ON A CLASS OF MULTIVARIATE
MULTISA}~LE
RANK-OP~ER
TESTS
by
PRANAB KUMAR SEN
University of North Carolina
Chapel Hill, North Carolina
and
MADAN LAL PURl
Courant Institute of Mathematical Sciences,
New York University
Institute of Statistical Mimeo Series No. 474
May 1966
*Research sponsored by Sloan Foundation Grant for Statistics, U. S.
Navy Contract Nonr-285(38) , and by Research Grant, GM-l2868, from the
National Institute of Health, Public Health Service.
DEPARTMENT OF BIOSTATISTICS
UNIVERSITY OF NORTH CAROLINA
CHAPEL HILL, NORTH CAROLINA
1
ON A CLASS OF MULTIVARIATE MULTISAMPLE RANK-ORDER TESTS*
Madan Lal puri* and Pranab Kumar Sen l
Courant Institute of Mathematical Sciences, New York University
and University of North Carolina, Chapel Hill
SUMMARY.
A class of rank-order tests for the multivariate
several sample location and scale problems is proposed and
studied here.
The principle of rank-permutation tests by
Chatterjee and Sen ([6],[8]) is utilized to make these tests
strictly distribution-free, and the well-known Chernoff-Savage
Theorem (cf. [9], [11], [16]) is generalized here to the
multivariate several sample problems.
This takes care of
the asymptotic power-properties of the tests.
1.
INTRODUCTION
Very recently, some attention has been paid to the
development of non-parametric tests in the multivariate
several sample problems.
The earliest test on this line
2
is the permutation test based on Rotelling's T statistic,
proposed and studied by
~Jald
and Wolfowitz [22].
However,
this is a value permutation test and is subject to the usual
limitations of this type of tests.
Chatterjee and Sen
([6],
[8]) have extended the idea of permutation tests to rankpermutation tests in the multivariate case and have considered
a class of genuinely distribution-free tests for location in
the bivariate two sample as well as p-variate c-sample case
*Research sponsored by Sloan
Fo~ndation Grant for Statistics,
U. S. Navy Contract Nonr-285(38), and by Research Grant,
GM-12868, from the National Institute of Health, Public.
Health Service.
10n leave of absence from Calcutta University.
2
(p,c> 2).
Anderson [3] has recently considered the
application of the principle of statistically equivalent
blocks (usually used in setting of tolerance limits)
multivariate non-parametric analysis.
in
His procedure contains
a little bit of arbitrariness of the choice of
Il
cu tting
functions" and moreover, it is really difficult to study the
properties of these tests when the null hypothesis is not
true, even in the asymptotic case.
On the other hand,
Bhapkar [4] has considered a class of location tests for the
multivariate several sample problem, which
only asymptotically.
is
distribution-free
Sen [21] has established the asymptotic
power equivalence of one of the tests by Bhapkar [4] and a
somewhat similar permutation test by Chatterjee and Sen [8].
A more general class of non-parametric tests for the various
types of problems that may usually arise in the multivariate
several sample case, has been considered by Sen ([19],[20],[21]).
These tests are all permutation tests based on appropriate
generalized U-statistics.
However, the above developments may not be regarded as
fully adequate.
First, mostly
the above literature deals
with the location problems, while the other problems seem to
be somewhat neglected.
Second, as in the univariate case,
the rank-order tests are supposed to play a very vital role
in the above development, and in the multivariate case,
practically no work has been on progress on this line.
The
object of the present investigation is to consider a class
of rank-order tests for location and scale in the multivariate
3
multisample case.
In this context, the well-knmm theorem
by Chernoff and Savage [9] (also see [11], [16]) is extended
here to the p-variate c-sample case, for p,c
~
2.
In view
of the fact that for multivariate distributions, the distributions of rank-order statistics mostly fail to be strictly
non-parametric even under the null hypothesis of the identity
of the different distributions, the principle of rank-permutations by Chatterjee and Sen ([6],[8]) has been adopted here
to achieve the desired distribution-freeness of the proposed
class of tests.
The aSYmptotic properties of tbese permutation
rank-order tests (PROT) are studied here with the aid of the
Wald-Wolfovlitz-Noether-Hoeffding-H~jektheorems
(cf. [12]),
and certain stochastic equivalence relations of PROT with the
multivariate extensions of Chernoff-Savage type of rank-order
tests (CSROT) have also been established.
2.
1fa(Ie)
Le t
(k)
= (Xla
PRELIMINARY NOTIONS
(k)
,. . ., Xpa. ), a
= 1,..., nk b e n k
independent vector valued random variables drawn from a p
variate universe with a continuous cdf (cumulative distribution
function)
Fk(~)'
for k = 1, ... ,c; all these c
being independent of each other.
(~2)
samples
It is desired to test the
null hypothesis
(2.1)
~,tbeing the class of all continuous p-variate
where F
€
cdf's.
lie are interested in translation and/or scale type
of alternatives, which may be posed as follows.
Let
4
Fk(x) = F(x + 5k )
""
""
""
where 51' ... ,5
""
""c
real p-vectors.
(2.2)
Then under the translation
type of alternatives, we are interested in testing
He: 21
= •.• = R. c = £
against the set of alternatives that these c p-vectors are
not all identical.
Again, we let
(2.4)
k = 1, ... ,c
are c non-negative p-vectors.
Then under scale
alternatives, we are interested in testing
HO: C1
"" l
= ... =
C1
""c
=
( 2 •5 )
I
""
(where I is the p-vector with unit elements)
against the set
""
of alternatives that
C1 , ... ,C1
"" l
""c
are not all equal.
Let us now combine the c samples into a pooled sample
of size
(2.6)
and denote these N p-variate observations by
Z,
"\Q
a=l, ... ,N;
Z
"0.
=
(Z,_, ••. ,Z
.w.
We then arrange the N values Zia' a
magnitude, and then denote by
set, for a
R~)
pa
)
= 1, ... ,N
the rank of
in order of
x~)
in this
= 1, ... ,nk ; k = 1, ... ,e; i = l, ... ,p. By virtue
5
of the assumed continuity of Fl, ••• ,F c the possibility of ties
can be ignored, in probability. Let now
(k)
(k)
I
(RIa , ••• ,Rpa ) ,
a
=
1,..., n k ; k
=
(2.8)
1,..., c ,
be the N random rank p-tuplets, and we define the collection
(-rank) Matrix by
_
(I)
RN - (R l
'"
'"
so that
~N
(c)
)
",n c
, ..• ,R
is a p X N random matrix:
R(l}
R(l}
ln l
11
RP)(N
",N
=
···
. ..
