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ON THE THEORY OF RANK-ORDER TESTS AND ESTIMATES FOR
LOCATION IN THE MULTIVARIATE ONE SAMPLE PROBLEM
by
Pranab K. Sen and Madan L. Puri
University of North Carolina
Institute of Statistics Mimeo Series No. 488
August 1966
This research sponsored by Sloan Foundation Grant for Statistics,
U.S. Navy Contract Nonr-285(38) and by Research Grant, GM-12868,
from the National Institutesof Health, Public Health Service.
DEPARTMENT OF BIOSTATISTICS
UNIVERS ITY OF NORTH CAROLINA
Chapel Hill, N. C.
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1
ON THE THEORY OF RANK-ORDER- TESTS AND ESTIMATES FOR LOCATION
IN THE MULTIVARIATE ONE SAMPLE PROBLEM*
by
Pranab K. Sen l
and
Madan L. Puri
University of North Carolina,
Chapel Hill
and
Courant Institute of Mathematical Sciences, New York University
O. Summary.
In the multivariate one-sample location problem,
the theory of permutation distribution under sign-invariant
transformations is extended to a class of rank-order statistics,
and is utilized in the formulation of a genUinely distribution
free class of rank-order tests for location.
ASYmptotic
properties of these permutation rank order tests are studied
and certain stochastic equivalence relations with a similar
class of multivariate extensions of one-sample Chernoff-Savage-
,
Hajek type tests are derived.
tests are studied.
The power properties of these
Finally, Hodges-Lehmann (1963) technique
of estimating shift parameters through rank-order tests is
extended to the multivariate case which includes among other
results, the results obtained by Bickel (1964 and 1965).
*Research sponsored by SlQan Foundation 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.
IOn leave of absence from Calcutta University .
2
Let ~_ = (x{l), ... ,x{p)) be a p-variate random
1. Introduction.
a
"'U
a
variable having a continuous p-variate cdf (cumulative distribution function) F{x,9) where x = (x(l), •.. ,x(p)),
"" ""
""
9 = (91' ... ,9 ), and where p is a positive integer which
""
p
in the sequel will be assumed to be greater than unity.
Let then
~l'
... '~N be N i.i.d.r.v. 's (independent and
identically distributed random variables) distributed
according to the common cdf F(x,Q).
Let 0 be a set of all
""""
.
continuous p-variate cdf's, and it is assumed that F
Let now
ill
€
o.
denote the set of all continuous p-variate cdf's
which are symmetric about some known origin which we may
without any loss of generality take to be x
""
= 0,
""
the
symmetry being defined in the following manner.
Let
L,
= P(x1 , ..• ,~)
be any set of points P
in the
p-dimensional Euclidean space Ep . Any point P belonging to
is denoted by P €
and the image of the point P is
z: '
L
denoted by P* = P(x *l , .•. ,xp* ) where xi* = -xi' i = 1, ... ,p.
The image set
* of
is denoted by
t-
L*
(1.1)
Then, if for any
(1.2)
L.
t
P(~
= (~
c. Ep
€
t:IF)
=
P(~
€
~IF)
we say that F is symmetric about O.
""
If, in addition, F is absolutely continuous having a
density function f, the symmetry of F may also be defined
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by the invariance of the density function under simultaneous
changes of signs of all the coordinate variates.
For the
convenience of terminology, (1.2) may also be termed as the
sign-invariance.
Excepting with the later section dealing
with the aSYmptotic normality and other properties of a
proposed class Of estimates and tests, we will drop the
assumption of absolute continuity of F.
Thus, for the time being let us frame the null
hypothesis
HQ
of sign-invariance as
The two particular classes of alternative in which we may
usually be interested are
(1.4)
F(x) is symmetric about some 6
'"
'"
~
0 .
'"
This will be the multivariate extension of the well known
one sample location problem.
H2 : F(x) is asymmetric about x = 0 though
'"
'" '"
it may have the location vector O.
'"
This is the multivariate extension of the univariate symmetry
problem.
In the univariate case, there is a third problem whose
multivariate extension would be the estimation of location
vector 6, assuming F(x) to be symmetric about x
'"
'"
'"
= 6.
'"
Tests of the type mentioned above in the univariate case
are due to Wilcoxon (1949), Govindarajulu (1960) and the
4
classical sign test.
Estimates in the univariate case are due
to Hodges and Lehmann (1963) and Sen (1963).
However, in the
multivariate case there are only a few suitable tests and
estimates.
In the bivariate case, Hodges (1955) has considered an
association invariant sign test on which some further work
has been done by Klotz (1964), and Joffe and Klotz (1962).
The test is strictly distribution-free but neither the null
nor the non-null distribution (for any suitable sequence of
alternative hypotheses) of the test-statistic appears ,to be
simple, and so there is little scope for studying the
Pitman-efficiency of such a test.
The only study of the
efficiency of this test is made by Klotz (1962) and that is the
Bahadur Efficiency.
Further, being a sign test it is
anticipated that as in the univariate case, for nearly normal
cdf's, the efficiency of this test may be appreciably low.
Some sign-tests are due to Blumen (1958) and Bennet (1962).
Bluman's test is also strictly distribution-free but it is
subject to the same drawback as Hodges' test.
test is distribution-free only asymptotically.
Bennet's
Some further
asymptotically distribution-free tests are due to Bickel.
His tests are based on generalizations of the univariate
sign-statistic
and Wilcoxon's signed rank statistic and are
distribution free only asymptotically.
However, his study
of the asymptotic power properties of these two tests provides
some useful information about the performance characteristics
of them.
Very recently, Chatterjee (1966) has considered a
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very interesting permutationally distribution-free bivariate
sign test which asymptotically agrees with Bickel's and Bennet's
sign-tests.
In this investigation this permutation argument
will be extended not only to the p-variate case for any p
~
2
but also to a much wider class of rank-order tests which
includes Bickel's tests as special cases, and which are
strictly distribution free under permutation arguments.
This extension also covers the normal scores and the other
type of multivariate score tests on which practically there
is no work in the literature.
In the problem of estimation
too, Bickel (1964) has extended Hodges and Lehmann (1963) type
of estimates to the multivariate case using only the median
and rank sum procedures.
In this investigation a much wider
class of rank order statistics will be used, and this will be
a further generalization of Bickel's work.
2.
The Basic Permutational Argument.
Let us denote the sample point by
(2.1)
and the sample space by
~.
Then under the null hypothesis
to be tested the joint distribution of
under the following finite group
~
remains invariant
~N of transformations gn
given by
(2.2)
ji = 0,1;
i= 1, .•• ,N,
6
where (-I)X
"tt
=
(-X(l), •.. ,_x(p)).
a
u
Thus there are 2N such
possible (gn) under which the distribution of
~N
remains
invariant. Hence, given ~N' we can generate a set SN of 2N
points ~N in ~N' such that all these 2N points will have the
same probability. Hence, conditioned on the generated set SN'
there are 2N = M points and the permutational probability
measure associated with each point is the same, viz., 2 -N ,
when
He
given by (1.3) is true.
Let us now denote by ~(~N) a test function l'Jhich with each
-tN associates a probability of rejecting HO •
NOH if w'e use
the known permutational probability measure of
~N
then it readily follows from the well known result
on SN'
of
Lehmann and Stein (1949) that a test function ~(-tN) based
on this permutational probability measure will possess the
structure SeE) (cr. Lehmann and Stein (1949)), and hence will
be a strictly distribution-free test. This characterizes the
existence of strictly distribution free test
o
<
E
<
1, for the hypothesis
He
of size E,
given by (1·3)·
Now in actual practice ~(-tN) has to be constructed with
special attention to the class of alternatives in mind, and
in most of the multivariate problems, ~(~N) depends on -tN
through a single-valued test statistic TN
to have a comprehensive test.
= T(~N)
in order
Due to these reasons we shall
consider the following procedure.
Let us rank the N values of IX(i)l, a
a
=
1, ... ,N, in order
of magnitude, and let R(i) denote the rank of Ix(i)1 in this set.
a
a
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7
(i)
-
(i)
Then (R l ' ... ,R N ) is a permutation of the numbers 1,2, ... ,N .
This procedure is adopted separately for each i = 1, ... ,p.
