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ON A CLASS OF RANK ORDER TESTS FOR INDEPENDENCE
IN MULTIVARIATE DISTRIBUTIONS
by
Madan L. Puri and Pranab K. Sen
Courant Institute of Mathematical Sciences, New York University
and University of North Carolina, Chapel Hill, N. C.
Institute of Statistics Mimeo Series No. 497
November 1966
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Research sponsored by Research Grant GM-l2868 from the
National Institutes of Health, Public Health Service.
DEPARTMENT OF BIOSTATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
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ON A CLASS OF RANK ORDER TESTS Fon INDEPENDENCE
IN MULTIVARIATE DISTRIBlJTIONSl(- t
by
Madan L. Puri and Pranab K. Sen
Courant Institute of Mathematical Sciences. New York University
and University of North Carolina, Chapel Hill, N. C.
~ummary.
1.
In this paper we offer nonparametric competitors of some classical
tests of multidimensional independence considered by Wilks (1935), Daly (1940)
and, Wa1d and Brookner (1941).
mutual independence of q subsets of the totality of p variates (p
~
q
~
2), and
(ii) pair wise independence of p variates [cf. Anderson (1957) Chapter 9] are
"
considered, 811 asalnlt appropriate classes of atochsatic dependence alternotives.
Some strictly distribution free permutation tests are developed here,
and their asymptotic properties are studied with the aid of a theorem on the
asymptotic distribution of a class of rank order statistics to the case of more
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than two variates.
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2.
••I
In this context, the problems of testing (i) the
!ntro~~~
Let~a,
1:1
(Xla!''''Xpa,)' aPl, ... ,n be n
LLd.r.v's (indepen-
dent and identically distributed (vector valued) random variables)having a pC> 2)
=
variate continuous cdf (cumulative distribution function)
(xl, ••• ,X ) ERP, the p-dimensional Euclidean space.
p
,.
F(~),
where
~
1:1
Let ~ be partitioned into
q(> 2) subvectors; that is
(2.1)
*
Research sponsored by Research Grant GM-1"2868 from the National
Institutes of Health, Public Health Service .
t Revised November 1966.
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where X(j) is of i. components; jal, ••. , q and
"'a'
1
J
...
be denoted by F(j)(x(j»
where ,..x(j)
€
R j,
q
E
Let the marginal cdf of X(j)
= p.
j .
.ral J
j-1, ••• ,q.
"'a.
Our first problem
relates to the stochastic independence of the q subsets of variates.
This may
be statistically framed by means of the null hypothesis
for all ,..x
(2.2)
€
RP•
Let us denote the marginal cdf of Xio. by F[i](x), i-l, •.. ,p; and the marginal joint
cdf of (Xio.' Xjo.) by F[ i, j] (x, y) for all i+ j= 1, ... , p.
X10.'
... ,
Xp:x. are said to be
totally independent if
..
(2.3)
Thus
for all ,..x
F(x)
H~P)
is a particular case of
H~q)
€
RP
•
when q=p, that is when, i{' .•. , ipal.
Similarly,
X ""'X
are said to be pair wise independent if
po.
10.
(2.4)
and for all ifj=l, ... ,p.
(It may be noted that (2.3) implies (2.4) but not the
vice-versa).
In the particular case of F(x)
.., being a multinorma1 cdf, pair wise independence
implies total independence and vice-versa.
implies independence and vice-versa.
Moreover in this case uncorrelation
Also, the multinormal cdf is completely
specified by its mean vector and the covariance matrix.
Thus, all the three
problems stated above reduce to certain specific structures of the covariance
matrix.
i
j
If the covariance matrix is partitioned into q2 submatrices of orders
x i.e; j,.e=l, ... , q, then the first hypothesis
H~q)
is equivalent to the
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hypothesis that all the off-diagonal
partitioned matrices are the null matrices,
while the hypotheses H(P) and H* are both equivalent to the hypothesis that the
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covariance matrix itself is a diagonal matrix.
considered in Anderson (1957, Chapter 9), and the reader is referred to it for
details of background and motivation.
The object of the present investigation is to explore the possibility of
generalizing the above problems in a completely nonparametric set up.
exist, and even if it exists, it may not play the fundamental role as in the case
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are defined for a much wider class of cdf's.
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Now, for
arbitrary (continuous) multivariate distributions the covariance matrix may not
of the mu1tinorma1 distributions.
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Tests for these hypotheses are
For this reason, we shall formulate a class
of association parameters which are regular functiona1s of the cdf
F(~)
and which
This will provide us with a suitable
nonparametric competitors of the classical covariance (or correlation) matrix
and also increase the scope of the methods.
Secondly, for mu1tinorma1 cdf's
uncorre1ation and independence are equivalent, but the same is not true for
arbitrary cdf's.
Thus, to be precise about the class of alternatives, we will
also define some dependence function with some emphasis on association alternatives.
The proposed tests will be shown to be consistent and reasonably efficient for
such alternatives.
Finally, for arbitrary cdf's, since pairwise independence
does not inp1y total independence (cf. Geisser and Mantel (1962», we shall also
consider the problem of testing H*, defined by (2.4), in some detail.
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In this
context, we shall extend a theorem by Bhuchongkul (1964) on the asymptotic distribution of (association) rank order statistics to the multivariate case, and
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subsequently use it to study the asymptotic properties of the proposed tests.
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3.
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~ormu~a~c1~~0t-associatjon~~~m~t~rs.
Apart from the necessity
of the existence of the second order moments, the estimator of the usual covariance
matrix are quite sensitive to outlying observations.
Here we shall consider some
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alternative measures of association which are really functiona1s of the parent
cdf's.
For this, we consider the marginal cdf's F[i] and F[i,j] (irj=l, ... ,p)
as in section 2.
Also, let J(i)(u), 0 < u < 1
(i=l, ... ,p) be some absolutely
continuous function defined on the open interval (0,1) and suppose that J(i)(u)
is normalized in the following manner:
1
(3.1)
Later on we shall impose certain regularity conditions on J(i)(u).
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for i= 1, ... , p.
and
oj J(i)(u) du = 0
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Let us
now consider the transformed variables
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(3.2)
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for i=l, ... ,p
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and denote
(3.3)
Y
,..
=
(Y , ... , Y )
P
1
Y. will be called the grade functional of X.: i=l, ... ,p.
~
The joint distribution
~
of Y
can be obtained from F(x),
but it will naturally depend upon the associa,..
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tion pattern of F(.x).
If J(i)(u) is not constant for all 0
< u < 1, (i=l, ... , p)
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it is easy to verify that independence of (X., X.) implies the independence of
J
~
(Y., Y.) and vice-versa.
~
J
Further this property is preserved under any monotone
transformation of X.; i=l, ... ,p.
~
So when we are interested in the problems of
stochastic independence we may work with Y , aF1, ... ,n, and suitably choosen
"'a,
J(i); i=l, ... ,p, often lead to some nice properties of such alternative measures.