R(c}
Inc
(2.l0)
...
R(l}
pI
R(l)
pn l
R(c}
pn c
where each row of this stochastic matrix is a random permutation
of the numbers 1, .•. ,N. Let then zi~;a = 1, if the ath smallest
observation of the i th variate values of the combined sample is
from the k-th sample, and otherwise let
k
=
1, ••• , c ; i
=
1, ••• , P .
Le t
zi~~a = 0
{E~~}, a =
= 1, ... ,N;
for a
1, ... , N; i
=
1, ..• , P
J
be given numbers satisfying certain restrictions to be stated
below (cf. Section 4).
We then consider the pc random variables
defined as
T(k)
Ni
=~
~
n ~
k
E(i}z(k) .
Na iN,a'
Let us finally define
k = l, ..• ,c; i = l, •.. ,p.
(2.ll)
6
_(i)
1
EN
= N
N
2::
a=l
(i)
Elb. '
i
=
(2.12)
1,..., P •
Then our proposed test is based on a statistic ~N which
is a positive semi-definite quadratic form in the pc stochastic
(k)
_(i)
variables TWi - EN
' i = l, ... ,p; k = l, ... ,c.
To formulate the test criterion in an appropriate way and
the rationality of the test procedure, we consider the basic
rank permutation principle, which is essentially discussed
in detail in ([6],[8]), and hence is
just sketched in brief.
Each row of the rank collection matrix
~N
defined in
(2.10) is a permutation of the numbers 1, ... ,N, and thus
~N
is a stochastic matrix which can have (Nl)P possible realiza-
* , are said to
Two collection matrices, say RN and RN
be permutationally equivalent when it is possible to arrive
tions.
at one from the other by only a finite number of inversions
of its columns.
matrix
~N
Based on this convention, the collection
defined in (2.10) will be permutationally
equivalent to another matrix
vectors as in
row of
*
~N
~N
* which has the same column
~N
but these are so arranged that the first
iscomprised of the numbers 1, ... ,N in the natural
order, i.e.
1
*
~N =
2
N
*
*
~12 R22
*
RIp
R*
2p
Thus, we have a set of N! collection matrices which are all
7
permutationally equivalent to
* and let this set be denoted
~N'
*
by S (~N)'
Now
*
p - l realizations and the set of all
can have (N!)
~N
these realizations is denoted bye;.
The probability distri-
* over ~N* will evidently depend on the parent
bution of ~N
distributions, even under HO defined hy (2.1). However,
* the conditional distribution
given a particular realization ~N'
of
~N
over the N! permutations of the columns of
be uniform under HO' i.e. if £N is any member of
under Ho '
whatever the parent cdf F may be.
* would
~N
then
Thus if we consider any
test depending explicitly on the collection matrix
formulate
-*
S(~N)'
~N'
and
a test function ~(~N) depending on the completely
specified permutational probability law (2.13), it follows
that such a test will be a similar test for the hypothesis HO '
In the next two sections, we shall consider in detail such
permutationally distribution-free rank order tests.
3.
PERMUTATION RANK-ORDER TESTS (PROT)
In this section, we shall consider a genuinely distributionfree test.
Let us denote by
~
the permutation probability
measure generated by the N! possible permutations of the
columns of
*
~N'
Then, following a few simple steps, we get
E(T(k) - E(i),rr) - a
Ni - N
N-
(3·1)
8
cov
= 1, .. . ,Pj
for i,j
k,q
= 1, ... ,c.
5kq is the Kronecker delta,
and
i, j
l, ... ,p
being the value of ~;} associated with the rank
E(k)
No, i
.
S - R(k}
-
=
Following then the same arguments as in ([6),[8)},
fa •
it seems reasonable to base a permutation test on the set
of p(c-l} contrasts
T~~}
-
~i}
If we denote by
*
Vij(~N}
j
i
*
~(~N)
= 1, .. • ,Pj k = 1, ..• ,c-l.
the p x P covariance matrix with elements
defined in (3.3), then it can be easily seen that the
permutational covariance matrix of the p(c-l} random variables
in (3.4) (taken in that order) is
1
(N-l)
where
®
«1L
n
5
k
kq
- I)}
Thus, if we assume
definite (for the given
-1 *
- ij *
(~N)
tV'\
~
V(R*)
~ ~N
'
stands for the Kronecker product of two matrices
(cf. [2, p. 347]).
~
k,q=l, ... ,c-l
= ~~v
*
~N)'
*
~(RN)
to be positive
and denote its reciprocal by
.
(~N}}i,j=l, ... ,p
then the reciprocal of the
p(c-l) x p(c-l} matrix in (3.5) comes out as
(N -1 )
nk
«N 5kq
nkn
+ .JL,9.)
) k,q=l, .•. ,c-l ®
n
c
¥.-1 (~~1* )
!
•
Then, using (3.6), we get after some essentially simple
steps that ~ can be taken as
9
which i s a lleighted sum of c quadratic forms in ( ~N(k) -~N)
,
k = 1, ... ,c
with the same discriminant, where
k
=
(3.8)
1, ... , c
and
From the remark made at the end of Section 2, it
follows that given RN, the permutation dis~ribution of ~
would be strictly distribution-free under HO defined by (2.1),
and hence an exact size
E
(0
< E <
I) test can be constructed
using this permutation distribution of ~N.
(For details, see
[6), [8)
l/lhich deal specifically with the rank-sum and median
tests.}
* is
In the above context, we have assumed that Z(~N}
positive definite.
As we shall see afterwards that under
very mild conditions on F
a very high probability.
~, this statement holds with
* } happens to be
However, if V(R N
'" '"
€
singular, we can work with the highest order non-singular
minor of
*
~(~N)'
and proceed similarly with the corresponding
subject of variates.
To apply the above test in practice, one would require
to stUdy all the N! possible permuted values of £N (under
~),
there being at most N!!TT~=l n k ! distinct values of ~ for
* Naturally, the labor involved in this scheme
any given ~N'
increases prohibitively with the increase in the sample sizes.
So in large samples, we are faced with the problem of
10
approximating the true permutation distribution of ~N by
some simple law and to reduce the computational labor by
that.
This study would be taken up in the next section.
4.
ASYMPTOTIC PERMUTATION DISmIBUTION OF PROT-STATISTIC.
We assume that for all N, the inequalities
(4.1)
for k = 1, ... ,c, hold for some fixed AO ~ l/c.
Also vIe require to impose some regularity conditions
on E(i)
Na
oefined in (2.11).
