.1, the
covariance matrix ~v is asymptotically equivalent to k(F)~ £v·
In the next section, we shall make use of this result for studying
the asymptotic properties of the test based on ~v in (2.17).
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We denote by
...* f;;..L ~q ~ S S '\')jj t (F) .Ckv
kt~v = > > v 'k V jtk t 'I
= j t =1 = k t =1 ' J ,
Asvmntotic nronerties of the test.~~ 'V'\J ~ 'V'\J'V'\J
(5.4)
5·
has asymptotically (under (1.1) and the assumptions of Section 2)
where
random variables, it follows that
Further, using Theorems 3.1 and 4.1, it is straight-forward
to show that under the assumptions made in Section 2,
,a non-central chi-square distribution with pq degrees of
freedom and non-centrality parameter
for all j,jl = 1, ... , p, where ~jjl IS are defined by (2.19),
(2.20) and (2.21). Then, using Theorem 4.1, Lemma 4.1,
(4.19) and some well-known result on the limiting distributions
of quadratic forms in asymptotically normally distributed
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Thus, we arrive at the following theorem .
THEOREM 5.1. Under (1.1) and the assumptions made in
Section 2, the statistic ~v has asymptotically a non-central
chi-square distribution with pq degrees of freedom and non
centrality parameter ~~, defined ~ (5.3).
Theorem 5.1 may be used to study the asymptotic power
properties of the test in (3.5) and to compare its efficiency
relative to standard parametric tests. Since in general
linear hypothesis testing problems, the different test-criteria
do not have uniformly good or bad performances, [cf. Anderson
(1958, Chapter 8)], it may be quite involved to give a neat
picture of the relative-efficiency of the proposed tests with
respect to all the parametric ones. Among the notable
parametric tests, Roy's largest characteristic root-criterion
is admissible and powerful against certain alternatives,
while the likelihood ratio test is so for certain other
alternatives. However, borrowing the ideas of Wald (1943),
it can be shown that in the sense of Wald, the likelihood
ratio test has asymptotically the best average power (on
suitable ellipsoids in the parameters ~jk's). In this sense,
the likelihood ratio test may be regarded as optimum. As such,
we will consider the asymptotic relative efficiency of our
proposed test with respect to the likelihood ratio test .
1, ... , p
20
Following the notations of Anderson (1958, Chapter 8),
we define the log-likelihood criterion (adjusted by a suitable
multiplying factor) by Tv (say,) and conclude from his Theorem
8.6.2 that Tv has asymptotically (under Ho in (1.2)) a chi
square distribution with pq degrees of freedom. Hence, the
same test procedure as in (3.5) applies to the likelihood ratio
test (with Zv replaced by Tv). We denote the covariance
matrix (assumed to be finite) of F(x) by r = ((p "k(F)))" k~ ~ sJ J, =
and assume it to be positive definite. The corresponding
reciprocal matrix is denoted by ,-1 = ((~jk(F))). Since the
likelihood ratio-test criterion is exclusively based on linear
estimators of ~, generalizing the method of approach of
Eicker (1963) and following essentially some routine steps it
can be shown that under the assumptions made in Section 2,
Tv has asymptotically a non-central chi-square distribution
with pq degrees of freedom and noncentrality parameter
It is seen that in multivariate situations, the asymptotic
relative efficiency (A.R.E.) of the test based on Zv with
respect to the one based on Tv' as measured by ~Z/~T' depends
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(i) If p = 1, no matter whatever be ~ and £v
for the two sample location problem. A few particular cases
may be worth mentioning here. First, if ~(u) = u (i.e., the
Wilcoxon scores case), (5.7) reduces to 12~11(F)( j' CD f 2 (x)dx)2,-00
which has known values (as well as lower bounds) for various
which happens to coincide with the usual efficiency expressions
matrices and as such
F(x). Secondly, if ~(x) is the inverse of the standard
not only on ((v, ,,(F))) and ((~, ,,(F))) but also on RandJJ . JJ. ~
£v. The following points are worth noting in this context.
for p = 1.
normal cumulative distribution function (i.e., the normal
scores case), (5.7) is always greater than or equal to unity,
the equality sign holds only when F is normal. This clearly
indicates the asymptotic efficiency of the proposed tests
it readily follows that both ~(F) and v(F) are diagonal"" ""
Hence, if we use the normal scores for ~Vj'S, again (5.9)
p(5.8) (ii) If F(~) =TT F[J'](X J')'
j=l
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becomes greater than or equal to unity, where the equality
sign holds only when F(x) is multinormal.