•e
Thus, conditioned on a given ZN' an observation X having p
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I'\.Q.
values is converted into
It,.,
'''''''''
Let nOl'J
= (R(l), ... ,R(P)) ,
a
a
a=l, ..• ,N.
(E ( i);
N,a a=l, ... ,N) be a sequence of N real
numbers; the sequence may of course vary over i=l, ... ,p.
The value of ~i)
corresponding
to the value a = R(i) is
,a
.
a
denoted by E(i) (i); a= 1, ... ,N; i= 1, ... ,p.
i~,Ra(EN(i);
a
,a
Thus,
= 1, •.. ,N); i = l, ... ,p, are
selected corresponding to R
in (2.3) we have a p-tuplet of
I'\.Q.
real values
(1)
(2.4)
~,a = (E
and, we denote by
~N
(1), .•. ,E
N,Ra
~ =
(1)
E
N,Ri l ) ,
=
~N
(1)
.. . ,
E
.. .
E
.
(p)
~,N
a=l, ... ,N,
(p));
N,Ra
the following p by N matrix
!N,l
(2·5)
(p)
E
N,Ri p )'
,
N R(l)
, -N
(p)
N,RAp)
is really a stochastic matrix, its i-th row being a
random permutation of (E~ii,
, . ... ,~i~)
, for i = l, ... ,p.
Thus there are (N~)P possible values of ~N'
~N
and
,
~N
~'Jo realizations
are said to be equivalent when it is possible to arrive
8
,
at
~
of
~N'
,
from
~N
by only a finite number of inversions of columns
Thus, we may always consider the first rO\J of
natural order.
~N
in
There will be thus (N!)p-l non-equivalent
realizations of ~N'
We denote this set of (N!)p-l realizations
~N' ~N will be termed a collection rank order matrix and
~N will be termed as the s ace of collection matrice~.
by
Let us now consider a sequence of N diagonal matrices
(C
, a=l, ... ,N) each of order p by p, where
I\(l
,
C = diag (C (l) , . . . , C(p ) )
(2.6)
a
'\.Q.
C~i)
= (_l)a i ;
if X(i)
(2.8)
a
ai=O,l;
>
0,
a
i=l, ... ,p;
and a i = 1
a=l, ... ,N,
if x(i)
a
<
0
fa
Thus, each
is a stochastic diagonal matrix, which
can have 2P realizations. Let us now define a p by pN matrix
which we term the collection sign matrix. This is again a
stochastic matrix which can have 2PN realizations. The set
of 2PN possible realizations of £(N) is denoted by tN' and
is termed the space of sign-matrices, while £(N) will be
termed
as the collection sign-matrix.
This-is again a
stochastic matrix which can have ~N possible realizations.
The set of ~N possible realizations of £(N) is denoted by
(N and will be called the space of sign-matrices, while
£(N) will be termed the collection sign-matrix.
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Let us now consider the pair of stochastic variables
(~N'£(N»
which lie in the product space
(~'£(N»
probability distribution of
(~N;(lJ) ~
The
on this product space
(defined on an additive class of (product) cr-fields of
subsets of (~N'£N»
depends on the unknown F{~), even when F€oo,
that is, even when HQ is true.
It follows from (2.2) that if we consider the finite
group ~N of transformations gn (containing 2N elements),
then the probability distribution of £(N) or (N remains
invariant under the transformations in ~~, that is, lf
(2.10)
gN£(N) = «-1)
-
j
Jl
jN
£1'"'' (-1)
~N)'
gN
GN '
€
J
j
... ,(_l) ac(p», a=l, .•. ,N
C = diag.«-l) ac(l),
"\.Q.
a
a
Ie
where (-1)
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distribution of C(N) remains invariant under any such
transformations gN' Again by definition of R (i) ,i=l, ... ,p,
.e
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Q
(with Ja = 0,1 for a
=
1, ..• ,N), then the probability
Q.
a • 1, ... ,N, and of
that
!N
~N
(in (2.3), (2.4) and (2.5»
we see
remains invariant under any transformation gN
or the original vector
~
€
~N'
€
~N
Hence if we consider the
stochastic matrix
(2.11)
*
(1)
gN~ = (~Nl (gN
(N)
£1), ... , ~NN(gN
£N»
g~i) =
(_l)J ij J =O,lj i=l, ..• ,Nj then it readily
i
follows that under HQ in (1.4), the joint distribution of
where
* conditioned on the given ~N {which implies that given
.
(~'£N» remains invariant under gN € ~N' Thus, if we
gN~
consider the set of 2N values
10
(2.12)
we get a set of 2N permutationally equ1valent (probab111s1tc)
points, and the permutational probab111ty measure on this
set is denoted by
09N,
~N' that i~ given
(!N,
Wh~Ch
is a conditional measure given
~N)·
We shall now develop a strictly distribution free test
procedure under this permutational probability measure.
Let us define
j = l , •.. ,p;
(2.14)
Thus, T~j) is the difference of the sum of the rank fUnctions
(~») for which x~j)
>
0 and the sum over the remaining set.
It follows eas11y that
(2.15)
E(:eN'
f N } = ° = (0, ... ,0)
~
for any given (~I£N) •
combinat10n j
~
Let us now define for each
(k)
. (4)
k (= 1, ..• ,p) four subsets Sjk , ... ,S jk '
where
(2.16)
c(k)
S(l)
jk
= (~
C(j)
<
0,
s(2)
Jk
= (~
C(j)
<
0, C(k)
a
S(3)
jk
= (~
c(j)
> 0,
S(4)
jk
= (~lb
C(j)
> 0,
a
a
a
a
<
0)
)0
0)
C(k)
<
0)
C(k)
> O}
a
a
a
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Of course, if j
null sets.
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Since under the transformation gN on
~,
both
c~j) and c~lC) change signs simultaneously, the product
E(jlj)
E(klk)
will have a positive sign for
s~~)
a::aa neg::tve sign for
remain invariant under gn
€
and
s~~),
G N.
S~~) and S~~)
and the same will
Hence it can easily be
shown that if we define
1
(2.17)
(j)
(k)
(j)
_
(k)
VN,jk = N (y- E (j)E- (k) +~ E (j)E- (k) - SfI)
NR
NR
gTIf) NRa NRa.
a
a
jk
jk
-
(k)
(j)
- ? E (j}E
Sf2)
jk
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then S3~) and s~~) will be the
= k,
NRa
(k) -~ E
S ( 3)
NRa.
N
(j}E
NRa
jk
1
(k)
(j)
(j) 2
vN, jj = N ~ [11c]
(2.18)
(k», j#k=l, ... ,p;
NRa
,
j=l, .•. ,p;
then
(2.19)
j,k = 1,. H'P.
Let us now denote by
v = ("
",N
_ N,jk ) j,k=l, .•• ,p
(2.20)
and assume (for the time being) that
XN
is positive definite.
(!N being a covariance matrix will be positive semidefinite
at least.
If XN is singular, then we can work with the
highest order non-singular minor of XN and work with the
12
corresponding variates).
conditions
~N
Later we will show that under certain
will be positive definite.
Thus, we consider
the following positive semi-definite quadratic form
(2.21)
where
-1
~N
is the inverse matrix of
~N.
If the null hypothesis is true, the center of gravity
of
will be zero, and conditionally on given ~N' that is,
N
on given(~'£(N»' there will be 2 permutationa11y equally
~N
likely (not necessarily all distinct) values of SN.
other hand, if
HQ
On the
is not true, it can be shown, as in the
univariate case that ~ T~j) converges in probabi~ity to
some non-zero quantity for at least one
j
= l, ... ,p and
hence SN/N will converge to a non-null quantity.
will be stochastically large.
right hand side tail of the permutation distribution of SN
(2.22)
~(~N) =
Thus the critical function
SN,E(~N)
1
if SN
aN,E
if SN = SN, E (~N)
0
if SN
>
<
SN,E (~N)
where SN,E(~N) and a N,£ are chosen such that
O<E<l,
provides a strictly size E similar region.
For small samples,
SN,E(~N)
-.
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Thus, SN
This suggests the use of the
as the suitable critical region.