Let us now define
00
(3.4)
00
J J J(i)(F[i](x»
-00
-00
J( .)(F[ .](y»
J
J
dF[ . . ](x,y);
~,
J
i, j=l, ... , p
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It may be noted that by (3.1), e.. = 1 for all i=l, ... ,p and le .. / < 1 for all
~~
i~j=l,
f
~J
Y. .
Ja.
In fact e .. is the product moment correlation of Y.
~J
10.
and
Thus, we define a nonparametric association matrix by
®=
«e.. » ..J= 1, ... , p
,.,.
~J~,
.
Naturally, the choice of the transformations J (i)' i= 1, •.. , p plays an
important role in the procedure to be considered.
We shall consider a general
class of such functionals and subsequently we shall briefly consider the problem
of selecting some particular measures.
Now, usually the cdf's F[i]' i=l, ... ,p are all unknown, and hence! is also
unknown.
However, (3.4) has a close analogy with the type of functionals considered
by Von Mises (1947), and following his line of approach we shall frame suitable
estimators of
I
(3.7)
••I
= O.
(3.5)
(3.6)
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=
Further, if X. and X. are independent, then from (3.1) and (3.4),
w.
Ja.
we see that e ..
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... ,p.
~J
®.
,...,
With this end in view, let us define
Fn[i] (x) = (number of Xia.
~
x, arl, ... ,n)/n; i=l, ... ,p
(x, y)
F
n[i,j]
= (number of (X. , X. )
~a.
Ja.
Then we may consider J(i)(Fn[i](x»
i=l, •.. ,p.
< (x,y),
=
aPl, ... ,n)/n; i1j=1, ... ,p.
as a natural estimator of J(i)(F[i](x»;
Further in actual practice, often we have a sequence of functions
viz. {In(i)(u)} which converges to J(i)(u) for all 0 < u < 1 as n~, and i=l, ... ,p.
Thus for the sake of more generality in presentation, we may consider the
estimators
(3.8)
Consequently, from (3.4), (3.6), (3.7) and (3.8), we may frame the estimator of
e ~J
..
as T .. where
n, ~J
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00
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00
(3.9)
=
1
n
E J n (.){F
J n (.)(F
n arl
~
n ['J{x.)
~
~~
J
n ['J(x.
J
J~
»;
i,j=l, •.. ,p.
It may be noted that T .. are nonstochastic (if J (.)'s are so) and by (3.1),
n, ~~
n ~
they converge to unity as
n~.
Our proposed tests are based on suitable functions
of the stochastic matrix T , when
... n
(3. 10)
T = «Tn, i'»i
. 1, ••• , p •
"'n
J
, J=
In the sequel, T
... n
®
,.,
and
lation dispersion matrices.
will be termed respectively the s.ample and popu-
It may be noted that many well known nonparametric
measures of association belong to the class considered above.
For example, if
we let
(3. 11)
(!{
In(i) ~
n+
= {.JL)t {~ _ n+l} ,.
n2-l
2
1»
for J{i)(u) =
Jl2
(u-t), 0
<
u
<
1;
i=l, ... ,p
e ~J
..
while
reduces to the grade
correlation coefficient (cf. Hoeffding (1948), pp. 318).
The measures con-
sidered by Blomquist (1950) and Bhuchongkul (1964) among others, also belong to
this class.
We have so far tried to provide a heuristic argument for measuring association
by the functionals 8 .. 's defined by (3.4).
~J
,
We may also justify the use of such
functionals if we adopt Hajek's approach (1962).
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arl, ••• , n: i=l, ... ,p
then T ., reduces to Spearman's rank correlation (cf. Kendall (1955»
n, ~J
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However, his study
specifically relates to linear regression alternatives whereas we are more
interested in association alternatives, and apparently his procedure seems to
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have very little impact on ours.
But, in spite of his logic, one may propose
some measures of association which are particular cases of the general class
of
@.
~
Again, in multinormal distributions any stochastic association is reflected
by the correlation matrix.
In general, deviation from independence may not be
reflected by correlation.
Thus to consider the class of alternative hypotheses,
we will also consider some dependence function (cf. Sibuya
(1959».
Denote
(3.12)
(3.13)
irj=1, ... ,p.
Both nand n .. are functions of the marginal distributions as well as the joint
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ones.
So we shall prefer to write them as n(F[l], ..• ,F[p]) and nij(F[i],F[j])
respectively.
Hereafter we shall term nand n .. as dependence functions.
1J
(It
may be noted that if Xl, ... ,X are totally independent (or if Xi and X are inj
p
dependent), then n(or n
ij
) will be equal to unity of all
Z
(or all xi' x j ) and
so, the divergence from unity on a set of points of measure non-zero relates to
stochastic dependence).
In subsequent sections, we shall make repeated use of
this type of dependence functions.
4.
Permutationally distribution-free tests for H(q).
It may be noted that if
X. and X. belong to two different subsets, then under H(q) in (2.2),
1
J
0
Let us now denote by
so that F(x)
...
€
~
p
•
Let us define
1r
e .. = O.
1J
the class of all r-variate continuous cdfts for r=l, ... ,p;
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F(j)
(4.1)
€
~
,
j= 1, ... , q}
1..
J
Thus
1p
0
is a subclass of all p-variate continuous cdfts for which the q specified
subsets of variates are mutually stochastically independent.
Thus, one may write
equivalently
F(x)
....
(4.2)
€
r& p(0)
and may be interested in the set of alternatives that F(x)
,.,
€
i p _~o.p
However,
such a formulation will give rise to considerable amount of mathematical complications since F(x)
is otherwise of completely unknown form.
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In this paper,
we shall restrict ourselves to the class of alternatives which relate to the lack
of independence through the values of the functional matrix
Let us now parti tion
®
...
-v
as
®u
®
,... 12
,..
@lq
®21 ®
,... 22
®2
... q
~
®
® defined in (3.5).
where
(4.4)
@ke
=
is of the order i
for
k
®q2
x i.e; k,.e=l, ... ,q.
1o/.e=1, •.. , q
under
Now
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N
®
.... q 1
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from (4.2), we have
i o q ).
We frame the class of alternatives as the set of all p-variate cdfts for which
the equality sign in (4.4) does not hold for at least one pair (k,£), ~£=l, ... ,q.
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Now recalling the definition of 8 .. in (3.4) and
~J
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co
co
co
co
n..
~J
in (3.13), we may write
(Here J' denotes the first derivative of J).
In most of the cases (as we shall see later on) J(i)(u) is monotone in
u, 0 < u < 1 for i=l, ... , p, so that if n.. (x, y) is uniformly (in x, y) greater
~J
than (less than) or equal to 1 with the strict inequality on at least a set of
rO.
points of measure nonzero, then 8 . .
~J
In many types of factorial dependence
(cf. Konijn (1956), Bhuchongku1 (1964) for the bivariate case), it is easy to
n..
verify that
~J
is greater than or less than one, depending upon the coefficients
of the stochastic factors, and hence such an alternative may be quite suitable.
Incidentally, we are not restricting ourselves to the particular type of the
alternatives of linear dependence as considered by Konijn (1956) and Bhuchongkul
(1964) (in the particular case of bivariate distributions), in fact, they can be
shown to be contained in the class of alternatives considered in this paper.