(iii) If we use the normal scores for ~Vj'S, and if the
parent cdf F(x) is a multinormal cdf, it readily follows from"'"
(2.19), (2.20) and (5.1) that ~(F) = ~(F), and hence, from
(5.3), (5.6) that 6~/ 6 T = 1. Or in otherwords, for parent
normal distributions, the use of normal scores leads to
asymptotically most efficient tests.
(iv) In general, for arbitrary F(~), 6 ~/ 6 T is bounded
below and above by the minimum and maximum characteristic
roots of ~(F)~-l(F), (the proof follows by a straight-forward. .
application of a theorem by Courant (cf. [11]) on the bounds
of the ratio of two quadratic forms). Now, the bounds of
£(F)~-l(F) have been studied in detail by the authors (Sen
and Puri (1967)) in connection with the multivariate one sample
location problem. As such, to avoid repetition, the details
are omitted.
6. AsvmDtotic oDtimalitv of ~ for certain F(x). We shall~~"""""""'''''''''''''' v""""""",,,,,,,,,,,~ "'"
study the asymptotic optimality of ~v for a class of parent
distributions. We assume that F(x) has the absolutely continuous"'"
density function f(x), and define,"'"
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for i = 1, ... , P .
I dfi[i](x) = dx f[i](x), for i = 1, ... , P •
We define the statistics
N~ (k) I (j), (1) (p)
UV,jk =~ GVi [fj(XVi ~Vi , ... , ~Vi )/f(~Vi)] ,
Let then
for j = 1, ... , P and k = 1, ..• , q. We also define
reciprocal matrix of !v' and
for all j,jl = 1, ... , p, and let T = ((T'k 'Ik')) be the'" . J,J
pqxpq matrix whose elements are cv,kk' X~jj" where £v is
defined by (2.10). Finally let T-l = ((Tjk,j'k')) be the"'v v
(6.2)
(6.4)
Then by an adaptation of the line of approach of Wald (1943),
*we can show that (i) under Ho in (1.2), Uv has asymptotically
a chi-square distribution with pq degrees of freedom (provided
((~jj')) is finite and positive definite) and (ii) the test
*based on Uv has asymptotically the best average power (on the
family of ellipsoids
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not aware of necessary conditions for the same.
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and consequently,
*to Uv )' will
Let us now define
Nv (k)' (") (")
VV,jk = ~ c Vi fj[j](Xvr )/f[j](Xvr ) ,
( 6.8)
We shall show that under certain conditions on F(x), the test'"
based on ~v has also asymptotically the best average power
on the same family of ellipsoids, provided ~jfS are chosen
suitably. Our treatment deals with a set of sufficient
conditions for this asymptotic optimality and the authors are
for j = 1, ... , p, k = 1, ... , q. If (6.9) holds, it follows
that UV,jk'S are linear functions of VV,jk'S,
the quadratic form based on VV,jk'S (analogous
*also have the same properties as that of Uv .
We now define
(6.11) ~j(u) = f;[j](F[~](U))/f[j](F[3]~u)) , 0 < u < 1,
j = 1, ... , P ,
(6.10)
where h jjl 's are real constants, not all zero, for j = 1, ..• , p.
(6.9) holds for (i) the multivariate normal distribution,
(ii) any coordinate-wise independent distributio~and it may
also hold for many other distributions.
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25
and define S 'k's as in (2.5). Following then H'jek's (1962)V,Jelegant approach it is seen that S - V converges inv,jk v,jkquadratic mean to zero for all j = 1, ... , p, k = 1, ... , q.
Consequently, on using Lemma 4.2 and some routine computations,
we obtain that for ~j'S given by (6.11), Lv in (2.17) is
stochastically equivalent to the quadratic form in VV,jk'S,
under (1.1) and the assumptions of Section 2. Since, under
*(6.9), this quadratic form in VV,jk'S is also equal to Uv in
(6.7), it follows that under (1.1), the assumptions made in
Section 2 and (6.9), (6.11),
(6.12)
Consequently, we arrive at the following.