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and aN,E are to be determined
from the actual permutational cdf of SN' while for large
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samples, we have proved in the next section that
2
SN , E(zN) and aN E converge in probability toJ(,p,E
2
and zero respectively as N -> 00, where}' E is
2
r"'P ,
the 100 (l-E )<to point of the
distribution
(2.23)
~,
'I-
with p degrees of freedom.
3·
Properties of
~N
and Asymptotic Permutation Distribution of SN.
and of (x(j),x(k))
Let us denote the marginal cdf of x(j)
a
a
a
and Fj,k(x,YiQj''k) respectively, and let
x > 0
o
-<
x,y
-<
Set
FN, j (x) = (number of ~ j) ~ x)/N
(3.5)
(3.6)
00
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-.
Finally, as in Chernoff and Savage, we write
a
=
1, •.• ,N,
=
j
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l, ... ,p,
where IN,j though defined only at l/(N+l), •.. ,N/(N+l) may
have its domain of definition extended to (0,1) by letting
it have constant value over (a/(N+l),(a+l)/(N+l)).
Further-
more, we make the following assumptions:
Assumption 1.
~oo IN,j(U) = Jj(U) oxists for
°
<
U
<
1
and is not a constant.
Assumption 2.
j[JN'/N~l l1f,jl - Jj{N~l Hjl]
1 2
dFN,j(xl=op{N- /
-00
Assumption 3.
J j (u) is absolutely continuous, and
IJj(i) (u)
I = I
d(i)J (u)
du
(r)
I
_
-5-i- 1.
~ K[u(l-u)]
2 ,
i = 0,1
for some K and some 5
00
>
0.
00
Let us now define
0000
v jk
=
JJ Jj[Hj(X;Qj)]Jk[Hk(y;Qk)]dHj,k(x,Yigj,Qk),
00
j,k= l, ... ,p
).
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15
and
v = (v
. jk ) j,k=l, .•. ,p •
We may remark that the assumptions 1 to 3 are needed
to prove the joint asymptotic normality ot the permutation
distribution or t N given by (~.14). Assumption 4 is required
to establish the stochastic convergence ot the permutation
covariance matrix
~N
to
Xwhich
we shall require inthe sequel.
Under the assumptions 1 to 4,
Thebrem 3.1.
probab11ityas N -> ~, where
Z=
(v
ZN
-+
:t in
jk ) is given by
+co +cd
(3.10)
vjk
Proof:
=
Jo J.tj[Hj{XjQj)] Jk[Hk(YjQk)] dHj,k(x,YjQj,Qk)
0
.
From (2.17), using (3.6) and (3.7) and the assumption
2, we rind that
~~
VN,j,k
JJ IN,j[N~l ~,j(X)] IN,k[N~l ~,k(Y)] d~,j,k(x,y)
=JJ J j[N~l ~, j(X)] Jk[N~l ~,k(Y)] d~, j,k(x,y)+
=
00
~
00
Now, writing
Jj[N~l ~,j(x)]
=
Jj(H j ) +
(~,j-Hj)J;(Hj) ~ ~il
J;(H j )
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(3.l4)
and proceeding exactly as in [14] and [26], we find,
omitting the details of computations that,
0000
JJ J j[ H (x;Q
vN, j, k =
00
j
j)] J k [ Hk(YiQk)
]dH j , k(X, Yi 9 j ,9 k )+op (1)
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Corollary 3. 1 ~ .i ;1:1'
I
where F j and Fj,k are symmetric about zero.
(ii) The assumptions of Theorem 3.1 are satisfied, then
~N
* in probab iIity as N....;..
-'> ~
00
,
where ¥, *= (v *jk)
is given by
+00
(3.16)
v;k =
+00
J,! J [2F (x}-l]Jk[2Fk (Y)-l] dFj,k(X'Y)
-00
Theorem 3.2:
j
j
.
- 00
Under the assumptions 1 to 4 N- 1/ 2TN -has
...;.--.;.~;.........;~~...;...;;;...~---.;;;~.;......;;;......;~-'
aSymptotically a p-variate normal distribution with mean
vector zero and covariance matrix
is given by
(3~10).
~
=
(V jk ) where (V jk )
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Proof:
To prove this theorem, it suffices to ShO\'1 that
for any arbitrary constant vector ~ = (6 1 , ... 6p )' the
distribution of N- l / 2 5 TN' is asymptotically normal under
'" '"
the permutational probability measure
~ ..
From (2.13)
we notice that
N
P
6 ·'T =~y- 5.
'" ",N
a= .1=.L J
I
(3.17)
r
where by definition
~a(~'£(N),R.)
~.p£N
depends upon
and
~.
Let us nO~1 consider the permutation distribution Generated
by 2N transformations gN in(;N defined in (2.2). It then
follows from earlier discussion that
~
and
2 remain
11
invariant under (;N or .tN' while £(N) may have 2
permutationally equally probable values:
(3.18)
which can only assume two values
i
(3.19)
with equal probability ~ for a
on the given
( 5 E (j)(j) , .j
j
~N,that
= 1, ... ,N.
is, on the given
= 0,1,
Thus, conditionally
(~'£N)'
)
remain fixed,
NR
while a (C(i)
a
' i = 1 , ••• , p ) can on1 y change s i gn s 1mu It aneous 1 y
(with in each a) for a =l,.... ,N. Thus, for the 2N, gN € G ,
=
1, .. . ,Pi a = 1,.... ,N
a.
N
we have the corresponding 2N values of
\'Jritten as
2'·TN which
may be
18
where (d N ) are mutually independent (under the permutational
a
probahility measure) and dN = +1 or -1 with equal probability
0:
1/2; and where conditionally on the given ~N' that is, on
the given q~N'£(N)) and~, IaN (~N'£(N) ,~) I, 0: = 1, ... ,N
0:
are all fixed. It is also easily seen that the permutational
average of (3. 20 ) is zero, and the variance N~ VN
So if we define WN,o:=
laNa(~N'£(N),~)ldN,o:,
then
~
I
•
unde~ ~,
(WN,a. ) are independent random variables with means zero
and liN ,a. can assume only two values -+1 WN,0: I each i'li th
probability 1/2. We will now apply the central limit
theorem to the sequence OlN,o.:
0:= 1, ... ,N) under
the Liapounoff's condition, viz.,
r·7
\ ;J • 01'
c.
•
Nhich reduces to showing that for some r
1
lim N-(=~-2)/2.
N->
00
>
N
2,
r
N ~ laN,o: q~N'£(N)'~) I
J:.
( -N
~
L-
0.=1
:=
2
(E
C
0
5) )r/2
aN ,a. ""N' ""!
(N)' ""
The denominator of the second factor of (3.22) is
2 ~N
~
,
which by Theorem 3·1 converges to some positive
quantity for any non-null 5
""
positive definite.
as
..' is assumed to be
""
So we require only
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to show that for some r
Since
aN,o.(~'£(N)'R.) is a linear function of ~N and R. (with
a
the sign of the coefficients being a minus or plus
depending on Q{N))' on applying the well known inequality
that
I LC=
i=l
(Cf. Loeve
ailr
~
pr-l
~
lai,r
i=l
[23J, pp. 155), we get
l
IaN (R_,C(N)' 5) Ir
N~
0.=1
,a ~N '"
'"
~
pr-l
~
i=l
Since, by using Chernoff-Savage conditions, we have for
2<
Ie
r < 2+5, 5 > 0,
00
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2, the numerator of the second
factor of (3.22) is bounded in probability.
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>
for i = l, •.. ,p ,
,
we see that (3.22) converges to zero in probability as N
-~
The proof follows.
Theorem 3.3.
Under the assumptions 1 to 4, the limiting
permutational distribution of SN
probability measure
=
1
-1'
N XN XN
!N
under the
~, is central chi square with p
degrees of freedom.
The proof of this theorem is an immediate consequence
of Theorem 3.2 and is therefore omitted.
Let us now define
-:z.
( "J
•
?'()
'"I
~i
+00
=
J
-00
Ji[Fi(X)-
Fi~-X)]
dF[i](X);
i=l, ... ,p)
00.
20
and ~ = (; l' ... , ~ ).
'"
p
Theorem 3.4.
The test based on SN is consistent against
the set of alternatives that;
'"
I o.