Now for the simplicity of presentation, we shall consider in detail the
tests for H(q) when q=2, and then we shall indicate briefly how the theory can
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be extended to the case of more than two subsets.
(4.6)
x =
"'a.
Let then
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where 1 :::;£, m=p-£ < p, and OFl, ... ,n.
We denote the sample point E
n
by
(1)'
~l
Pxn
E
= (Xl"'''' X') '"
(4.7)
""n
""
"'n
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(1)'
' ... ,
!n
(2)'
(2)'
~l
' ... , ~n
The joint distribution function of lOIn
EPxn is given by
n
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n
(4.8)
G(E
Nn )
=
F(x
... ex. )
OFl
(2)
and by (2.2), under H
o
n
n
(4.9)
G(E
.... n )
=
_I
OFl
Let now
by
~n'
R
",n
= (Rl,R_, .•. ,Rn )
-~
be any permutation of (1,2, ... ,n) and we denote
the set of all the n7 permutations of (1,2, ... , n).
Furthermore, let
us also denote
(1) ,
X
...n
(4.10)
E(R )
~ ..n
=
(2)'
?£.
(4.11)
1
S(E
) = {E(R
)
",n
tv ",n
R
, ",n
(2)'
€
{R
"'n
, ... ,~
n
R
....n
E
:R)
•
....n
Then, it follows from (4.9) and (4.10) that the joint distribution of '\I
E(R
)
Nn
remains invariant under the group of transformations R
"'n
E
R
-n
if H(2) holds.
0
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Consequently, we shall call S(E
), the permutation invariant set of ~n
E.
~n
now
~
~n
be the n p dimensional Euclidean space, so that E £
~n
~.
~n
Let
Now under
H(2), the probability distribution of E over the set of n1 points in S(E ) is
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"'n
"'n
uniform, and hence, if
~(E
(4.12)
HE*)
",n =
..n
L.
E*€S(E )
for all E
...n
€
.... n
"'n
~,
then
...n
) is any test function choosen in such a way that
n'
€
E (2) {~(~n)} = €,
(4. 13)
the level of signification.
H
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(EH(q){'} denotes the expectation of {.} under
H~q».
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Now to formulate
~(E
..n
) in a suitable manner and to make it invariant under
monotone transformation of the variables, we proceed as follows.
For each integer
n, let
(i)
(4.14)
(i)
i=l, ... ,p.
(Ln, 1 ' " ' ' L n, n ) ;
be the p sets of real constants, where
(4.15)
L (i)
n,a.
= i i ) (~n);
n
i= 1, ... , p.
o;=l, ••• ,n;
and where the functions J(i) have the same interpretation as in section 3.
n
this notation we can rewrite Tn,ij (cf. (3.9»
(4.16)
Tn,1.' j
=
1
n
as
n
L.
0;=1
i, j=l, ... , p
With
-12 -
where R.
is the rank of X.
1~
X.
1~
1~
and X.
J~
among (X. , ... ,X. ) for i=l, ... ,n.
11
1n
are. stochastically independent for all i=1, •.. ,£ and j=£+l, ... ,p.
.»
Let us now partition T = «T .
"'n
n,1J
q=2.
Under H(2),
a
i,j=l, •.. ,p as we did
®
-
in (4.3) for
We then write
«T(l~:»)
«T(l,.~»)
«T(2, ~»)
n, iJ
«T(2~~»)
n,1J
(4.17)
T =
",n
n,1J
n,1J
note that
«T(l,. ~») = «T .. »
n, 1. J
n,1J
i, j=l, •.• ,£; «T(l,.~») = «T .. »
n,1J
n,1J
i=l, ... ,£
j=£+l, •.• , P
(4.18)
.»
«T(2,. :») = «T .
n,1J
n,1J
«T(2~~») = «T .. »
. n,1J
n,1J
i=£+l, .•. , P
j=l, ... ,.t
i=.t+1, ... , p
j=.t+1, •.. , p
Let us now denote by ~, the permutational probability law generated by the
n
nr equally likely (conditional)
realizations of E over S(E).
""n
"'n
It is then
easily seen that
Wn ) =
(4.19)
E(T .. /
n,1.J
(4.20)
Cov(T . j' T ','
n,1
n,1 J
0
for all i= 1, •.. ,£
,lIP n ) = ...L
n- 1
for all i,i l =l, .•• ,£
and
and j=l+ 1, ... , p.
T .. T
n,11' n,JJ'
j,j'=£+l, ..• ,p.
. a compre h
'
test f or H(2)
(.1.e., wh en q= 2) ,
·
. t eres t e d 1n
S 1.nce
we are 1.n
enS1ve
0
following some simple steps we may consider the test statistic
(4.21)
v(J)
n(2)
= (n-1)
.e
L:
£
L:
P
L:
P
L:
i= 1 i 1=1 j=.B+1 j I=Z+l
•• 1
11
T ., T . . T
n, 1 J n, l ' J I n
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T~~~)iS
where
in
the (i,i')th term in
«T~:~~»)-l and T~~~)iS
the (j,j')th term
«T(2~~»)-1. However, we shall find it more convenient to work with the
n,lJ
test-statistic Sn(2) defined as
S(J)
n(2)
(4.22)
where 11·1: is the determinant of the matrix «.».
S
n
is analogous to the usual
likelihood ratio test (cf. Anderson (1957), pp. 233) where instead of T ., I s,
n,lJ
the sample product moment correlations are used.
relation between
S~~~)
and
We shall first consider the
V~~~).
THEOREM 4.1. If ~ has a nonsingular distribution in the sense that
by (3.4) and (3.5), is positive definite, then
1\
~ n~,
®
\I
...
provided EI T .. 1 (2+5)/2 <
n,ll
00
\I Tnll
®,
....
defined
converges in probability to
for some 5 > O.
The proof of this theorem follows precisely as that of Theorem 4.2 of Puri
and Sen (1966), and is therefore omitted.
THEOREM 4.2.
Under H(2) defined in (2.2),
o
IT n,lJ
··1
(4.23)
_1
= 0 (n 2) for all i=l, .. . ,t; j=£+l, ... ,p
p
The same result also holds under the permutational law ~ .
n
Proof.
First note that under H(2), E(T 'j) = 0, i=l, ... ,£; j=£+l, ... ,p.
n
0
n,l
Furthermore, since
1
n
2
*~ J~(i)(~/n)
is finite for all i=l, ... ,p; and since by
0;=1
(3.1),- ~ J (.)(~/n)..i71 for all i=l, ... ,p, we find after a few simple steps that
n ar=l n
(4.24)
1
VarH (T
o.
c
.. ) = -n-1 ' c
n,lJ
<
00
for all i=l, ... ,p; j=£+l, ... ,p.
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Hence using (4.19), (4.20) and (4.24) and Tchebyscheff's inequality we obtain
I
I
the desired result.
THEOREM 4.3.