Extending the idea
of Chernoff-Savage [9] to the multivariate case, He write
E~)
(4.2)
= IN(i) (a/(N+l)); a = 1, ... ,N •
While IN(i) need be defined only at l/(N+l), ... ,N/(N+l),
we shall-find it convenient to extend its domain of definition
to (0,1) by letting IN(i)
to be constant on (a/(N+l),
(a+l )/N+l) ), a =0,1, ..• ,N.
Let us now define
F~~~](X) = ~k
(number of
x~) ~ x);
1=1' ... 'P;k=1, ...
,
()
~c
(k)
( k)
HN[i] x = L--k=1 AN
FN[i](x), i = 1, ..• ,p.
,~:·3)
(4.4)
Also let
F~~~,j](X'Y) = ~k
(number of
(X~)'X~))~(X'Y));
k=l, ... , c .
i
1=
j
=
1, ... ,p.(4.6)
11
Finally, He define the marginal cdf of X~) and of
(k) (k)
(k)
(k)
.
(X ia ,X ja ) by F[i](x) and F[i,j](x,y) respectlvely, and define
~c
( k)
( k) ( ) .
( ) = ~k=l AN
H[i]_x
F[i] x
,
i = 1, ... ,p
(4.8)
(k) (x) = -1 (number of X(k)
FN
rv
n
"0.
k
x) , k = 1, ... ,e
<
-
rv
(4.10)
-c
(k)
()'
H()
~ = ~k=l AN
Fk ~
(4.11)
.
Then for the study of the asymptotic permutation distribution
JeN,
of
we shall make the following assumptions:
(1) J(i)(H) = lim IN(i)(H) exists for 0
-
<
N->- co
H
<
1 and is
0
<
not a constant.
co
(2)
J
[IN(i) (N~l HN[i])
-00
for all k = l, ..• ,c and i = 1, ... , p .
(3) J(i)(u) is absolutely continuous in u:
Id~~
J(i)(U)
I
=
IJ!~l(u) I.::
for r = 0,1, and some 5
independent of i
CD
(4)
>
= l, ... ,p.
-CD
-
~ j
1, and
0, where K is a constant,
co
for all i
<
K[u(l-u)]-r- 1/2 +5,
JJ [IN(i) (N~l ~[i] )IN(j) (N~l ~[j])-
-00
u
= 1, ... ,p and k = 1, ..• ,c.
12
It may be noted that condition (4) will be satisfied
if for all i = 1, •.. ,p,
Q)
~+l
J[JN(i)(
J(i)(N~l
HN[i]) -
dF~fl](x)=op(1)(4.l2)
l\r[i])]2
-00
for all k = 1, ..• ,c.
In practice, it is easier to
verify (4.12) than condition (4).
So long we have assumed that Fk € jf, k = l, ... ,c,
where
is the class of all continuous p-variate cdf's.
G1
We will nOi'J impose some restriction on:::f-.
Let us define
1
~i=JJi(u)dU~
o
1
v ij (F) =
.
J
JJ
J(i) (u) du -
0
=
i=l, ... ,p
~i,
(4.13)
i=j=l, ... ,p
(4.14)
+00 +00
-00
-00
J(i){F[i]{x))J(j)(F[j](Y)) dF[i,j]{x,y)-
~i~j
.
il=
j
=
l, ... ,p,
and denote
X,{F) = ~(Vij{F)))i,j=l, ... ,P·
Let
~
(4.l5)
denote the class of all continuous p-variate
cdf's for i'1hich ,....,
'v{F) is positive definite and assume that
Fk(~)
€
~o
for all k = l, ... ,c.
(4.l6)
It may be observed that (4.l6) implies that
(4.17)
that is, t{H) is positive definite uniformly in
satisfyinG (4.1).
A~l), ..• ,A~c)
Further, it may be noted that conditions (1),
(2) and (3) are sufficient for the asymptotic multinormality
13
of the joint permutation distribution of the p(c-l) random
variables in (3.4).
Condition (4.16) or (4.17) is required
to ensure the non-singularity of the above asymptotic distribution, as the same is required, in turn, for the asymptotic
distribution of
ot
N in
(3.7).
Condition (4) is required to
establish the convergence (in probability) of the (random)
* ) to rvv(H), which will be
permutation covariance matrix V(R
rv rv N
required in the sequel.
Before we proceed to consider the main theorem of this
section, we present the following two theorems, as they
will be required subsequently.
If conditions (1), (2) and (3) hold and if
THEOREM 4.1.
* ) is asymptotically non-singular, then the joint
V(R N
rv ""
permutation distribution of the p(c-l) random variables
[T~~)_~i);
i = l, •.• ,p; k
=
l, ... ,c-l] defined in
(2.11) and (2.12) is asymptotically a p(c-l)-variate normal
distribution.
PROOF.
It is sufficient to show that under the stated
regularity conditions, any arbitrary linear function of
(1) . .
(c-l) (
( 8))
!N '···'!N
defined in 3·
, when standardized,
has a permutation distribution which is asymptotically a
p variate normal one.
If now ~l' ..• '~c-l'~c (=0) be any
c-l arbitrary constants, then
> ~=l ~k :t~k)
=
(4.18)
£N = (UNl ,·· .,UNp )',
where
U
Ni
(i)
cN:x
= >N
----a=l
-c
= L-k=l
c (i) R( i) .
NO. -Ie'
i = 1, ... , p ;
(k)
~k ZiN,a ' a
=
1, ••• IN;
(4.19)
i
=
1, ... ,p. (4.20)
14
It is then easily shown that for any given ~l' ... ,pc;
(i)
(i)
(i)
.
~N
= (c Nl , ... , c
NN )_sa t~sfies the Wald-\'folfowi tz condition
(ao],p.236) of the Wald-Wolfowitz-Noether-HoeffdinG-Hajek
theorem ([12],P.522)
= 1, ... ,p.
for each i
Also, using the
conditions (1), (2) and (3), it is readily seen that
(i)
(i)
(i)
~
=
tion
(U~,p.236)
-
--
(E Nl ,' ... ,ENN ) satisfies the Noether-Hoeffding condiof the
same theorem
for each i
=
l, ..• ,p.
Hence, on applying the above mentioned theorem (as extended
to the vector-case by Hajek ([1 2],p. 522)
under the assumed
* )), the desired result follows
non-singularity condition on V(R N
"" ""
readily.
Hence, the theorem.
THEOREM 4.2.