In particular, if
'"
F[1](X) is symmetric about some 5 i , 2 = (5 l ,···,5 p )'
then; I 0 for any 5 I 0, and hence the test is consistent
-
'"
'"
'"
'"
against any shift in location of a symmetric distribution.
Proof:
It is easy to show that ~ T~i)
as N -+
00
J.""'or all i
probability as N
assumption.
->
;i in probability
1 SN..... s
I:
•
Thus, N
· -11:s I ln
'" '" '"
where v is positive definite by
= 1, ... ,p.
-> 00,
'"
Consequently SN will be stochastically
2
indefini tely large as N
as N ~
00.
Again, SN ,e +'xp ,ex
-> 00.
the test is consistent against ;
~
O.
Furthermore, if
'"
Fi(X) is symmetric about some 5 i F 0, then it is easy to
show that ~i I o. Hence the theorem.
Asymptotic Normality of
XN
for Arbitrary F.
In this section we shall prove that under a certain
set of conditions the random vector
XN
(p))
= ( TN(1) , ... ,TN
has a joint normal distribution in the limit.
For the
sake of convenience we shall consider the statistic
(4.1)
where
-.
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Hence
and since the right hand inequality tends to one as N --> co,
4.
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21
N
TN,j = 1 L: R(j) Z{j)
N i=l ~,i N,i
(4.2)
z~;I = 1 if
~;i; i = 1, ... ,N;
where,
xi j )
>
= 1, ... ,p,
j
z~;i
0, and
= 0 otherwise.
are the constants satisfying
the assumptions (1) to (3) of the previous section.
* is related to the
N ~N* - N -EN where
reader may notice that the vector
~N
by the relation
=
~N
_
2
1
The
N
~N
(1)
N
vecto~
(p)
EN=N(LENi, .. ·,~ENi) ,
~1'
~1'
~'.
and so it suffices to consider the equivalent statistic
:t~.
The main theorem of this section is the following:
Theorem 4.1.
Under the assumptions (1) to (3) of Sectlon :2.:
1/2
the random vector N
-1(.
(~N-~N(Q))
has asymptotically a
p-varj~t2
normal distribution with mean vector zero, and covariance
matrix (aN,jk)' where
00
IJ.N , j
=
Jo
J j [ Hj (x; 9 j )] dF j (x; Gj) ,
J=l, ... ,p,
and where (aN, jk) is given by (4.14) and (4.15)
Proof:
and
as
N
\vriting I N, j (N+l ~,j) as
respectivel~.
22
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and,
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and, simplifying, we can express TN,j as
N
TN, j = Il N , j + B1N , j + B2N , j + ~ CiN, j
(4.4)
where
=
J
( 4 . 5)
Il N , j
(4.6)
B1N ,j =
J
Jj(Hj(X,Qj))d(FN,j(x)-Fj(X,Qj)
(4 . 7 )
B2N , j =
J'
(I1J, j - Hj) J ~ (Hj) dF j (x )
(4.8)
C1N ,j = - N;l
( 4 •9)
C2N , j =
J
(4.10)
C3N ,j =
J [Jj(N~l HN,j)- Jj(H j )
J j ( Hj (x, Q j )) dF j ( x, Q j )
J I1J,j J;(H) dFN,j(X)
(I1J, j
-
el
Hj) J; (H j (x )) d (FN, j (x ) - F j ( x) )
-
(N~l
11-J,j-H j )
J~(Hj))dJN,j
(4.11)
Proceeding as in Govindraju1u et a1. (1965) it can
be shown that the C-terms are all
Op(1/N1/2)~
Hence the
difference 1IN(TN,j-IlN,j)-~(B1N,j+B2N ,j) tends to zero
in probability. Thus to prove this theorem, it suffices
to show that for any real 5 i , i = l, ..• ,p, not all zero,
N1/ 2 ~ 5.(B1N .+B 2N j) has normal distribution in the limit.
j-1 J
,J
,
Now pr~ceeding as in [4), we can express N1/~5.(B1N j+B 2N .)
J~l J
,
,J
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23
as a sum of independent and identically distributed random
variables having finite first two moments.
The proof follows.
To compute the variance-covariance matrix of B1N ,j+B2N ,j'
we note from (4.9) and (4.10) that
(4.12)
B1N ,j+B 2N ,j=
~(FN,j~X)-Fj(XjQj))J;(Hj(XjQj))dFj(-XjQj)
-
~ (FN,j(-X)-Fj(-X;Qj))J;(Hj~X;Qj))
dFj(X;G j )
This has mean zero, and variance
= E (
-
=
~~FN,j(X)-Fj(X;Gj»J;(Hj(X;Gj»
~
(FN,j(-X)-Fj(-XjQj»J;(Hj(X;Qj»
dF j (XjQj»)2
~ ~~ Fj(X;Qj)[l-Fj(YjQj)]J~(Hj(XjQj»J;(Hj(Y;~j»dFj(-XjQj)
o<x<y<oo
(4.13)
+
~
~
dF (-yj9 )
--
j
~~ Fj(-y;Qj)[l-Fj(-XjQj)]J;(Hj(X;Qj»J;~Hj(YjQj»
00
dFj(XjG j ) dFj(Y;G j )
~ ~'Fj(-YjGj)[l-Fj(Xj9j)]J;(Hj(X;Gj»J;(Hj(Y;Gj»
x=O Y=O
dF
j
(-X j 9 j
) dFj(YjG j )
[Note that the application of FUblni's theorem permits the
interchange of integral and expectation.]
Hence
j
o<x<y<oo
00
-
dFj(-XjG j )
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-.
Simj.larly;
00
(4.15) O'N,jk=
00
J J [Fj,k(x,YiGj,Gk)-Fj(X;Qj)Fk(YiGk)]J;[Hj(X;Gj)J
x=o Y=O
00
00
-J J [Fj,k(x,-Y;Gj,Gk)-Fj(XiGj)Fk(~YiQk)]J;[Hj(XiGj)]
x=o y=o
,
Jk[Hk(Yi Gk )] dFj(-X;G j ) dFk(YiQk)
00
00
-J J [Fj,k(-X,Y;Qj,Qk)-Fj(-X;gj)Fk~y;gk)]J~[Hj(X;Qj)]
x=o y=O
00
+
00
J fiFj,k(-X,-y;Qj,Qk)-Fj(-X;Gj)Fk(-Y;
Gk)]J;[Hj(X;Gj)~
x=o Y=O
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5.
Asymptotic Distribution Under Translation Alternatives.
From this section onward, we shall concern ourselves
p
with a sequence of admissible alternative
hypothesis HN,
which specifies that for each j,k = l, ... ,p,
g
F.(Xi Q .) = Fj(X- .=.J.);
J_
J
yIN
(5.1)
w~ere Fj~~)
g.
Fj,k(x,y;Qj;Qk) = F.
J, k
Q
k
(x- --.J..,y_ - )
1"N
11N
and Fj,k(X'Y) aresymmetric about zero, and
G = (G l , ... ,Gp ) is unknown.
We shall also assume that the constant ~ji,
, i=l, ... ,N;
I"IJ
j=l, ... ,p,
is the expected value of the i-th order statistic
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25
.e
of a sample of size N from a distribution function ?fJj(x)
given by
x
I
about zero or uniform over (0,1).
Jj(x)
I
-1
= ¥j
It may be noted that
implies that the function
*-1 l+x
(x) =?/J j
(~) •
We first prove the following lemma.
If J j
Lemma 5.1.