If IT
n,
.. / = 0 (n-b) for all i=l, ... ,i; j=.e+l, ... ,p then
p
~J
IVn(J)
(2)
(4.25)
(J)
_
+ n log Sn(2)' - 0p(n
-1
0-
), provided ~ is positive
definite.
~.
By Laplace's expansion of the determinants (cf. Ferrar (1941»
(4.26)
E
1::;ql<q2<' . ·<qt~P
2
is the determinant consisting of the rows 1,2, ... ,£
where IIT(l
ql
-'
2
...
I,~ II
is the complimentary
deter-
q2 ... qg
minant.
By virtue of Theorem 4.2, we have the following results.
If only one of ql' q2' ... , qi is different from 1,2, ... ,1"
... lJ
2
(4.27)
liT (
ql
q2 . .. q,e
_1.
\I = o p (n 2).,
111"(
q2
ql
then
l~ I
2
_1.
= o P (n 2) ,
qz
and,
i f r of ql' q2' ... , ql, are different from 1,2, .. . ,1"
2
(4.28)
IIT(ql
-
for r > 1
... lJ
q2 • •. ql,
I\=op(n-
r
then
2
/2 );1\1'* (
ql
q2
~
_.
r
II = o (n- !2)
"
P
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Consequently, from (4.26), we obtain
£
+l:
(4.29)
£
l:
P
p
l:
T
l:
n, ij
i=l i t=l j=£+l jt=£+l
T
n, i'j'
IIT(l)
n, i i t
1IIIT(2)
n, jj'
II
where
\IT(l~.
1I
n,~~'
(4.30)
= cofactor of T
(cf. (4.18»
..
n,~~
and
o
cofactor of. T
n, JJ
~n
o'
T(2,2)
..
n,
~J
(cf. (4.18».
Now by definition
(4.31)
and
jj'
(2)
_
Tn (2) -
(2,2)
liTn, JJ
... II / liT.
n, ~J 1I
0
Hence, from (4.22), (4.29), (4.31) and Theorem 4.1 we obtain
(4.32)
-n log S(J)
n(2)
=n
£
l:
l
l:
p
P
l:
l:
i=l i'''l j"P,+l jt=l+l
ii'
Tn~J
.. Tn,ltl'
.
Tn (l)
0
'
which proves the required result.
THEOREM 4.4.
of
v~~~)(or
~
Under the assumptions of Theorem 4.4, the permutation distribution
-n log
s~~~»
converges asymptotically to a chisquare distribution
£(p-Z) degrees of freedom.
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Proof.
By virtue of (4.21) and Theorem 4.3 it suffices to show that any linear
combination
un =
(4.33 )
.1..
n2
£
p
~
~
i=l j=Z+l
d .. T ..
1J
n, 1 J
has asymptotically (under ~ ) a normal distribution in the limit with mean
n
zero and finite variance.
That the mean is zero and variance is finite follows
by using (4.19) and (4.20).
To prove the asymptotic normality of U, let us
n
denote
(4.34)
Z (R )
ex. "'n
=
~
i=l
i. d ..
j=£+l 1J
L
(i~
L(j)
n, iex. n, R.Jex. '
ar=l,.••. ,n
so that
U
n
(4.35)
Now for any E
"'n
E
=n
n
1
r.
-"5
ar=l
Z (R ) •
ex. ""n
values obtained by letting
ex. "n
Bn
€
Essentially then by the same method of
~n'
proof as in Wald-Wolfowitz-Noether theorem (cf. Fraser (1957), p. 237) it is
easily seen that
i f r=2k
(4.36)
0(1)
for k=0, 1,2, ...
i f r=2k+l
Hence the theorem.
Thus for large samples, the permutation test based on
v~~~)(or
reduces to the following rule:
(4.37)
If
v(J) or -n log S(J)
n(2)
n(2)
< X2 t
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I
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I
I
I
_I
S(F. ), defined in (4.ll),Z (R ) can assume nl equally likely
til n
_.
(p-t ),ex.'
accept H(l)
0
-n log
S~~~»
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where X2
r,~
is the
100(1-~)%
point of a chi-square distribution with r degrees
of freedom.
We will not discuss briefly the case of q(q > 2) subsets.
tion X as in (2.1),
w~
@
~
as in (4.3), and T accordingly.
Nn
Here we parti-
Thus
. .. ,
(4.38)
T
Nn
=
... ,
where
«T~~~~»)
is of the order i k x it; k,.t= 1, ... , q.
Then, by analogy with
parametric likelihood ratio test statistic, for the problem of testing
(4.39)
H(q):
o
n ®'nXJ, k =
®k
...,XJ =
III
,.,.0
kf.t=l, ... ,p.
We consider the test statistic
s(J)
(4.40)
n(q)
=
liTn, 1...111
Proceeding as in Theorem (4.3), we can show that -n log
s~t~)
asymptotically equivalent to a positive definite quadratic form in
is
p
~
i
O
•
k<t= 1 k 1..t
statistics Tn ke0where ~~ and X.t~ (k<t=l, ... ,p; O?l, ... ,n) belong to two dif-
,
ferent subsets of variates.
The permutation argument considered earlier readily
extends to the present case, where we shall have (nl)q-l equally likely realizations.
Further as a straight forward generalization of theorem 4.4, we shall
arive at the conclusion that under the. permutation model, -n log
s(J)
n(q)
s~t~)
where
is defined in (4.39) has asymptotically a chi-square distribution with
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L. i
Thus we have a test criterion very similar to
i.e degrees of freedom.
k
k4
(4.37) with the only difference that the degrees of freedom will be equal to
p
L.
i
.
k4=1 k ~.e'
As a special case, we consider q=p.
Here i = ... = i = 1, and the probp
l
lem reduces to that of testing the hypothesis of total independence of all the
p variates, (cf. (2.3».
v(J)
(4.41)
n(p)
Here we may work with the test criterion'
=
P
(n-l)
L.
i<j=l
i . ./T ..
n,~J
n,~~
T
..
n,JJ
(which is asymptotically equivalent to -log S(J) where S(J) is defined by
n(p)
n(p)
(4.40) by putting q=p).
In this case the permutation distribution of
v~~~)
coincides with the unconditional null distribution, and following the lines
of theorem
4.4~
it can be shown that under
H~P)
defined in (2.3),
v~~~)
has
asymptotically a chi-square distribution with p(p-l)/2 degrees of freedom.
5.
Asymptotic Normality of T
for arbitrary F(x).
,.,.n~N
In this section we shall
establish the joint asymptotic normality of the statistics {T
.. ; lSi<jSP}
n~ ~J
defined by (3.9) for arbitrary F(x).
IV
We shall make the following assumptions:
Assumption 5.1.
n~
In(i)(u) = J(i)(u) exists for 0 < u < 1 and is not con-
stant for i=l, ... , p.
Assumption 5.2.
J( .)[F[ .](x.)]} dF [ . . ](x.,x.)
J
J
J
n ~,J
~
J
where
=
_1
0
p
(n 2)
_.
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II
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(5. 1)
1
In(i)(l) = o(n4);
Assumption 5.3.
i=l, ... ,p.