If the conditions (1), (2), (3) and (4) hold,
- . (1)
--i.
then for all Fk €~, k = 1, ..• ,c, and uniformly in AN
(satisfying (4.1))
(Note:
He shall throughout use the notation
p
->
(c)
, •.. ,A N
to mean
convergence in probability as N -> 00.)
PROOF.
By virtue of conditions (2) and (3), and the
well-kno\'1n results in the univariate case (cf. [9]), it
follo\'JS that
E~i) ~>
1
J J(i) (u) du =
o
lJ.
i
for all i
= 1, ... ,c. (4.21)
Hence, usinG the condition (4), it follows from (3.3) and
(4.21) that
15
Vij(~~)
~i~j
+
=
JJ
R
2
+
J(i)
(N~l
-
N
HN[i]) J(j)(N+l ~{[j])dHN[i,j](x,y)
(4.22)
0
(1)
p
where R2 denote the two dimensional real space.
So our
problem reduces to that of showing that the intecral on the
right hand side of (4.22) converges in probability to
(4.23)
A = jJ'J(i) (H[i]) J(j)(H[j]) dH[i,j](X'y) .
R
2
Let us now define
(v.
l 3
l 31 . i
of... N- / -< H[ i] (x) -< 1 - N- /
I N (i) --
J'
=
1
, ... , p
(4.24)
j
=
l, ... ,p.
Then, using Cauchy-Schwarz inequality, (4.24) and the well-known
result (cf. [9]) that for each i
N ~[i](X) - H[i](X)
s~p N1/21 N+l
= l, ... ,p,
I
is bounded in probabilit~(4.25)
we can write the right hand side of (4.22) as
(4.26)
where
AN
J'J
=
J(i)(H[i])J(j)(H[j]) dHN[i,j](X'y)
(4.27)
IN(i,j)
BIN
jrr
= J
N
N
J(i)(N+l ~[i])[J(j)(N+l HN[j]) -
IU(i,j)
- J(j)(H[j])] dHN[i,j](X'y)
B2N =
Jf
J(j)(H[j])[J(i)(N~l ~[i])­
IN(i,j)
(4.28)
16
Now, usinG (4.25) and the condition (3), it is easily seen
that for all points in IN(i,j)
+ (l-~)H[i])
I} , 0
<
~
(4.30)
<
1
= 0p ~l)
uniformly in x
JI
€
IN(i,j)'
J ( i) (N ~1 HN[ i] ) I
IN(i) -
Also, noting that
d~ [ i] (x)
<
co
( 4 • 31 )
.
for all i = l, ... ,p
(by condition (3)), we obtain from
(4 .28 ), (4 . 30) and (4. 31) t ha t
Similarly, it is easily seen that under the stated
regularity conditions
JI
is bounded
J (i) (H[ i] (x))
IN(i)
in probability for all i
(4.29), ( l~ .30) and (4.33)
I
dHN[ i] (x)
= 1, ... ,p.
Hence, from
it follows that
(4.34)
Finally, An in (4.27) can be written as
(Ie)
(
dFN[i,j] x,y
)
17
where
if ~X,y)
=
€
~(i, j)
(4.36)
otherwise
0
h
an d , were
1\(X(k)
ia' X(k)).
ja ' a
= 1 , ... ,nkj?
are independent and
identically distributed random variables (i.i.d.i.v.)
having the bivariate cdf
(k)
F~i,j](X'Y).
Now by virtue of
condition (3), it is easily seen (by using the elementary
inequalities of ([9],p. 986)) that
1 5
sup
E 'I gN (X~~
, YJa
. )1+
N
.LU
~
IF[.
(k) .].5?
.J.,J
< 00
for k=l, ... , c,
where 5 is a positive quantity defined in condition (3).
Also, {gN~Xia,Xja)' a = 1, ... ,nk }
are n k independent and
identically distributed random variables, and hence it can
easily be shown by using (4.37) that
(k)
(4.38)
dF[i,j](x,y)
for k = 1, ... , c .
Nm'1 from (4.8), (4.35) and (4.38) it follO\lS that
p
AN
A
->
where A is given by (4.23).
From (4.22), (4.26), (4.32), (4.34) and (4.39), \'Je arrive
at the result
i
F
j
= 1, ... ,p.
Similarly, it can be shown precisely on the same lines as in the
* is independent of ~N* and
univariate case that vii(~N)
18
i=I, ... ,p.
Hence the theorem.
THEOREM 4.3.
Fk €
If conditions (1), (2), (3) and (4) hold and if
cfo for
all
k
=
tion of the statistic
l, ... ,c, then the permutation distribu-
dfN (defined by
(3.7)) asymptotically
reduces to a chi-square distribution with p(c-l) degrees of
freedom.
PROOF.
The proof follows more or less trivially from the
preceding
two theorems, and hence is omitted.
By virtue of Theorem 4.3, the permutation test procedure
based on;t;r simplifies in large samples to the following rule.
2
If oZN
,j) >
_ xE,p(c-l)' re j ec t H0'
(4.40)
<
2
XE,p(c-l)'
accept HO '
where X2 r is the lOO(l-E)~o point of a chi-square distribution
E,
with r d.f. (degrees of freedom).
In order to study the power properties of the test
considered above, we require to study the unconditional
distribution of
hypotheses.
dCN ,
under appropriate classes of alternative
This, in turn, requires the study of the joint
distribution of the rank order statistics defined by (2.11)
and the same will be considered in the next section.
The problem of the choice of suitable [
will be considered in a later section.
~~ i),
i = 1, ... , p
J,
19
5.
ASYr~TOTIC MULTINORMALITY OF T~~); k = 1, ... ,c,
i
= l, ... ,p, FOR ARBITRARY Fl , ... ,F c '
It follows from (2.11), (4.2) and (4.4) that we can
write T(k) equivalently as
Ni
T~~)
+00
J IN(i)(N~l HN[i](x))
=
dF~[l](x)
.
-00
If Fk € ~ for all k = l, ... ,c, ,and if IN(i)
satisfies the conditions (1), (2) and (3) (cf. Section 4)
THEOREM 5.1.
for all i
= l, ... ,p, then the pc-vector with stochastic
1/2
(k)
(k)
_
.