THEOREM 6.1. Under (i) the model (1.1), (ii) the- ~
assumptions made in Section 2, and (iii) (6.9) and (6.10)
Lv has asymptotically the best average power on the family
of ellipsoids in ~'k's, defined by (6.8).---=---- - J -
In particular, if F(x) is normal, the condition (6.9)1'\1
.. -is satisfied and (6.11) lead to the coordinate-wise normal
scores. Hence, ~v based on normal scores statistics leads to
asymptotically best test for the family of ellipsoids in
(6.8). A special case of this theorem (that is, for p = 1)
is dealt in an interesting power of Matthes and Truax (1965).
26
7 - A characterization of the multivariate multisamole location'"~ tV'\J IV'V'v~~~
~- Let ~ 1<1' ••• , Xknk
be nk independent and identically
distributed p-variate random variables having a continuous
p-variate cumulative distribution function Fk(~) for k::::l, •.. ,c(>2).
In the multivariate multisample location problem [cf. Puri and
Sen (1966) and the other references cited therein], we may let
Fk(X) = Fl(x - e), K:::: 2, ... , c, e l :::: 0'" '" "'k '" '"
We define N :::: n l + ... + nc ' and consider a sequence of N
p-vectors ~l' .•• ' ~N' of which the first nl observations are
from the first sample, the next n2 from the second sample and
so on. Thus, (7.1) will correspond to (1.1), if we let
q :::: C - 1, ~jk :::: ej,k+l for k = 1, ... , c-l, and where ~ViIS are
null vectors for an 1 < i ~ nl , £Vi = (0, 1, 0, ••• , 0)1
for nl + 1 ~ i ~ nl + n2 , and so on. Thus, the results
derived in this paper, also generalizes the results of Puri
and Sen (1966). The statistic Lv in (2.17) can be shown
to be identical with the corresponding LV in Puri and Sen (1966),
when (7.1) holds. Consequently, the efficiency considerations
made in Sections 5 and 6 also applies to the multisample
multivariate location problem.
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[2]
[3]
[4 ]
[5]
[6]
[8]
REFERENCES
Anderson, T. W. An introduction to multivariate
statistical analysis. New York: John Wiley, 1958.
Chatterjee, S. K., and Sen, P. K. (1966). Nonparametric
tests for the multivariate multisample location problem
S. N. Roy memorial volume. (Edited by R. C. Bose et al).
University of North Carolina (in press).
Chernoff, H., and Savage, I. R. (1958). Asymptotic
normality and efficiency of certain nonparametric test
statistics. Ann. Math. Statist. 29, 972 - 994.
Eicker, F. (1963). Asymptotic normality and consistency
of the least squares estimators for families of linear
regressions. Ann. Math. Statist. 34, 447 - 456.
H'jek, J. (1961). Some extensions of the Wald-Wolfowitz
Noether theorem. Ann. Math. Statist. 32, 506 - 523.
H{jek, J. (1962). Asymptotically most powerful rank
order tests. Ann. Math. Statist. 33, 1124 - 1147.
H§jek, J. (1966) Asymptotic normality of simple linear
rank statistics under alternatives, I. (to be published).
Matthes, T. K. and Truax, D. R. (1965). Optimal
Invariant Rank Tests for the k-Sample Problem.
Ann. Math. Statist. 36, 1207 - 1222.
Puri, M. L., and Sen, P. K. (1966). On a class of
multivariate multisample rank-order tests. Sankhya,
Ser A. 28, 353 - 376.
[10]
[11]
[12]
Rao, C. R., Linear statistical inference and its
applications. New York: John Wiley, 1965.
Sen, P. K., and Puri, M. L. (1967). On the theory of
rank order tests for location in the multivariate one
sample problem. Ann. Math. Statist. 38, 1216-1228.
Wald, A. (1943). Tests of statistical hypotheses
concerning several parameters when the number of
observations is large. Trans Amer. Math. Soc. 45, 426
482.
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