Assumption 5.4.
IJ~i)(u)1 ~ K[u(l-u)]
-1
jJ(i)(u)/ ~ K[u(l-u)]-Z
where
Jl i )
and J(i) denote the first and second derivatives of J(i) respectively,
K is some constant, and i=l, ••• , p.
Finally, we shall denote
00
00
(5.Z)
(5.3 )
(5.4)
3
n cr {(.
n
.) (
~,J, r,s
)}
3
= t=lJ'=l
E
E
Cov(U
(a)
(i,j);t'
( )
)
(r,s);t'
U a
where
i f t=l
+00 +00
= J J [F 1X (xr )
ra
-00
-00
- F[r] (x r )] J(s) [F[s](xs )] J(r) [F[r](x r )]
d F[
r, s
] (x , x ), i f £=2
r
s
I
_.
-20-
00
00
if 1=3, r < s=l,. •• ,p
and, where
FIX
(5.6)
(u)
=
1 if X
sa,
sa,
< u and zero otherwise.
=
We shall now prove the following
THEOREM 5.1.
Under the assumptions 5.1 to 5.4, the random vector with elements
.», =
1
n"2[ (T (. ,) - Jl (,
1 < i < j < p] where Jl (. .) is given by (5.2) has a
n 1,J
n ~J
n 1,J
limiting normal distribution with mean vector zero, and covariance matrix ~ =
=
-
n«on{(i,j),(r,s)}»
given by (5.3) and (5.4) respectively.
Proof.
We write
(5. 7)
J n (.)[F
1
n [.](x.)]J
1
1
n (,)[F
J
n [,](x.)]
J
J
=
{Jn (.)[F
1
n [,](x.)]J
1
1
n (.)[F
J
n [,](x.)]
J
J
and, (by Taylor's theorem)
+ ~[Fn[i](xi) - F[i](x i )]
2
J(i)[SFn[i](x i ) + (l-S)F[i](Xi )] J(j)[SFn[j](x j )
+ (l-S)F[j](x j )]
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+ (l-8)F[
'J(X,)]~
,
J
J
J(' ,)[8F ['J(X.)
J
nJ
J
o< e <
1
and
(5.9)
= d[Fn [.1,J'J(x"x,)
1 J
dF [, 'J(X"x,)
n 1,J
1 J
- F[, 'J(X.,X.)J
1,J
1 J
+
dF[. 'J(x"x,),
1,J
1 J
Then, proceeding as in Bhuchongku1 (l96'4), we can express T ., (d. (3.9»
n,1J
T .,
n,1J
(5.10)
=
i 1Arn(i, j) +
r=
as
~
B(i, j)
1
r= rn
where
(5.11)
(5.12)
(5.13)
A~~,j)
00
=
~~,j) =
00
J J
-00
-00
00
00
_! _!
[Fn[iJ(x i ) - F[iJ(xi)JJ(j)[F[jJ{xj)JJ(i)[F[i](xi)J
dF[i~j](xlxJI
[Fn[j] (x j ) - F[j]{xj)JJ{i)[F[i]{Xi)]J(j)[F[i](xj)J dF[i,jJ(xrx)1
-.1.
and, the B terms are all 0 (n 2); (cf. Bhuchongku1 cited above).
p
1
3 (' ,)
The difference n 2 [ T ,. - r. A 1, J ] tends to zero in probabi l i ty as n
n,1J
r=1 rn
tends to infinity, and
l
80,
I
vectors n2 [T . , 1::; i
n,1j
by a well known theorem [Cramer {1954 p. 299)J, the
n
(i j)
< j ::; pJ and n'2[ r. A '
r=1 rn
,
1~ i
<j
~p] have the same
-22 -
limiting distributions, if they have one at all.
suffices to show that for any real Aij (1
I I
A, ,(A(i,j) + A(i,j) + A(i,j»
i=l j=l 1J 1N
2N
3N
~
i
<
Thus, to prove the theorem, it
j
~
p), not all zero,
has normal distribution in the limit.
Now
I:
I:
3 (' ,)
~ A 1,J can be expressed as
r=1 rn
3
~ A
(5.14)
C1, J,) = 1. "n [u(ex.,),
L...
n a;=
1
r=1 rn
(
1,
J) ;1
+
(ex.)
U(,1, J')'2
,
where the U(, ,)' s are given by (5.5).
1,
(5.15)
I'-1 '-1
I A,1J,( ~ 1Arn(i, j)} = 1.
J-
1-
r=
n
i<j
p
" ["
GXr' 1 i =1
L...
(ex.)
U(,1, J')'3
,
]
Hence
J
n
+
L...
I:
p
(a,) ,) 1
~ A, , (U(,
j= 1 1 J
1, J ;
i<j
The right hand side of (5.15) is the average of n independent and identically
p
distributed random variables, each having mean
third moments.
p
~
~ A, ,
i= 1 j= 1 1 J
i<j
Tha asymptotic normality follows.
n
, J
Furthermore using (5.14),
it is easy to check that the variance-covariance matrix
is given by (5.3) and (5.4).
(i ,) and finite
~
~
...
= n«cr {(' ,) (
)}»
n 1, J , r, s
The theorem follows.
The following theorem gives a simple sufficient condition under which the
assumptions 5.1, 5.2 and 5.3 hold.
THEOREM 5.2.
If In(i)(ex./n) (i=l, •.. ,p) is the expected value of the ex.th order
statistic of a sample of size from a population whose cumulative distribution
is the inverse function of J(i)' i=l, .•. ,p and if the assumption 5.4 of theorem
5.1 is satisfied, then the assumptions 5.1, 5.2 and 5.3 are also satisfied.
I:
el'
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The proof of this theorem is analogous to that of theorem 2 of Bhuchongkul
(1964) and is therefore omitted.
With the use of this theorem it is easy to verify that if In(i)(a!n) is the
expected value of the ath order statistic of a sample of size n from (i) the
standard normal distribution, (it) the logistic distribution, (iii) the double
exponential distributions, (iv) the exponential distribution, and (v) the uniform
1
distribution, then the vector
n"2[ (T .,
n, ~J
-}l
(.
n~,
"»,
J
1 < i < j < p] has a limi t-
-
ing normal distribution.
We shall use the results of theorem 5.1 in deriving large sample power
properties of the statistics associated with the (i) tests of independence of
q
~
2 sets of variates, and (ii) tests for pair-wise independence.
For the
simplicity of presentation, we shall first consider the problem of testing independence of two sets of variates.
6.
Testing Independence of Two Sets of Variates.
Let the p component vector
X be partitioned into two subvectors X(l) and X(2) as defined in (4.6).
"'a
""0.
-0.