( 5 • 1) 4
II
+1I(
x=o
= ?/J -1
, then
j
F j (x )[ 1-F j (y ) ] J ; ( 2F j (x ) -1 ) J ; ( 2F j ( Y) -1) dF j (x) dF j ( Y)
-
O<x<Y<~
~
.e
E~;l
the above definition of
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0 ,
*
where ?fJj~X)
is a distribution function either symmetrIc
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-
~
1-F j (x) )( l-F j ( y) }J; ( 2F j (x) -1 ) J ; ( 2F j ( Y) -1) dF j ( x ) dF j(Y )
y~o·
==
~
II
II j (x) [ 1-Hj ( Y) ] J
.
j (2F j (x) -1 ) J ; ( 2F j ( y) -1 )
d H . (x ) dII. ( y)
J
.J
O<x<y<oo 1
+ %(
-
=
i!J~
o
JI
~
(5·2)
J j (x) dx) 2
0
(" )
-
ax ·
~
x=O y=O
00
I
[F j, k(X' y) -F j (x)Fk(y)] J; (2F j (x) -l}J~( 2F k (y) -1 )dF j (-x )dFk(-y)
-
~
-
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00
00
-J J
[F j , k ( - x, y) - F j ( -x) Fk ( Y) ] J ~ ( 2F j (x) -1 ) J ~ ( 2Fk ( Y) -1 )
x=O y=O
dF (x) dF ( -y)
j
1c
00
00
JJ
[F j , k ( -x, - y ) - F j ( - x) Fk ( - y )] J ~ ( 2F j ( x) -1) J ~ ( 2Fk ( y ) -1 )
x=o y=O
dF j (x) dF k ( y )
+
00
=
%
00
JJ
-00
J j ~ 2F j ( x ) -1) J k ( 2Fk ( Y) -1) dF j , k ( x, y) ,
-00
Proof:
To establish the first equality of (5.1), we note
that Hj~X)
=
2F j (x)-1, and so the right hand side of the
first equality can be written as
4
fJ"' (2Fj(X)-1)(1-Fj(Y))J~(2Fj(X)-1)J;(2Fj(Y)-1)
O<x<y<oo
-
+
~
(J
dFj(X) dFj(Y)
1 ·
J j (x) dX) 2 ,
o
which equals
4
Jf
Jf
F j (X)[1-F j (Y)]J;(2F j (X)-1)J;(2F j (Y)-1) dFj(X) dFj(Y)
O<x<y<oo -
- 4
(l-Fj(x)) (1-F j (Y))J;(2F j (X)-1)J;(2F j (Y)-1) dFj(X)dFj(Y)
O<x<y<oo
+~ (
J
o
that is,
1 J j (x) dx) 2
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27
JJ
o
4
Fj (x)[ l-F j (Y)] J ; (2Fj (x) -1 )J; (2F j (Y)-1 ) dF j (x) dF j ( y)
<x<y<oo .
00
00
+2JJ
(l-Fj(X)) (1-F j (Y))J;(2F j (X)-1)J;(2F j (y)-1)dF j (X)dF,1(y)
x=o yeO'
J1
00
-4
00
x=O yeO
I!
(1-Fj(X))(1-Fj(Y))J;(2Fj(X)-1)J;(2Fj(Y)-1)dFixldFj(Y)
.
.
.
2
+ 1j: (
J j (X ) dx ) •
- 0
The last two integrals cancel out, and this proves the first
equality.
To prove the second equality, note that we can
rewrite the second equation as
1
JJ
x(l-y)J' (x)J' (y) dx dy + ~( J Jj(X) dx)2,
o<x<y<oo
.
0
~
and the result follows as in Chernoff-Savage (1958).
To prove (5.2), we note that when
t
is uniform over
(0,1), the left hand side reduces to
00
00
IIFj,k(X'y) dFj(-X) dFk(-y)
x=o yeO
00
-
-oi
00
11
F j,k(X'y) dFj(-X) dFk(y)
7;
-b!f
x=o yeO
00
00
J'
J
Fj , k (-x, y) dF x ) dFk(y) -~
x=O yeO
+ J J Fj,k(-X'-y) dFj(X) dFk(y) - -d: .
x=o yeO
-
j (
00
00
Noting that Fj(X) = 1 - Fj(-x), making simple transformations, and integrating by parts, the above expressions equal
22
-.
~~
JJFj (x) Fk (Y) dFj , k (x, y ) - i
-00
-00
The proof of the case when Jj(X)
= ~jl(x)
where ?j(X)
is given by (5.2) follows by using the relations
j=l, ... ,p,
and proceeding as above.
Theorem 5.1.
(i)
If, for
ever~
j
= l, ... ,p,
the conditions of Theorem 4.1 are satisfied
9
(1.i) Fj(.X;QJ
o. )
=
Q
r
1"N
1/N
k
Fj(X- .:..J.); Fj(X,YjQj,Qk) = F.(x-.=..1 ,y- ._.
)
IN
J
*
then N"1./2 (!N-~N)
has aSYmptotically a p-variate normal
distribution with mean vector zero, and covariance matrix
('t"jk) where
't"jk =
The proof of this theorem is a direct consequence of
Theorem
)~.1
and Lemma 5.1 and is therefore omitted.
It may be noted that the limiting distribution of
N1/2(T*_~
) is nonsingu1ar if and only if the f~Llnctions JJ.
""N I'CN
and F j are such that the moment matrix of
is non-singular.
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(Jj(2Fj~X)-1~j=lJ.. _P}
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The following theorem gives the conditions under which
the limiting distribution of
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is singular.
Let Xj , j = 1, ... ,N, be random vectors satisfying the assumptions of, Theorem 5.1.
Then the
Theorem 5.2.
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Nl/2(~;_~N)
asymptotic distribution of
k
<
P if and only if
Nl/2(~;-~N) is k-variate normal,
a.s. F.
= ~ a kJ k [2Fk (y)-1]
Jj~2Fj~X)-1)
[a
+ constant,
k are some constants].
The proof of this theorem is analogous to that of
Theorem 3.3 of Bickel (1964) and is therefore omitted.
6.
The Proposed
Class of Asymptotic Tests and its Limiting
Distribution.
Ive
no\'l
assume that the covariance matrix ~*
defined by (5.3) is non-singular.
hypothesis
He
= ('1" jk)
Then for testing the
given by (1.3), we propose to consider the
* defined
- as
test statistics SN
.
A
*
* (0) --*SN = N(~N-~N ) ~
(6.1)
1
*
(~N-~N(O))
,
'"
where
TN*
1\
is defined by (4.1) and (4.2);
z:..*
-
'"
*-1
consistent estimator of L.
'
and
'"
(6.2)
where
00
=
J
x=O
J j ( 2F j (x) -1)
dF j (x) ,
-1
is a
30
Lemma 6.1.
(i)
If
the conditions of Theorem 5.1 are satisfied,
(ii) the assumptions of Lemma 7.2 of Puri (1964)
hold for each function F j and J j
;
J
=
l, ... ,p
then, Nl/2(!;-~N(O)) has aSymptotically a p-variate
normal distribution with mean vector,
(6.4)
where
00
cJ = - 2
Jo J;(-
and covariance matrix
[Here
fj~X)
2Fj (x )-1 )f J (x) dFj (x)
2:'* =
jk ) given by (5.3).
'" of Fj(x).]
is the density
(T
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The proof of the lemma follows by noting (that by
I
assumption (ii))
(6.6)
and then applying Slutski's theorem ([12], p. 254).
A consequence of this lemma is the followinG
Theorem 6.1.
If the assumptions of
then for N -+ 00,
*
6.1 are satisfied,
the limiting distribution of
*-1 *
N{lN-~N(O))2:
Lennna
(TN-~N{O))
,
is non-central chi square with
p degrees of'" freedom andnoncentrality parameter
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where a is Given by (6.4) and (6.5), and ~**
= ~T;k)
j.s
gj.'y en bX
(6.8)
/\".-1
No\'l since
N~
is a consistent estimator of ~ *
""
then for 11
~s
~
-1
""
have the following
If the assumptions of Lemma 6.1 are satisfied,
Theorem 6.2.
*
SN
2:"
00
,
_the limiting distribution of the statistic
non-central chi square
wit~
p degr~~s of freedom,
* siven b~ (6.7).
and non-centralit0~~a~et~~5(SN)
/\ -1
*
From Theorem 6.2, it is clear that the choice of>
'-'"
is of no i.mportance in the limit. k17 consistent estin:ator
*-1
of L
\·Jill pre8erve the asymptotic distribution of the
""
i{..
statistic S11.
One such
con~istent
estimator is provided
by the permutational covariance elements in Section
3.
He m3.Y c.lso use a theorem by Bhuchongkul (1964) to propose
a si.milar class of esti.ma tes.
HO'ilever here condltions are
rela tively mo:"e restrictive (since they relate to
2.S~·:lptotic
normality) than the ones in Theorem 3.1, and for our
purpose, \'le need not bother about the conditions as we
simply require here the asymptotic convergence (not
normality) of the estimates.