_
'',
Ni - IJ. Ni ), i - I , ... ,p, k - 1, ..• ,c] has
a limiting normal distribution with zero mean vector
(k,q)
and covariance matrix ~ =((C1N[i,j]"' where
elements [H
(T
+co
IJ.~~)
=
J
J ( i) (H[ i] (x))
dF [ ~~ (x) ; i = 1,..., p; k= I, .. . , c. (5 . 2 )
-00
(r)
(k,q)
C1N[i,j]
dF[i](Y)
[r]
dF i
+
11
(k)
Ai
(r)
(x) dF[i](Y)
(r)
(y,x)dF[i] (x) dF[i]
-oo<y<x<oo
(if k
(s)
= q,
i
= j);
J
(y~
20
(if k = q, i ~ j);
Jf
A~k)
(r)
(q)
(x,y) dF[ i] (x) dF[ i] (y)
-oo<x<y<oo
-oo<y<x<oo
(if k ~ q, i
= j);
21
c
(r)
= - ~ AN
r=l
c
(r)
- LAN
r=l
JJ
JJ
-00
. (r)
+ L:= AN
r=l
(5.6)
-00
+00 +00 ( q )
-00
c
(k)
(q)
(r)
Bi , j (x,y) dF[i](X) dF[j](Y)
+00 +00
+00
(k)
(r)
B.~,J.(x,y) dF[i](x) dF[j](Y)
-00
+00 (
)
J JBi:j~X,y)
-00
(q)
(k)
dF[i](x) dF[j](Y)
-00
(if k
~ q,
i
~
j);
where
(a)
(a)
(a)
(a ) ,
,
Bi,j(t,U) = [F[i,j](t,U)-F[i](t)F[j](U)] J(i)[H[i](t)]Jj[~j](U)]
(5·8)
Proof:
AS.in the univariate case (cf. [11, Theorem 5.1]) we
Oc)
rewrite TNi
as
. where
~~~)
is given by (5.2),
(5·10)
and the
(k)
erN(i)' r = 1,2,3,4
are all 0p(N-
1/
2); for
i = 1, ... ,p and k = 1, ... ,c. The difference
1/2 (k) .(k)
1/2 (k)
(k)
N (TNi - ~Ni ) - N (B1N(i)+ B2N (i)} tends to zero in
22
probability for all i = 1, ... ,p
and k = l, ... ,c
and so the
vectors
1/2
[N
(k)
(TNi
and
(k)
- ~Ni ) , i
(k)
1/2
= 1, ... , p; k = 1, ... , c)
(k)
= 1, ... ,p; k = 1, ... ,c)
(BlN(i)+ B2N (i)), i
[N
have the same limiting distribution if they have one
at all.
Thus, to prove this theorem, it suffices to show
that for any real non-null 5 = (51' ... ,5 c )' the random vector
1/2
N
c
(k)
(k)
5k [ ~lN + ~2N )
k=l
2.-
(k)
(k)
(k)'
(where ~rN = (BrN(l),···,BrN(p)) , r = 1,2), has
asymptotically a p-variate normal distribution.
Now. proceeding as in the univariate c-sample case
(k)
(k)
(cf. [1 6 ] and[[ll],Section 5]), we can rewrite B11J(i)+ B2N (i) as
_c_
(q) { I
AN
q=l
~k
N
q
where
(i)
(q)
Bkq (1) (X:Ia ) =
~
(1)
-
(q)
Bkq(l) (X ia ) a=l
-
1
nk
nk a=l
_
(i)
00
(q)- (q) ,
(k)
[Fl[i]- F(i)]J(i)(R[i]) dF[i)(x)
00
(q)
(k)
(k) ,
[Fl[i]- F[ i]]J (i) (R[ i)) dF[i)(x)
J
(k)
Bkq (2) (X ia
.
l
)j~
(5.13)
-00
(i)
(k)
Bkq (2) (X:Ia ) =
J
-00
(k)
(k)
Fl[i] being the empirical cdf of Xia ' whose true cdf is
(k)
F[i](X)' for q,k = l, ..• ,c;
i = 1, ... ,p.
If we now define
23
(k)
c
=
=
(k)
Lq=l
fk
5 q Bqk(l) (X1a. )
c
(q) (i)
(k)
AN B kq (2) (X ia )
~
fk
(i)
(k)
= B
(X
Ok
ia
)
and
1
= N
2'
c
k=l
~
~
L-
0.=1
(1)
~N
then
(i)
(k)
(X
Bk
ia
(p)
= (BN , ... ,BN )
(5~12)
)
'
i
=
l, ... ,p,
,
can be written as Nl / 2
p-vector with elements B~i)
(i
=
~N'
which is a stochastic
l, ... ,p) which involve c
independent sums of independent and identically distributed
Bk(i)( X(k) ), a. = l, ... ,n .
= l,'... ,c .
, k.
k
ia
Further, as in the univariate case (cf. [11, Section 5]) it
random variables
can be easily shown that
E
(i)
{
for all i = l, ... ,p; k
IBk
=
(k) 2+5}
(X ia ) I
1, ... ,c,
< 00
(5·20)
where 5 is defined under
Condition 3 in Section 4.
Hence, the asymptotic multinormality of (5.12) readily
follows from (5.18), (5.19) and (5.20) through an application
of the (vector case) Central Limit theorem.
24
Further, it can be shown by somewhat lengthy algebraic
manipulations that
(q)
(i)
cov
(j)
(ql)}
Bkq(l)(X ia ), Bk I q r (1) (X ja
{
)
if
= 0
J JBi, j (x, y ) dF[ i] (x) dF [
+00 +00 (q)
=
-00
(k)
q
f.
qr ,
(kl)
if q = q I; i f.
J] ( y)
j;
-co
JJ
=
-oo<x<y<oo
JJ
+
(k)
(q)
Ai
(k l
)
(y, x ) dF[ i] (x) dF[ i] (y)
if q
= ql,
i
=
j,
-oo<y<x<oo
(q)
where
Ai
(q)
(x,y)(A 1
(y,x»
( )
and Bi;j(X,y) are given by (5·7)
and (5.8) respectively.
(i)
(k)
(j)
(k l
Bkq (2) (X ia ), Bkrq r (2) (X ja
{
COY
)
}
)
if k f. k'
= 0
J -J
+00 (k)
+00
=
-00
= -
(q)
(ql)
B1, j (x, y) dF[ i] (x) dF [ j]
(y)
if k
=
k I; i f.
j;
-00
JJ
(k)
(q)
(qr)
Ai (x, y) dF[ i] (x) dF [ i] (Y)
-oo<x<y<oo
+_
JJ
(k)
(q)
(q')
Ai (y,x) dF[i](x) dF[i] (y)
-oo<y<x<oo
ifk= k l
;
i=
j.