Let the population dispersion matrix
@
defined in (3.4) and (3.5) be
'"
partitioned as we did the sample dispersion matrix T . defined in (3.10). Then
-n
we have
(6.1)
Note that
®11 is of order £
x £,
@12
is of order £ x m.
lem of testing H(2)
o
(6.2)
H(2):
o
® 12 = ®21 = ,..0
I
"OJ
.-.J
Then, for the prob-
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s~t~)
we propose to consider the statistic
in rejecting
H~2)
if
H~2)
below that if
s~t~)
defined by (4.22).
exceeds some predetermined number hE
is true, the statistic -n log
distribution with tm degrees of freedom as
n~.
t h 1S S(J)
n(2) test W1t h a 1 arge samp 1e approx1mat1on
0
The test consists
0
0
0
s~t~)
We shall prove
has a limiting chi-square
This provides the user of
0
f th e vaue
]
0
f hE f or any
O<E<l.
We shall now study the asymptotic distribution of
s~t~)
assuming a sequence
of alternative hypotheses (H(2), n=1,2, ••. } which specifies that
n
i=l, •.. ,t, j=t+l, ••• ,p
(6.3)
where 0ij is defined by (3.13), wij is some function of (F[i],F[j])' and wij
for all (i,j); i=l, ..• ,t; j=t+l, •.. ,p.
of the alternative hypotheses {H
(2)
n
Cl:tl
(6.4)
~.
63 12
=
«f3 ij »
0
It may be pointed out that the sequence
, n=1,2, ... } defined in (6.3) implies that
~2 = ~ 1 = n
where
f
_1. tu
2
\D 12
is an t x m matrix where elements are
(6.5)
i=l, •.• ,t; j=Z+l, ... ,p.
The following lemma is an immediate consequence of the fact that var{T oo}
n,1J
tends to zero as
Lemma 6.1.
(a)
T
"'n
~
n~
for all F
€
":f p •
Under the assumptions of theorem 5.1,
®
'"
in probability as n~ for all F E ~.
p
_.
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n~
in probability as
for all F
E ~
P
i,it=l, ... ,t
(c)
jjt
jjt
Tn (2) ~ 8(2) in probability as n~ for all F
E
~p
j,. j t=.t+1, •.• , p.
where
iit
jjt
8(1) and 8(2) are the (i,it}th and (j,jt}th elements
(6.6)
of~~ and~~
respectively.
If, for each i < j = l,. •.. ,p
Lemma 6.2.
(0
(ii)
the conditions of theorem 5.1 are satisfied
the hypothesis (H(2)} defined by (6.3) is valid then, the matrix with elen
1
ments n"2[(T
,,-}l
n,. 1J
(,
,»,
n 1, J
i=1, ... ,t; j=J+l, ... ,p] has a limiting normal dis-
tribution with means zero and variance-covariance matrix E* = «cr~('
t
1,
,) (
J ,
where
(6.7)
if
(6.8)
i=l, .. . ,t; j=.e+l,.
i, pl, 000'£; j, s=Z+1, • 00' p.
(note that 8"11 = 1).
00' p
r, s
)}»
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The proof of this lemma is an immediate consequence of theorem 5.1, (6.3) and
(3.1), and is therefore omitted.
THEOREM 6.1.
Under the assumptions of lemma 6.2, the limiting distribution of
(J)
-n log Sn(2) is non central chi-square with t m degrees of freedom and noncentrality
parameter
(6.10)
Proof.
p,
1,
p
P
ii t
jjt
= I:
I:
I:
I:
13 iJ· 13 i,J·, 8(1) 8(2)
n(2)
i=l il=l j=Z+l jt=l+l
.6S (J)
1
Under the assumed conditions n2 T .. is bounded in probability for all
n, ~J
i=l, ••. ,1,; j=t+l, •.. ,p:
Hence applying Laplace expansion for the determinant
liT
\I and proceeding exactly as in theorem 4.3, we obtain
...n
1,
(J)
(6. 11)
p
1,
p
I:
I:
I:T .. T . .
-n log Sn(2) =nI:
n ~J n, ~ I J I
i=l il=l j=£+l j'=1,+l '
Using lemma 6.1, we notice that -n log S~~~) is asymptotically equivalent to
V'k
n
where
I-
(6. 12)
P,
p
p
V*=nI:
I:
I:
I: T .. T . .
n
n 'J n"IJ'
i=l il=l j=.e+l j'=1,+l'.L
.L
iii
jjl
where 8(1) and 8(2) are defined in (6.6).
The result now follows by an application
of lenuna 6.2.
We shall consider the special forms of the function .6 (J) for suitable
S
n(2)
choices of the function J in section 8 where we consider the asymptotic properties
of the
S~~~)
tests in relation to their parametric competitors based on the
likelihood ratio test [cf. Anderson (1947), p. 233J.
In the passing we may remark that forthe problem of testing independence
of q sets of variates against the sequence of alternatives {H(q), n=1,2, ... }
n
which specify that
n~~)
~J
is of the form (6.3) for all i,j belonging to different
sets, it can be shown by the use of theorem 5.1 that -n log S(J) (cf. (4.40»
n(q)
_.
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p
has asymptotically a non-central chi-square distribution with
~
i it degrees
k
k<C=l
The details are omitted because of the intended brevity of the
of freedom.
presentation.
In a relatively simple case, when q=p, the statistic (4.41) has under the
sequence of alternatives (6.3), asymptotically, a non-central chi-square distribution with p(p-l)!2 degrees of freedom and non-centrality parameter 6 (J)
S
n(p)
given by
~(J)
(6.13)
n(p)
where
7.
~
...
P
2
~
~ij
i<j=l
.. is defined in (6.5).
1.J
Pair wise Independence.
In view of the fact that the pair wise independence
does not in general imply total independence, we shall consider this problem
seperately in this section.
(7.1)
H*:
a
To be precise we consider the problem of testing
F[. '](X"X')
1., J
1.
J
= F[ 1..](x.)
1.
. F[ '](x.) for all pairs (i,j)
J
J
against the sequence of alternatives {H*, n=1,2 ... ) which specifies that
n
_1
F.j(X.,x.) = F.(x.) •
(7.2)
1.
1.
J
1.
1.
FJ.(x ,) {l + n "2 w.. (F.,F,)}
1.J
1.
J
J
where w.. (F.,F.) is not identically equal to zero (a. e.), for at least one pair
1.J
1.
J
(i,j); 1 < i < j
~
p.
The test-statistic proposed for this problem is based on
the statistic
(7.3)
'"
in(J) = n
.,.1
~ r
,
T*
Nn
where
(7.4)
T* = (Tn, 12' Tn, l3,···,Tn, (-1
""n
p , p»
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-28-
'"
r -1
and
is a consistent estimator of
1
variance matrix of n2
THEOREM 7,1,
r -1 which is the inverse of the co-
~~,
n
If, for each i < j = 1,.,.,p
(i)
the conditions of theorem 5.1 are satisfied
(ii)
the hypothesis H* defined by (7.2) is valid
n
then, the statistic ~n(j) defined by (7.3) has asymptotically a non~central chi
square distribution with p(p-l)/2 degrees of freedom. and non centrality parameter
defined as
6
~n(J)
A
L:>
(7.5)
~n(J)
=
A
L:>
r- l
A
L:>
'V
where
(7.6)
(The
~,
Proof. The proof of this theorem is an immediate consequence of the fact that
under the given assumptions, the random vector ~n Nn
T* defined by (7.4) has a
limiting normal distribution with mean vector 61\1 given by (7.6) and the co-
r_ = «'t"{(,1,J,) , ( r,s )}»
1,
where
if i=r, j=s
Eo [J(,)(F,(x,»J(,)(F,(x,»J(
1
1
1
J
J J
r )(F( r )(x r »J( s )(F s (x s )]
i f ifr, jfs
(7. 7)
't"{(i,j),(r,s)}
=
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,'s are defined in (6.5».