.!~~~em_~:2'
and the
Th9_.Ecrmutation test based on SN .fjivenJ>Y (2.21)
asymp~oticallY
*
non-pal"ametric test based on SN
,g~vel1
32
33
.Q:l. (6.1) are asymptotically power equivalent for the sequence
of translation type of alternatives defined by (S.l).
Proof:
From Theorem 5.1, it follows that for any such
translation type of alternatives, the covariance matrix of
the proposed vector of rank order statistics converGes
asymptotically to
_*
L- ,
of which
'"
.1\*
2:
is a consistent estimator.
'"
Again by Corollary 3.1, it readily follows that under any
such sequence of translation type of alternatives, the
permutation covariance matrix in Theorem 3.2 also converges
*
to ~.
Thus by looking at (2.21) and Theorem 6.2, lIe
obs;ve that under (5.1), SN
cally equivalent.
~
S; where
~
means stochasti-
The proof follows.
I
I
-•
I
I
I
I
I
I
_I
By virtue of the stochastic equivalence of the tests
* in the next section.
of the unconditional test based on SN
I
I
Special Cases:
I
*
SN and SN'
A.
Vole
Let Jj(u)
*
shall only consider the asymptotic properties
= u,
* test reduces to the rank-sum
then the SN
-
SN(R) test (which may be regarded as a p-variate version of
the univariate one sample test) considered in detail by
Bickel (196S).
In this case the non-centrality parameter
(6.7) reduces to
5 (R) =
where y
=
(1 jk ) is given by
Q
1- 1 9
I
I
I
I
I
-.
I
I
I
.-
I
I
I
I
I
I
Ie
I
I
I,
I
I
I
I
.e
I
(6.10)
J f~(X)dX)2
1/12(
,j
=k
+00 +00
JJ
-~
F j (x) Fk ( Y) dF j I k ( x, y ) - 1/4
-~
~
k
--------------- , j r
+00
+00
(J r~(X)dX) if r~(Y)dy)
-co
-co
B.
Let J j be the inverse of a chi-distribution with one
* test reduces to the normal
degree of freedom. Then the 3 N
* (lN) test which may be regarded as a p-variate
scores 3 N
version of the univariate one-sample normal scores test.
In this case the non-centrality parameter (6.7) reduces to
5(}) = 9 ~-l g'
(6.11)
(6.12)
~jk
where ~ = (~jk) is given by
j
=
;
= k
j ., k
.
35
7·
Asymptotic Relative Efficiency.
The asymptotic efficiency of one test relative to another
in the present context can easily be obtained by a method
given explicitly by Hanan (1956).
as follows.
It may be stated roughly
If the two test statistics have, under the
alternative hypothesis, non-central chi square distribution
with the same number of degrees of freedom, the asymptotic
relative efficiency of one test with respect to the other
test is equal to the ratio of their non-centrality parameters
after the alternatives have been set equal.
For detaiis,
the reader is also referred to Bickel (1965), Puri (1964)
and Sen (1965) among others.
Our main interest will be
to study the relative performance of (i) the S~(~') test,
(ii) the S;(R) test and (lii) the Hotelling's T~ test,
which too has asymptotically a non-central chi square
distribution with p degrees of freedom, and non-centrality
parameter
,
Q
where
11 =
n",
(a jk ) is the covariance matrix of the underlying
-
distribution function F.
Thus, applying the definition of Hanan, and denoting
eT,T* as the asymptotic efficiency of a test T relative
to T*, we have (cf. (6.7))
e
where ~**
*
2
SN,TN
=
(9
L
**-1 ,
= (T * ) is given by
jk
covariance matrix of F.
9) /
(q
-1 ,
11
(6.8) and
9 )
TT =
(cr jk ) is the
I
I
-.
I
I
I
I
I
I
el
I
I
I
I
I
I
I
-.
I
I
I
,e
Special Cases:
(a)
Normal Scores and Rank Sum Tests.
From (7.2), we find that the efficiencies of the
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
normal scores
*
sNell
*
test and the rank sum SN(R)
test
relative to T~-test are
(7.3)
e
where A =
(7.4)
*
~Ajk)
e
2
SN(lJ ,TN
*
(Q"A-
=
l'
Q )/(Q
is given by (6.12) and
2
=
-(Q
r
-1'
Q )/(Q
SN(R),T
-N
where r = (r jk ) is given by (6.11).
,
1T- 1Q)
TT =
(cr jk ).
-1,
1T
Q)
We may remark that the expression (7.4) is the same
as the one obtained by Bickel (1965) for the p-variate
one-sample problem, and by Chatterjee and Sen (1964)
for the bivariate two-sample problem.
For the study
of the various aspects of the efficiency (7.4) in
special situations the reader is referred to the
interesting papers of Bickel (1965) and Chatterjee and
Sen (1964).
(b)
Totally SymmetriC Case.
A bivariate random vector
(X,Y) is said to be totally symmetric if (X,Y), (X,-y),
(-X,Y) and (-X,-Y) have the same distribution function.
It can be shown following the lines of the argument of
Bickel (1965), that a sufficient condition for the
asymptotic independence of the components of
total symmetry of (Xij),xi k » for every pair
* is the
~N
(j,k)~
37
Thus in the event of totally symmetric case A,
all diagonal matrices, and hence we have
""
11 ,
y are
(7.6)
Applying a theorem of Courant (cf. footnote) Bickel
I
I
-.
I
I
I
I
I
I
(1965) proved that
ttl
....1
where~+
is the family of all totally symmetric p-variate
distributions whose marginal densities exist.
While the authors were working on the study of
bounds of the efficiencies (7.S) and (7.7), the results
of Bhattacharyya
T~e~~em
(Courant)
x.Ax '/xB:c I ,
\'1h~:;,e
(1966) who was considering a two-sample
The maximal and minimal values of
A and B are non-negative definite,
§nd B Js non-s!ngular, are given by the maximal and
minimal
eigenvalu~s
of AB
-1
.
I
I
I
I
I
I
I
-.
I
I
I
.-
multivariate problem were made public.
I
I
I
I
I
I
variate two sample problem.
Ie
I
I
I
I
I
I
I
•-
I
It turns out that
the efficiency expressions (7.5) and (7.7) are the same
as in the case of the corresponding tests for the multiThus for the ease of reference
we give below some of the results for the proofs of which
the reader is referred to [2].
(7.9)
(7.10)
In the passing we may remark that when the components
of F are totally symmetric as well as identically distributed,
then the expressions (7.5), (7.6) and (7.7) become independent
of 9's, and the results are the same as in the case of
the corresponding univariate one-sample problems.
(c)
Normal Case.
Let us now assume that the underlying distribution
function F is a non-singular p-variate normal with mean
vector zero and covariance matrix
1T =
(Ojk).
Then it
can easily be checked that
(7.11)
e
*
2 =
sNell ,TN
1 .
This means that in the case of normal distributions,
the property of the univariate normal scores test relative
to the student's t test is preserved in the multivariate case .
39
This is interesting in the sense that the same is not the
case with the multivariate rank sum test as Bickel (1965)
has shm'm that
(7.12)
and
(7.13)
inf inf e *
2 = 0
FE~ Q
SN(R)"TN
for p
~
3
.2
for p
=
2 •
<
Tr-
e *
2 ~ 0.965
SN(R),TN
From (7.11) and (7.12), it follows that
for p ~
3
inf inf e
9
*
> 1
for p
~
2.
SN(l),S;(R)-
[~ is the family of all nonsingular p-variate normal distributiorn.]
The above results indicate that in case the underlying
distribution is normal, the normal scores test is always
preferable to the rank-sum test.
8.
Estimation Problem.
One Rample Case:
(1)
(p)_
As before, let X
= (X
, •.. ,Xa. , a. - 1, ... ,N be
"-Q
a.
a sample from a p-variate cdf F(X I - Q1 , •.. ,X p -Qp) where
Q = (9 , ... ,9 ) is unknown, F is symmetric about zero,
p
1
and F is continuous. Our aim to find suitable estimates
for the parameter Q.
I
-.
I
I
I
I
I
I
and Bhattacharyya (1966) has proved that
FE~
I
Suppose that the Hodges-Lehmann
statistic h [(5.1) of [18]] is calculated for every univariate
_I
I
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I
I
I
I
I
-.