25
Finally
(k)
(i)
(k')}
(j)
cov { Bkq(l) (X ia ), Bk'q' (2) (X ja
=
(q)
11
-J-<;o -J-<;o
=
)
if q 1= k';
0,
(q')
(k)
if q = k'; i 1=
Bi, j ( x, y ) dF [ i] (x) dF [ j ] ( y )
-00
-00
rr
JJ
=
j;
(q)
Ai
(k)
(q')
( x, y ) dF [ i] (x) dF [ i] ( Y )
-oo<x<y<oo
+
11
if q = k'; i=
j.
-oo<y<x<oo
Using (5·13), (5.14), (5.15), (5.21), (5.22) and (5.23)
and following somewhat lengthy algebraic computations we
obtain the covariance terms given by (5.3), (5·4), (5.5)
and (5.6).
The theorem follows.
It may be noted that the aSYmptotic normal distribution
derived in Theorem 5.1 is singular and the rank of this distribution can at most be equal to p(c-l).
If the null hypothesis
(2.1) is true, then it is readily seen that
(5·24)
is the kronecker delta and
1
1J~
o
11
1
i) (x) dx - [
00
1
J ( 1) (x) dX] 2
1f i = j
0
00
J ( 1) (F [ 1] (x) ) J ( j ) (F [ j] (y)) dF [ 1, j] (x, y)
-m -m _
[!J(1)(X) dX) [ ! J
o
0
1j
(Y) dy) i f 1,£ j
26
F[i] and F[i,j] being the marginal cdf's Xi and (Xi,X j )
respectively when X = (X1, •.. ,Xp ) has the cdf F(~).
Thus, if we assume that F
E.
j- 0' where fO is the class of
all continuous cdf's for which (4.15) is positive definite,
then it readily follows that the asymptotic normal distribution
in Theorem 5.1 is of the rank p(c-l), and by simple argument,
we may conclude that
(5.26)
where ~-l(F)
=
((Vij(F))-l
=
((v ij )) has under the conditions of
Theorem 5.1, asymptotically a chi-square distribution with p(c-l)
d.f. when HO defined by (2.1) is true.
r.
1\
If now ~ = ((V ij )) is any consistent estimator of ((V ij ))
and if ~-l = ((~ij)), then it follows from (5.26) and some simple
""'
arguments that
has also under the conditions of Theorem 5.1, asymptotically a
chi-square distribution with p(c-l) d.f. when HO defined by
(2.1) is true. Hence an asymptotic test may be based on the
f\
statistic ~N' and such a test will be termed as a CSROT (see
the concluding part of Section 1).
27
6.
THE LIMITING DISTRmUTIONS OF PROT AND CSROT FOR
SEQUENCES OF TRANSLATION AND SCALE TYPE OF ALTERNATIVES.
In this section, we shall concern ourselves with a
sequence of admissible alternative hypotheses {~1 which
specifies that for each k
(k)
=
1, ... ,c,
(k)
Fk(~) = F(x1N , ... ,XpN )
with F
€
~
(6.1)
where
(k)
x iN
=
(6.2)
, i = l, ... ,p.
We shall use the notation
and assume that at least one of the following two equalities
It may be noted that if 5(1) =
is not true.
~
..
. .. =
we have the usual location problem, and if g(l) =
~
(c)
5
,
~-
. ..
- g(c) ,
~
we have the scale problem; while if both the equalities hold,
then F
= ... =
Fc . Furthermore we shall denote as before
l
the marginal cdf's of the i-th variate, and that of the
i-th and j-th variate, corresponding to the distribution
function F(X l , ... ,Xp ) by F[i] and F[i,j] respectively.
Next, let us denote
+00
B( i) (F) =
.J
-00
~X
J ( i) (F [ i] (x»
dF [ i] (x) , i
= 1,..., p
(6.4) .
28
+00
C(i) (F)
=
--J x ~ J ( i) (F[ i] (x»
dF [ i) (x) , i = 1,..., P
( 6. 5 )
-00
(k)
(k)
11
(6.6)
k = l.l""c
11 i
(k)
=
(111
"-
(k)
)
' ••• ,1') p
,
k
=
1, •.• , c
Then, we have
THEOREM 6.1.
(i) for all k
-If
= l.l •.. c ; lim ~~k) = ~(k) exists and is
.I
N->oo
positive fraction less than unity,
(ii)
conditions of Theorem 5.1 are satisfied;
. ")
-
then under t.he hypothesis ';I
1r
the random vector
j>defined by-l.6.1) and (6'.2),
[Nl/2(T~~)_~~~», i = l,: •. ;p; k = 1, ... ,c]
has a lir:iting normal distribution with mean vector zero
and covo.riance matri~:)
rv
* =(((a(k,q)))
[ i, j]
(k.lq)
5k
O'[i,j] = (~ - l)Vij(F) ,
uhere
(6.8)
i,j
= 1.1" .,p;
k,q
=
1, ... ,c
where 5kq is the usual kronecker delta and Vij(F) is given by
(5·25).
The proof of this theorem is an immediate consequence
of Theorem 5.1 and some routine algebra, and is therefore
omitted.
29
THEOREM
6.2.
If F
€
~o and if in addition to the conditions
of Theorem 6.1, the conditions of Lemma 7.2 of Puri [16]
are satisfied by F[i](X) for all i
the sequence of alternatives
;eN
HN
= l, •.. ,p,
then under
in (6.1) and (6.2),
(defined in (3·7)) has aSymptotically a non-central
chi-square distribution with p(c-l) degrees of freedom and
the non-centrality parameter
PROOF.
It follows from Theorem 6.1 that under {f~J the rank
of the asymptotic normal distribution in Theorem 5.1 is p(c-l).
Further, it follows from Theorem 4.2, (4.11), (4.14) and
Theorem 6.1 that under the stated conditions
v(H) - > v(F)
'"
'"
as N
'"
-> 00
,
(6.10)
v(F) •
'"
where'" indicates that the difference of the two sides tends
to zero in probability.
Finally under
iHN3
it is easily
seen (by using (4.2) and (5.2)) that
[IJ.
(k)
Ni
.
- E(i)] ->
N
n
(k)
'Ii
for all i = 1, .•. , p; k = 1, ... , c. (6 .11 )
The rest of the proof of the theorem follows from
Theorem 6.1 and by an application of the well-knol'm resul ts
on the limit distributions of quadratic forms of asymptotic
normal distributions, and hence is omitted.
30
~-l
Now since A
~
1
is a consistent estimator of A- , we have
~
the following
THEOREM 6.3.