1J
variance matrix
_.
2
E [J(,)(F.(x,»J(,)(F.(x,»J( )(F (x »] i f i=r, jfs
o
1
1
1
J
J J
s
s s
2
i f ifr, j=s
Eo [J(.)(F,(x.»J(
111
J,)(F.(x.»J(
J J
r )(F r (x r »]
2
i f i=s, jfr.
Eo [J(,)(F,(x,»J(
1
1
1
J,)(F.(x.»J(
J J
r )(F r (x r »J
A_1
1
and, the fact that r
is a consistent estimator of r- .
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From theorem 7.1, it is clear that any consistent estimator of r
preserve the asymptotic distribution of the
statistic~n(J)'
-1
will
A natural con-
sistent estimator of (7.7) may be obtained readily by replacing J(i)(Fi(x
i
»
by J n (.)(F
(')(x.l.CL » for all i, and the expectation by sample averages.
1.
n 1.
8.
Asymptotic Relative Efficiency.
In this section we shall make large sample
(J)
tests, and their parametric
power comparisons between the Sn(2) tests , ae(J)
n
competitors based on the likelihood ratio test.
particular cases of the
(a)
S~~~)
Special cases of the
~
Let J(i)(u) =
The -n log
S~~~)
~
-1
and
S~~~)
~~J)
First we consider the interesting
tests.
tests.
(u) where ~ is the standard normal distribution function.
test then reduces to the normal scores -n log
non-centrality parameter in this case is
~S(~)
S~~~)
test.
where
n(2)
£
I,
p
P
i i i jjl
= ~
~
~
~
~iJ' ~ilJ't 6(1) 6(2)
i=l i 1=1 j=l,+l j 1=1,+1
(8.1)
where
1
(8.2)
(c,f. (6.5»
and
,
(8.3)
•• t
61.1.
,
6
jjt
are the (i,il)th and (j,jl)th terms of
respectively, and where
(8.4)
In an important case, when F(x)
is
ow
where
N(~,~)
"'..,
The
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~
(8.5)
=
« P(n») pxp'•
and P
ij
(n)
=
1.J
p ..
1
. r=
'" n
7\.., i = 1, ... , t; j=.e + 1, ... , p
1.J
or i=t+l) •.. , p; j=l, .. . ,f,
(n)
= p .. otherwise.
ij
1.J
We have
[
(8.6)
(n)]
<P i / x, y, Pi j ) _ 1
<P. (x) <P .(y)
1.
J
where <P •• (x,y,p~~» is the normal cdf of (x.,x.) where X. is N(O,l), X. is N(O,l)
1.J
1J
1 J
1
J
'
d '1ng '"~ .. ( x,y,p (n»
an d p (n).
.. 1S t h e corre l
at1.on
coe ff"1.C1.ent b etween X . an d X
.' E
xpan
..
1J
1.
J
1.J
1.J
~)
about p..
1.J
~
= 0, and neglecting the terms of order o(n 2), we obtain (also see
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el
(8.7)
where
~i
is the density of <Pi; i=l, •.. ,p.
Hence, when F(x) is normal N(~,~)
"'" '"
(8.8)
13 •• = 7\ .. ;
1J
1.J
i=l, ••• ,.e; j=.e+l, ... ,p.
(8.9)
8
i=l, •.. ,t; j=l, ... ,.e; or i=t+1, .•. ,p; j="+l, ... ,p.
ij
= Pij;
...
Hence from (8.1), in case F(x) is normal
(8.10)
I,
p
t
P
iii
6 (<P) = L:
~
L:
7\
..
7\'1'1
L:
PO)
S
n(2) i=l i 1=1 j="+l j 1=1,+1 1J 1 J
jjl
P(2)
where
•• 1
(8.11)
•• 1
p(~) and p(~) are the (i,il)th and (j,jl)th terms of
-1
f 11
-1
D-1
-1
= «P ij ) ~ . ; J 22 = «P ij ) ~ .
1, J=l, .• • ,t
1, J=£+l, •.. , p
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a(iQ.
Let J (i) (u) = .[12 (u-!), then the -n log
S~~~)
test reduces to the rank
(R)
sum statistic -n log Sn(2) which is a multivariate analogue of the Spearman's
rank correlation statistic [cf. Kendall (1955)].
The non-centrality parameter
in this case reduces to 6S(R) , where
n(2)
£
£
P
P
ii' jj'
6 (R) = r.
r.
r.
E
13 .. 13.,., 8(1) 8(2)
S
i=l i '=1 j=£+l j '=£+1 ~J ~ J
n(2)
(8.12)
where
00
00
and
ii'
j j, .
.
()
tal-I
))
6(1),6(2) are the (i,i')th and j,j' th terms of~l· «8 ij i,jIll1, ... ,t and
·»· J=,e,
. D+1, ••• , p'
H 2 =
~
l
«8 i J
.
respect~vely and, where
~,
00
00
f f
(8.14)
(Fi(x) - !)(Fj(y) -!) dF[i,j](x,y)
-00-00
In case F(x) is
-
and
P
(n)
ij
1
N(ll,r.)
"".....
= .[n Aij
1o1here
r. =
.".
«p ~~»), p ~?)
1.J
1.J
for i=t, .• • ,1,.; j=£+l, ..• , p
=
p ..
1.J
or
for i,j=l, ... ,£ or i,j=t+l, •.. ,p
i=£+l, •.• ,p; j=l, .•. ,!'
the non-centrality parameter 6S(R) reduces to
n(2)
(8.15)
£
£
p
P
r.
E
r.
r.
i=l i '=1 j=l,+l j '='+1
ii'
jj'
A.. A.,., P*(l) p'ij 2 )
1.J
1.
J
"
where
(8.16)
ii'
P~l)
jj'
and P~2) are the (i,i')th and (j,j')th terms of
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-32 -
-1*
«(§. Sin-l Pij»-l
P22 = '7r
2
i, j=£+l, ... , p.
respectively.
Special Cases ouf,(J) tests.
(b)
-
n
--
Then the (f (J) reduces to the normal' scores
8(b.l)
ce n (rD)
n
test.
The non-centrality parameter is then (cf. (7.5»
~ (rD)
n
(8.17)
where
£-1 =
«T(i,j),(r,s)}»
In a special case when F(x) is
N(~,E)
tv
'"
is given by (7.7).
where E =
N
(n)
«P ij
»,
pairwise indepen-
dence is equivalent to total independence and, hence from (8. 7),
~ (rD)
(8.18)
(8b.2)
n
=~
~'
Let J(i)(u) =.fi2(u-t), then the£'n(J) test becomes the rank sum~(R)
The non-centrality parameter, for the case when F(x) is
test.