I
I
I
40
,e
I
I
I
I
I
(j).sample Xl(j) , ... ,XN
' j - l, ... ,p.
hi
,e
I
hJ(xiJ),···,x~J» for the value obtained from sample
Xlj), ... ,x~j) on the j-th variate.
1
(8.1)
N
h j =- W ~=1
(j)
~,1
Thus we have
(j)
ZN .. 1
where E(j)
N,i is the expected value of the i-th order statist1c
of a sample of size N from the distribution function given by
(8.2)
for x
-0
>
.
Let
sup (Qj
( (j) -Qj, ... ,x ( j ) -Qj)
hjX
N
l
> ~)
Q**
j = inf (Qj
( (j)
(j))
h j xl -Qj'···'XN -Qj
< ~)
Q*
j
(8.3)
I'
Ie
I
I
I
I
I
I
I
=
We shall write
=
and
where
~
is a point of symmetry of the distribution of
h. when Q.
J
J
= o.
It was shown in [
18]
that the estimate
A
(Qj* + Q**
j )/2 = QN,j of Qj hasNmore robust efficiency than
the classical estimate j = ~ Xj /N.
a=l
a
In this section we shall consider the properties
X
of the estimate
(8.4)
of Q, and compare it with the normal theory estimate
-X = (Xl,.·.,X
- p ")
where -.JL
Xj = ~=1
considered by Bickel ~964).
(j )/N,
~
and the estimates
I
I
41
Regularity Properties.
8(a) .
The following theorems are immediate consequences of the
theorems proved by Hodges and Lehmann (1963) for the case p
=
1,
and are therefore stated without proofs.
A
A
"-
The distribution of the estimate QN= (QN,1, ... ,9n,p)
is (absolutely) continuous if F is (absolutely) continuous.
Theorem 8.1.
.A.
A
N(xl+a, ... , xN+a) = QN(2£1' .•. , xN)+a
(a l ,··. ,ap ) is any 1 )( p vector of constants.
Theorem 8.2.
..!:!here il:.
=
Theorem 8.3.
(1)
Q
The distribution of
A
N is symmetric about Q, if
Q
F is symmetric about zero,
-.
I
I
I
I
I
I
_I
Theorem 8.4.
For every vector
a
= (al, ... ,ap )
I
I
8(b) •
Asymptotic Normality.
Consider a sequence of sample sizes N
= 1,2, ... ,
and let Q be a sequence of values of Q. We shall indicate
N
the dependence of h j and ~ on N by writing h jN and ~N.
Theorem 8.5.
Let
of constants.
~
= (aI' ... ,ap ) be a one by p vector
Let c l '
~N = -~cN = -(al/c N,
~et ~be
,c N' ...
be constants, and let
, ap/c N)·
a continuous p-variate distribution function
I
I
I
I
I
-.
I
I
I
42
,e
whose marcinals have means zero and variances one, and
lim PN(CN(hjN-~) ~ u j ; j = 1, ••. ,p)
N->oo
up +apBp
u1 +a l Bl
G
=
(
A'···'
A
)
1
P
I
I
where PN indicates that the probability is computed for the
(j
parameter values QN and where h jN stands for h jN ( xl( j ) ' ... ,xN
.J
I
I
I,
Then for any fixed Q
, ... , ~
A)
(8.6)
p
By Theorems (8.2) and (8.4)
Proof:
Ie
= lim P
N->oo
I
I
I
I
suppo~
.Q
"" ~ ~)
(C~N
= lim P (h (x(j)-
N->oo 0
= lim
1
~
(j):J))
C , .•. ,xN - C ~ ~N
N
N
(j)) - ~N ~ 0)
Po{(
h j .xl(j) , ••• ,xN
N->oo
=
j
.
alB l
G(-A- , ... ,
1
~.
A
p
).
Now combining Theorems (8.5) and (4.1) we have
Theorem 8.6.
Under the assumptions of Theorem 5.1,
I
Nl/2(~_Q) has asymptotically a p-variate normal distribu-
I
I
tion with mean vector zero and covariance matrix
,e
I·
2:**
'"
*
= (T~k)
.
given by (6.8).
J
Let J j = tjl where t j is R(O,l), then the
A
A
estimate QN will be denoted by QN,(R) and in this case the
Remark 1.
))
.
covariance matrix
2:**
* reduces to
= (Tjk)
Z = (Yjk)
defined by (6.11). '" [See also Bickel (1964).]
-1
Let J j = W
j where W
j is given by (5.2);
A
A
then the estimate QN will be denoted by QN,l and in
this case, the covariance matrix ~** = (T *jk ) reduces
Remark 2.
to ~
=
8(c).
~Ajk)
defined
by (6.12).
I
I
-.
I
I
I,
Asymptotic Efficiency.
To obtain an idea of the relative performance of one
estimator with respect to another, we employ the notion
of "the generalized variances" a concept introduced by
Wilks (see1'1i1ks (1962), p. 547).
The "generalized variance"
of a p-variate random vector (X 1 , ... ,X p ) with non-singular
covariance matrix 2: = (P.kcr.crk) is defined to be
2
2
'"
J
J
cr l ... crp det II Pjkll
where "det" denotes the determinant.
var X
=
T and T' of ~ with asymptotically non-singular matrices
~T
= (T
T T) and L~T ' = (T'
T' T ' ) require Nand N'
LPjkcrjcr
Pjkcrjcr
k
k
observations to achieve equal asymptotic generalized
Then the asymptotic relative efficiency of T/
with respect to T
eTI T
,
is defined as
~_ U~criT ' ... cr~);LdetIlPIkIlJP
T'
T'
_
- lim N N->oo
cr1 ... crp
I
fl
_I
I
Suppose that the two asymptotically unbiased estimates
variances.
I
detll Pjkll
Now from (6.8), the generalized variance of "GN is
'I
I
I
I
I
I
e-I
,I
I
I
,I
I,
44
(8.8)
var
where
I
I
I
I
Ie
I
I
I
I
,e
I
JJ
P***
jk
J j ~ 2Fj (x) -1 ) J k ( 2Fk (Y)-1) dF j , k (x, y )
-co -co
=
j=k,
1
and, the generalized variance of the mean estimator
XN
(8.10)
=
(XN, l' • • • , ~ , p) is
var
(~)
=
fr
j = 1
C1~
J
. det I!Pjk ll
where ~ C1 jkC1jC1k ) is the covariance matrix of X j'
Hence, we have
I
I
I
+00 +00
t
***II p•
detllPjk
(8.11)
detl\ Pjkll_
Our main interest is the study of the relative performance
A
'"
_
of the estimators QN A, QN R and~. [For the efficiency
A
,~
,
comparisons of QN,R' ~ and the estimates based on the
medians of the observations, the reader is referred to
Bickel (196lj)].
From (8.11) it is easy to deduce
p
iT
(8.12)
~1
J
j=l
12(j~(
f~(X)dX)2
*
@et~Pjk~
I
I
-.
p
~et p jk~
-co
where
~
~
[J
*
Pjk
JFj(X) Fk(y) dFj,k(X'y)-
~]
f. k
if j
-co -co
=
(8.14)
where
~ +00
1
J Jj- [
1
F j ~ x) ]}- [Fk (y)] dF j , k (X, Y)
j
f.
k
-co -co
if j
1
= k.
(8.16)
[When p = 1, the efficiencies (8.12), (8.14) and (8.16)
reduce to the familiar expressions in the one-sample univariate
case.]
We shall study the above efficiencies in different
situations.
Case 1.
A
'"
Total Symmetry.
QN,l' QN,R and
XN
_
Let us assume that the estimators
have pairwise independent coordinates.
In this situation the covariance matrices reduce to diagonal
matrices, and we have
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
-,
I
I
46
I
.e
I
I
I
I
1\
I
(8.17)
(8.18)
(8.19)
Hence from the results of Hodges-Lehmann (1961), ChernoffSavage (1958) and Mikulski (1963), we have
(8.20)
Ie
I
I,
,e
I
F€J-
(8.21)
(8.22)
I
I
t
I
I
inf
where
inf
F€J-
2t is
7r
=
b'
the set of all totally symmetric absolutely
continuom p-variate distributions.