If the assumptions of Theorem 6.2 are satisfied,
then for N
co, the limiting distribution of the statistic
->
A
iN
defined by (5.27) is non-central chi-square \'lith p(c-l)
d.f., and noncentrality parameter ~ot defined in Theorem 6.2.
It now follows from Theorems 4.3, 6.1,6.2 and 6.3 that
under the sequence {~"3 of alternative hypotheses
LN
P
~
A
/N
(6.12)
and hence, using a well-known result of Hoeffding ([[14],p.172]),
it follows that for testing HO against the set_of alternatives
[HN\the PROT and CSROT are asymptotically power-equivalent.
7.
CHOICE OF {J(i)' i = 1, •.. ,p1 AND RELATED AS~MPTOTIC
POHER EFFICIENCY OF THE RANK-ORDER TESTS.
In this section, we shall consider specifically the
problems of location and scale considered in Section 2, and
for each problem, we shall consider some specific tests and
study the resulting asymptotic efficiency.
(a) Location problem.
We shall consider here the following tests.
Median test.
This is characterized by
l' if a.
~ [(N+l )/2] ,
{ 0, otherwise,
where [s] denotes the largest integer
~
s.
This implies that
31
if u
<
1/2
if u
>
1/2
The function J(i)(U) has a single point of discontinuity at
u
= 1/2, and if the cdf F[i](X) has a uniquely defined median,
the Lebesgue measure of this point will be equal to zero.
If we work with (7.1) for all i
= 1, ... ,p, the test will
be termed the multivariate mU1tisamp1e median test.
( i1)
Ranlc-sum tes t •
Let
I N(i)(a/(N+l))=a/(N+1); a.
which implies that
J(i)~u)
= 1, ... ,N
= u: 0
weight-function is used for all i
<
U
<
1.
If this
= 1, ... ,p
the corresponding
test will be termed the multivariate rank-sum test.
Both the median and rank-sum tests have been studied
in detail by Chatterjee and Sen ([6],[8]), and hence, here
these will not be considered again.
(iii)
~-Score
test.
Let
~(x)
be any specified continuous
cdf, and we define
IN(i)(a./(N+l)) = ~-l(a./(N+l»; a=l, ... ,N
This implies that J(i)(U) = ~-l(u): 0
<
u
<
1.
(7.3)
If we use
(7.3) for all 1 = 1, ... ,p, the corresponding test will be
termed the
~-score
test.
In particular, if
~
is a standardized
normal cdf, the test is termed the normal scorretest.
be noted that often we define the
~-scores
It may
in a slightly
32
different but asymptotically equivalent way.
Let aN ,a be the
expected value of the o.-th order statistic in a sample of
size N drawn from a universe having the cdf
IN(i)(o./(N+l)) =o.N,o.;
0.=
~(x).
Then we take
1, ... ,N.
For a wide class of cdf's [cf. Hoeffding [15]), the use of
(7.3) and (7.4) leads to aSYmptotically equivalent results.
The conditions (1), (2) and (3) of Section 4 are known to be
satisfied by (7.3) and (7.4) for a wide class of
~(x)
(cf. [9], [11], [16]), and so we require only to show that
condition (4) or (4.12) is also satisfied.
To do this
we rewrite (4.12) as
where 1 ..:: 0. 1 < ••• < a
< N are any n distinct integers,
nk
, , 'k.
.
and ~(;N,o.) = a /(N+l), for a = 1, ••• ,N. Now (7.5) cannot
be larger than
We shall then prove the following.
LEMMA 7.1.
(i)
J
Let
~(x)
be any continuous cdf, satisfying
2
x d'l'(x) = 0,
x d~(x) < co,
(ii)
~(x)
for all x
<
J
is symmetric about x = 0; 'l'(x) is convex
0 and concave for all x
>
0, (or vice versa).
Then
l1:m
N->oo
=0
33
The left hand side of (7.7) can be written as
PROOF.
(7.8)
It has been shown by Ali and Chan [1] that if
concave (convex) for x >0, then for i
~
~(x)
is
-
[(N+l)/2],
aN,i ~ ~~) ~N,i and opposite inequalities hold for i
Thus, if
~(x)
is convex for x
<
° and
concave for x
>
<
[(N+l)/2].
0,
we readily get from the above result that
and the opposite inequalities hold if
for x
<
° and convex for x
1 N
/it
2
N~ (U- N i-~N i) ~
i=1 "
0.
>
1
~(x)
is concave
Thus, from (7.7), we get that
2
1 N
2
i - N L ~N i
i=l '
i=1
'
N
I N~O~
l.
(7.10)
Again it follows from Hoeffding's results [15] that
1 2::
N
N
i=l
aN2
'
i
--;>
Jx
2 d'l.'(x) <
00
,
and by the fundamental theorem of integral calculus
1
N
2
N ~ ~N,i ~
J
x
2
d'l.'(x)
< 00.
From (7.10), (7.11) and (7.12), we readily conclude that
(7.7) converges to zero as N
-> 00.
Hence the lemma.
In particular, if 'l.'(x) is a standardized normal cdf,
it is convex for x
<
° and
result of Lemma 7.1 holds.
concave for x
>
0, and hence the
The same lemma holds for logistic,
double
eA~onential,
rectangular and many other cdf's.
Finally, we require to show that v(F) in (4.15) is
'"
positive definite under certain conditions.
v(F) will cease
'"
to be positive definite when there exists one or more linear
relations among the variables J(i)(F[i](x», i
=
I, ..• ,p.
This in turn requires that the scatter ,of the points of the
distribution F(Xl, ••• ,xp ) is confined t. a (p-l) or lower
dimensional) hyper-curve whose equation is
(or
more than one such equations).
Thus, if the cdf
F(Xl, •.• ,x) is non-singular in the sense that there is
-
p
no (p-l) or lower dimensional sUbspace of the
p-dimensional Euclidean space Ep whose probability mass is
exactly equal to unity, then (7.13) cannot hold, in
probability, and hence (4.15) would be positive definite.
In actual practice this is practically no restriction on F.
In the parametric case, under the assumption of
non-singular multinormal distribution of each of Fl , ••• ,F c ' the
likelihood ratio test for location is given in Anderson ([2], Sec.
8.8) and Rao ([18], Sec. 8d.4), and it is easily seen that under
[~5 (when ~(k) =
0 for all k
= l, ••• ,c), this statistic
has asymptotically a non-central ~2 distribution with
p(c-l) d.f. and the non-centrality parameter
~U =
1::.
A(k) [(i(k)}
k=l
where
> = (( 0' ij»
'"
:L--1 (~(k»,]
(7 .14 )
'"
is the common covariance matrix of all the cdf's.