'"
~
(8. 19)
(R)
n
(c)
Parametric Theory.
N(~,E),
I'U
is
tV
= ;'2 "t ~I
In the parametric theory a commonly used test for the
hypothesis H(2) is based on the statistic [see Anderson (1957), p. 242»)
o
(8.20)
where
_.
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(8.21)
A
(8.22)
A. ,
=
n
E (x(J,
0;=1
~
-x), (x,.,a,-x)
'""'"
i=1,2
1J
(8.23)
It is easy to see, by an application of a theorem due to Wald (1943) that
under the sequence of alternatives of the type H(2), -n log V* has asymptotically
n
a non-central chi-square distribution with tm degrees of freedom and non-centrality
parameter
~
given by
n
(8.24)
t
t
P
P
1'1"
",
JJ
E
E
E A., "i'j' PO) P(2)
i=l i '=1 j=.t+l j '=£+1 1J
~ = E
n
ii'
."
where P(l) and p~i) are defined in (8.11).
Furthermore, as mentioned earlier, since under the normal theory model,
pair wise independence is equivalent to total independence, therefore the tests
for H(P) and H* are based on the same statistic, viz.
o
(8.25)
0
u* = .J!L.
fr a.,
i=l
where A
= «a1J
.. ». '-I
1, J- , ..• , p
11
is defined in (8.21), and
n
n
(8.26)
a, ,= E (x. -x.)(x, -X );
1J
0;=1 10. 1
Jo. j
x, =
1
E
0;=1
x1.Jn.
It turns out, (again by an application of Wald's theorem (1943»
that -n log U*
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-34-
has under the sequence of alternatives (7.2) (which in the case of normal theory
.
(u)
1
model 1.mply p .. = -"1\ .. for all i,j=l, •.. ,p) asymptotically a non-central chi1.J
1.J
s/l'lare distribution with p(p-l)/2 degrees of freedom and non-centrality parameter
!'i*
"1\'
-U = "1\.......
(8.27)
where
~
..r;;
is defined in (8.17).
Now it is well known [cf. Puri and Sen (1966)] that if the two statistics
have, under the alternative hypothesis, non-central chi-square distributions with
the same number of degrees of freedom, then the asymptotic relative efficiency
of one test with respect to the other test is equal to the ratio of their noncentrality parameters after the alternatives have been set equal.
Hence, denoting e[T,T*]' as the asymptotic efficiency of a test T relative
to '1'*, we have
~
are defined in (8.1) and (8.29), respectively.
(8.29)
where 6 (R) , and ~ are defined in (8.12) and (8.29), respectively.
S
n(2)
In particular, where F(x) is N(~, «p~~»», it is easy to notice from (8.10),
N
(8.15) and (8.29) that
(8.30)
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(8.28)
where 6 (¢) and
S
n(2)
--I
1.J
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(8.31)
e[ S(R) , V*] =
n(2)
;2
.£
t
p
E
E
E
P
E
i=l i '=1 j=.£+l j ·=t+l
p
t
t
P
E
E
E
E
i=l i'=l j=£+l j ·=t+l
"..
~J
i.i'
"i'j'
p~'
"ij "i'j' P
ii'
jj'
p')'(
p
jj'
ii'
Jj'
ii'
jj'
where P(l)' P(2)' Pel) and P(2) are defined in (8.11) and (8.16) respectively.
From (8.30) and (8.31) we note that whereas in the case of normal distributions, the property of the bivariate normal scores test of independence (cf.
Bhuchongkul (1964»
relative to the likelihood ratio test is preserved in the
multivariate case the same is not true in the case of multivariate rank sum
S(R) test.
n(2)
However, for special cases of the type dealt in Sen and Puri (1966)
one may consider finding the bounds of (8.31).
We do not pursue this problem in
this paper.
Finally, it is easy to notice that under the normal theory model, in the
case of pair wise independence (or total independence),
=
(8.32)
e (\1J)
(8.33)
9
~(R), U* = 71"2
n
£n '
U*
1
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_.
REFERENCES
1. . Anderson, T. W. (1957).
and Sons" Inc.
Introduction to Mul tivariate Analysis, ·John Wiley
2~
Bhuchongkul (1964), A class of nonparametric tests for independence in
bivariate populations, Ann. Math. Statist., 35, 138-149.
3.
Blomqvist, N. (1950), On a measure of dependence between two random variables.
Ann. Math. Statist. -..I
21, 593-600.
4.
Chernoff, Herman ~nd Savage, I. R. (1958)" Asymptotic normality and efficiency
of certain nonparametric test statistics, Ann. Math. Statist. 29,972-994.
5.
Cramer" Harold (1954)"
Press.
6.
Daly" J. F. (1940), On the unbiased character of likelihood-ratio tests
for independence in normal systems, Ann. Math. Statist., !J, 1-32.
7.
Ferrar (1941),
8.
Fraser, D. A. S. (1957), Nonparametric Methods in Statistics,
and Sons, Inc.
9.
Hoeffding, Wassily (1948), A class of statistics with asymptotically
distributions" Ann. Math. Statist., --.i
19, 293-325.
-
Mathematical Methods of Statistics, Princeton Univ.
Algebra, Oxford University Press.
John Wiley
norm~l
10.
Hoeffding, Wassily (1948), A nonparametric test of independence, Ann. Math
Statist., -.J
19, 546-557.
11.
Kendall, M. G. (1955),
12.
Konijn, H. S. (1956), On the power of certain tests of independence in
bivariate populations, Ann. Math. Statist., IW
27, 300-323.
13.
Puri, Madan L. and Sen, Pranab K. (1966), On a Class of Multivariate Multi
Sample Rank Order Tests. Institute of Statistics Mimeo Series, No. 474~
University of North Carolina, Chapel Hill.
14.
Sen, Pranab K. and Puri, Madan Lal (1966), On The Theory of Rank-Order Tests
and Estimates for Locatio~ in the Multivariate One Sample Problem, Ann.
Math. Statist., (Submitted).
15.
Sen, Pranab K. (1966), A Class of Permutation Tests for Stochastic Independence, Institute of Statistics Mimeo Series, No. 458, University of North
Carolina& CRapel Hill.
16.
Sibuya, M. (1959), Bivariate Extreme Statistics, Ann. Inst. Statist. Math." 11,
'"
195 -210.
Rank Correlation Methods, Griffin, London.
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17.
Wa1d, A. (1943), Tests of Statistical Hypotheses Concerning Several Parameters When the Number of Observations is Large, Trans. Amer. Math. Soc.
54, 426 -482.
~
18.
Wa1d, A. and Brookner, R. J. (1941), On the distribution of Wilk's statistic
for testing the independence of several groups of variates, Ann. Math.
Statist., g, 137-1,:>2.
19.
Wilks, S. S. (1935), On the independence of k sets of normally distributed
statistical variables, Econometrika, 2, 309-326.