Now suppose that the components of F are totally
syrrooetric as well as identically distributed, then the
efficiencies are independent of p, and hence the same
as in the case of one-sample univariate situations.
Case 2.
I
I
Equally Correlated Distributions.
Let us now assume that the distribution of (X~j),X~k))
is independent of
j
and k when Q
= O. Then, it turns out
-.
I
·1
I
I
,I
that
(8.23)
(8.24)
I
_I
(8.25)
I
Ca3e 3.
Normal Case.
Let the underlying distribution function F be a
nonsingular
p-variate normal distribution function with
mean vector zero, and covariance matrix::
= (Pjkcrjcrk )
then, from (8.12), (8.14) and (8.16) we obtain
(8.26)
I
I
where
00
(8.27)
P;k =
00
J Jlj[~] lk[~]
-00
-00
j
II
I
I
t
dlj,k(x,y) .
-,
I
I
I
Ie
I
I
I
I
I
I
I'e
I
48
(8.28)
* is defined by (8.27).
where Pjk
Now Biclcel (1964) has proved that for p
(8·30)
I
I
I
I
,.
I
inf
F€~
e~
>
2,
=0
GN,R;XN
where ~ is the set of all nonsingular p-variate normal
distributions and, since
it follo\'1s that
(8.32)
e
I
detllp *jkll ]l/p ,
detll Pjkll
(8.29)
Thus, we find that when the underlying distribution
function is normal, the multivariate normal scroes estimator
A.
QN,! (which is as good as the means estimator) can be
infinitely better than the multivariate Rank-sum estimator
"-
N,R for p > 2. This leads to the question of examining
the relative performa.nce of these two estimatora vj.z.
Q
A
"-
GN,l and GN,R for the case when the underlying distribution
2 2
function F is bivariate normal N(O,Ol,02'P)'
The
efficiency behavior of '"
GN,l and ~
GN,R is given by the
following
Theorem 8.7.
"
The efficiency of QN,R with respect to -'\QN,I
is independent of cr1
~
cr 2 and is given by
1 - P
2
]
1/2
I
I
-.
I
I
=
.2 [ _ (1+2 cos ~ (v+u))
2
u(v-u)
where u is determined by
eg
. Q'
N,R' N,l
p
]
= 2 cos (u+rr)/3. The function
is monotone increas ins for 0
symmetric about p
lim
Ip 1->1
1/2
< p <
-
1 and (2 )
= 0 and hence unimodal. Finally,
e""
g
~
.g
N,~'
=(
N,l
3
v sin 1I
)1/2
= O~91.
3
The proof is the same as in Bickel (see Theorem5·2 [ 5 ])
since when the underlying distribution is normal,
I
I
I
I
_I
I
,
I
I
I
I
I
-I
I
I
I
,e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
50
References
1. Bennet, B. M. (1962).
~ts.J.
On Multivariate Sign
Roy. Statist. Soc. Sere B 24 195-161.
~
2. Bhattacharyya G. K.
(1966).
Multivariate Two
Sample Normal Scores Test for Shift.
Doctoral dissertation,
University of California, Berkeley.
3. Bhuchungkul, S.
(1964).
A Class of Nonparametric Tests
for Independence in Bivariate Populations.
Statist.
~
Ann.
r~th.
138-149.
4. Bhuchongku1, S. and Puri, Madan L. (1965).
On the Estilnation of Contrasts in Linear Models.
Ann. Math. Statist.
5. Bickel, Peter J.
~
198-202.
(1964).
On Some Alternative Estimates
for Shift in the P-variate One Sample Problem.
Ann. Math. Statist.
~
1079-1090.
6. Bickel, Peter J. (1965).
On Some Asymptotically Non-parametric
Competitors of Hote1ling's T2 . Ann. Math. Statist. ~
160-173.
7. Blumen, I. (1958).
A New Bivariate Sign Test.
J. Amer. Statist. Assoc. ~
8. Chatterjee, S. K. (1966).
Location.
448-456.
A Bivariate Sign Test for
Ann. Math. Statist.
To appear.
9. Chatterjee, S. K. and Sen, P. K. (1964).
Nonparametric Tests for the Bivariate Two Sample
Location Problem.
Calcutta Stat. Assoc. Bu1l.!Z
10. Chetterjee, S. K. and Sen, P. K. (1965).
49-50.
Non-parametric
Tests for the Multi-sample Multivariate Location Problem.
Ann. Math. Statist.
Submitted.
51
11. Chernoff, H. and Savage, I. R.
-,
(1958).
Asymptotic Normality of a Certain Non-parametric Test
Statistics.
Ann. Math. Statist.
12. Cramlr, Harold
(1946).
~
972-994.
Mathematical Models of Statistics.
Princeton University Press.
13. Govindarajulu, Z. (1960).
Central Limit Theorem and
Asymptotic Efficiency for One-Sample Non-parametric
Procedures.
Mimeographed doctoral dissertation.
University of Minnesota, Dept. of Statistics.
14. Govindarajulu, Z., Lecam L. and Raghavachari, M. (1965).
Generalization of Chernoff-Savage Theorems on Asymptotic
Normality of Non-Parametric Test Statistics.
Proceedines Fifth Berkeley Symposium
15. Hannan, E. J. (1956).
Math. Statist. Probe
The Asymptotic Power of Tests Based
J. Roy. Stat. Soc. Sere B. 18
Upon Multiple Correlations.
~
227-233·
16. Hodges, J. C., Jr. (1955).
Math. Statist. 26
~
A Bivariate Sign Test.
Ann.
523-527.
17. Hodges, J. L., Jr. and Lehmann, E. L. (1961).
Comparison of Normal Scores and Wilcoxon Tests.
Proc. Fourth Berkeley Symp. 1
~
307-318.
18. Hodges, J. L., Jr. and Lehmann, E. L. (1963).
Estimates of Location Based on Rank Tests.
Ann. Math. Statist. ~
598-611.
19. Joffe, A. and Klotz, J. (1962).
Null Distribution and
Bahadur Efficiency of the Hodges Bivariate Sign Test.
Ann. Math. Statist.
~
803-807.
I
I
I
I
I
I
I
I
el
I
I
I
I
I
I
I
-.
I
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
52
20. Klotz, J. (1964).
Sign Tests of
Small Sample Power of the Bivariate
~lumen
and Hodges.
Ann. Math. Statist.
~
1576..1582.
21. Lehmann, E. L. (1959).
Testing Statistical Hypotheses.
Wiley, Ne\'l York.
22. Lehmann E. L., and Stein, C. (1949).
On the Theory of
Some Non-parametric Hypotheses. Ann. Math. Statist. 20
23· Lo'eve, M. (1955).
""""
Probability Theory.
28-45.
D. Vn Nostrand Co., New York.
2!!·.
Mikulski, P. W. (1963).
On theEf'ficiency of Optimal
Non-parametric Procedures in the Two Sample Case.
A~~.
Math. Statist.
22-32.
~
25. Puri, Madan L. (1964).
of c-Sample Tests.
Asymptotic Efficiency of a Class
Ann.
~~th.
Statist.
~
102-121.
26. Puri, Madan L. and Sen, Pranab K. (1965).
On a Class of Multivariate Multi-Sample Rank Order Tests.
27. Sen, P. K. (1965) .
On a Class of' Multi-Sample Multivariate
Non-parametric Tests.
Jour. Amer. Statist. Assoc.
Submitted.
28. Sen, Pranab K. (1965) .
Non-parametric Tests.
On a Class of Two-Sample Bivariate
Proc. Fifth Berkeley Symp. f.1ath.
Statist. Probe (in press).
29. Sen, Pranab K. (1963).
On the Estimation of Relative
Potency in Dilution (-direct) Assays by Distribution-free
Methods.
Biometrics.~,
532-552.
53
30. \'la1d, A..and Wo1fowitz, J. (1944).
Statistical Tests Based on Permutations of the
Observations.
Ann. Math. Statist.
31. Wilcoxon, F. (1949).
Procedures.
~
358-372.
Some Rapid Approximation
American Cyanamid Company, Stamford, Conn.
32. Wilks, Samuel S. (1962).
John Wiley, New York.
Mathematical Statistics.
I
I
-.
I
I
I
I
I
I
eI
I
I
I
I
I
I
I
-.
I