35
Thus
the relative asymptotic power-efficiency of~-test
with respect to the likelihood ratio U-test is given by
e[ofNJ:i
U
r=
li
/li u
It follo\'lS from (7.3), (7.4), (4.15), Theorem 6.2 and (7.14)
that if F is a p-variate non-singular normal cdf, then
lit:!.
fer all
~(k),
k = 1, ••• ,c.
rv-
= li U
So asymptotically, for normal
cdf' s, the normal scores test is
;'1.S
the U-te-st.
8,8
efficient
In the univariate case, Hodges and
Lehmann [13] have shown that asymptotically normal scores
test is always at least as good as the student's t-test
for all cdf F €~.
The generalization of such a statement
to the multivariate case seems to . . quite difficult to prove
and dubious too.
It
tur~s
(7.15) depends not only on Bi(F),
multivariate case,
i
= 1, ... ,p
out that for the
(defined in (6.4)), but also on the matrix
v(F) in (4.15) as well
rv
as on rv
g(k), k
=
l, ... ,c.
Using
the results of Hodges and Lehmann [13], it is easily seen
that for the normal scores test
Bi(F)/cr ii .::. 1
for all i = 1, ... ,p, and all F
€
~O. (7.16)
So, if the p-variates in F are all uncorrelated, it is
easily seen that (7.15) .::. 1 for all F
of rv
g (k) , k
=
l, ... ,c.
€
~O' independently
On the other hand, if these p variates
are correlated, we may proceed precisely on the same line as
in Bickel ([5, (4.3)]), and express (7.15) as the product of
36
two factors, namely the univariate and the
factors.
rnu1.t:hmr1::lh:::
The univariate factor is independent of
~(k),
'"
k = 1, ... ,c, and is easily shown to be at least as large
as 1, while the multivariate factor depends implicitly on
the characteristic roots of
g(c).
~
-1
L
and explicitly on g
(1)
'"
'"
, ••. ,
In the bivariate two samp1: case, Chatterjee and Sen
'"
[6] have obtained some asymptotic
power-inequalities of the
median test and the rank-sum test, and Bickel [5] has
obtained some bounds for the maximum and minimum power
of the same tests in the single sample case.
Essentially,
the same technique may be applied to study the bounds for
(7.15) in the particular case of p
in the general case of p,c
~
=
c = 2.
However,
2, it seems considerably
difficult to establish the bounds for (7.15) or to evaluate
(7.15) exactly.
cdf's
In fact, for some non-normal multivariate
(7.15) can be shown to be greater than one, but
we believe that (7.15) cannot be greater than or equal to
1 for all F
€
jto.
Some counter-examples may be provided
for that, especially in the limiting non-degenerate case,
where one has essentially to follow Bickel's [5] technique
with multisamp1e extensions.
This is however not considered
here.
(b).
Scale problem.
Here, we shall consider the following
rank-order tests.
(i)
Let I N(i)(a/(N+1))=
so that
IN~l - ~I for a =
J(i)~U) = ju - ~I for 0
<
u
<
1 and i
1, ... ,N; i = 1, ... ,p
(7.17)
=
l, ••• ,p.
37
(7~18)
(ii) Let I N(l) (a/(N+l)) = [ N+l - 1]2
2
for ex = 1, . . . , N; i= 1, . . • , p
a
so that
1 2
J ( i) = (u - 2)
(iii) Let
LLN ,a
for 0
<
u
and
< 1
i = 1, ... , p •
be defined as in Section 7 (a) and let
IN(i)~ex/(N+l)) =a~,ex '
ex = 1, ... ,N;i=l, ... ,p
(7.19)
so that
o
< u <
1; i= 1, . . . , p. (7. 20 )
In the univariate case, the rank-order test corresponding
to (7.17) has been considered by Ansari and Bradley, and
Siegel and Tukey (cf. [17]); the rank-order test corresponding
to (7.18) by Mood and
the one corresponding to (7.19)
by Klotz, among others (cf. [17]).
Here also, in the
multivariate case, the last test (for the case when
~(x)
is the standard normal cdf) will be termed the multivariate
normal score test for scale.
Proceeding then precisely on the same line as in the
location case, it is easily seen that under essentially
similar conditions, a strictly distribution-free permutation
test and a large sample non-parametric test can be
constructed for each of the three types of weight functions
defined in (7.17), (7.18) and (7.19).
In the parametric case, under the assumption of
non-singular normal distribution of each of F l , ... ,F c '
some optimum likelihood ratio test is available for testing
the identity of dispersion matrices of Fl, ... ,F . If, however,
c
we are interested only in testing the identity of the variances
for each of the p variates, we can have a similar test
whose test-statistic (say V) has asymptotically under
H
N
in (7.1) and (7.2) (where ,...,
g(k) = ,...,
0 for all k = l, ... ,c), a
non-central
It 2
distribution with p(c-l) d.f. and the
non-centrality parameter
where r = (( y ) ) .
'
,...,
. ij i,J=l, ... ,p
with
i=l, ..• ,p
(7.22)
i
~
,...,
I
j
=
l, ... ,p,
= ((cr ij » being the covariance matrix of F.
Hence the
a~mptotic
efficiency of the
~
test with
respect to the V-test is given by
In particular, when the underlying distribution function
F is a non-singular p-variate normal, the multivariate normal
scores test for scale is asymptotically as efficient as the
V-test.
The efficiency of the first two scale tests can be
expressed as the product of two factors, the univariate
factors are known (for the normal cdf, they are equal to
0.61 and 0.76 respectively), while the multivariate factors
39
will depend on ~ as well as on 5(k), for k = 1, ... ,c.
-""
""
As in the location test, nothing can be said about the
lower bounds for the asymptotic efficiency of the normal
scores test or the other two tests for the non-normal cdf's.
So far,
we have considered rank-order tests for
which IN(i) does not depend on i.
Evidently, it is always
possible-to use a rank order test with different IN(i) 's
for different i
= l, ..• ,p.
In the event when there are
sufficient reasons to believe that for different
i's
the F[i] 's are appreciably different from each other, some
specific IN(i) will be suitable, such a test may be
recommended~
ACKNOl:ILEDGMENT
Some of the ideas of this paper evolved in the course
of collaboration of one of the authors with Dr. S. K. Chatterjee
of Calcutta University, while they were studying the
rank-permutation tests considered in ([6],[7],[8]).
The authors would like to express their sincere thanks to Dr.
J. Sethuraman
for pointing out some errors in the
original draft and making useful suggestions.
40
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