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ON A ROBUSTNESS PROPERTY OF A CLASS OF
NONPARAMETRIC TESTS BASED ON U-STATISTICS
by
Pranab Kumar Sen
University of North Carolina Chapel Hill,
and University of Calcutta
Institute of Statistics Mimeo Series No. 513
February 1967
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Work supported by the National Institute of Health,
Public Health Service, Grant GM-12868.
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DEPARTMENT OF BIOSTATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
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ON A ROBUSTNESS PROPERTY OF A CLASS OF
NONPARAMETRIC TESTS BASED ON U-STATISTICS*
By
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Pranab Kumar Sen
University of North Carolina Chapel Hill,
and University of Calcutta.
1.
Summar
and introduction.
A class of nonparametric tests based on Hoeffding's
(1948) U-statistics is shown to be robust for heterogeneity of the distributions
of the sample observations; the theory is illustrated by means of some known
tests.
Let Xl, ... ,X be independent (real or vector valued) random variables
n
having cumulative distribution functions (cdf's) Fl(x), ... ,Fn(x), respectively.
Let n be the set of all cdf's closed under the following (countable) set operation:
if Fi€n for i=1,2, ... , then (l/n)~i~lFiEn for all n > 1.
It is assumed that
F = (F1, ... ,F ) is an element of the product set nn, and the null hypothesis
-n
n
states that
(1.1)
H :
o
F
~n
n
€wcn.
n
Let now
(1.2)
~ = {F:
un
Nn
Fl= ... :FnEnJ, w*n = wn n
&n and
w**
n = wn -w*.
n
When w* is some hypothesis of invariance of F (under certain finite group of
n
-n
transformations that maps the sample space onto itself), then granted certain
conditions, tests based on Hoeffding's (1948) U-statistics are essentially distribution-free [e.g.,. the signed-rank test by Tukey (1949)(for the hypothesis
* Supported by the National Institute of Health, Public Health Service, Grant
GM-12868.
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of sign-invariance (i.e.,
symmetry»~
the tests by Kendall (1938), Hote11ing
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and Pabst (1936) and others [cf. Hoeffding (1948~ pp. 316-321)J (for the
hypothesis of matching-invariance (i.e.? bivariate independence»J.
However,
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if F EW** (even when w is some hypothesis of invariance), tests based on U"'n n
n
statistics are, in general, not distribution-free.
E =(X ' ... ,X ) and 0 ~
"'n
n
l
* ~ 1)
A test *(E
) (where
-n
is termed robust for FEW if
Nn n
(1. 3)
Q(O
< Q < 1) being the preassigned level of significance of the test.
If there
is no confusion, *(E
.... n ) satisfying (1.3) only for large n will also be termed
robust.
The object of the present investigation is to show that under certain
conditions on w
(to be stated in section 2), the usual nonparametric tests for
n
F EW* based on U-statistics are robust for F EW.
"'n n
"'n n
Some illustrative examples
are also considered.
2.
Statement of the main theorem.
Corresponding to a symmetric kernel
of degree
m(~
(2.1)
U = (n) -~S~(X. , ... ,X. );
n
m
~l
~m
~(Xl'··
1), Hoeffding's (1948) U-statistic is defined as
S
=
(l
<
i
1
< ... < i m< n}.
Following Hoeffding, we define
m
(2.2)
8. , ... ,.
~l
~m
= J ... J~(x1, ... ~xm) Tr dF (x.),
i
i
j
J
1 ~ 1
Then, U is an unbiased estimator of 8(F ), defined by
n
Nn
1
< ... < i m ::; n.
-.
.,X )
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8(F )
(2.3)
"'n
n -1
= ()
m
L. S8. ~ ... ~ . .
~l
~m
U is said to be strictly distribution-free for F €w*~ if its distribution is
n
"'n n
independent of F (EW*); this implies that
""n
n
8(F )
(2.4)
",n
= 80
(known) for all F EW*.
""n
n
Since for F EW*, Fl; ...SF sF, 8(F ) reduces to
""n
n
n
"'n
m
8(F)
(2.5)
=
J ... J¢(xl, ... ,x ) rr dF(x.), for all F EW*.
m
1
J
"'n
n
There are certain situations where U may not be properly distribution-free
n
(for F EW*), but (Z.4) holds and further it is possible to find an estimator V
"'n
n
n
_1
]
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of the variance of U such that V 2[U -8 converges in law to a standardized
n
n
n 0
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assumption F E ~ (implicit in w*) is redundant if n is not small and W satisfies
Nn
n
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normal distribution for all Nn
F EW*;
Un is then said to be asymptotically distrin
bution-free for F EW*.
....n
some conditions.
(Z.6)
¢l(x;FZ)
n
In either case, our contention is to show that the
To formulate the same, we define for any (Fl~FZ)En2
= J... J¢(x
m
~x2,.··,xm) ~ dFZ(X
Y
8(F l ;FZ)
=
J¢l(x;FZ)dFl(x);
(Z.7)
Zm-l
(2.8)
rr
1
dF(x.) - 8 2 (F)
J
=
-4-
where
~l (F)
is defined by Hoeffding (1948, p. 298).
Also, let
(2.9)
Then, we have
n
U
~
~
w* -w(i)C
w(i)C w .
n
. n
. n
n
(2. 10)
Now, concerning w ' we assume that (i) for some 5 > 0
n
)1
m
J ... JIHxl, ... ,x
(2.11)
uniformly in 1 <_ i
(2.12)
2+5 m
rr dF.
~j
1
(x.) < co,
J
< ... < i m -< nand .....Fn €w n ; and (ii)
l
~l(F;;F) >
0 uniformly in F Ew(i), i=l, ... ,n.
"'n
.L
n
In conjunction with (2.8), (2.12) implies that
~l(F)
(2.13)
> 0 uniformly in /Un
F EW*.
n
Now, according to Hoeffding (1948, p. 299),. (2.13) implies that e(F) in (2.5)
is stationary of order zero for all F EW*.
/Un
n
By analogy with this, we say that
(2.12) means that e(F ), defined by (2.4), is stationary or order zero for all
"'n
F E Uw (i).
"'n . n
~
Further, i f U is strictly distribution-free for F €w*, we have from
n
"'n n
(2.13) that
(2.14)
Let us now define
~10
(known) > 0, for all F €w*.
"'n n
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(2.15)
We say that w is a resolving subset of On if
n
(2. 16)
F E:W n
""n
+ F( n ) (1, ... , l}Ew*.n
Further, if in addition to (2.16), we have on defining 8(F ;F ) as in (2.6),
l 2
(2.17)
8
o
for all F E:w ,
""n
n
n
we say that wn is an affine resolving subset of 0, the affinity being with
respect to 8.
(2. 18)
Finally" we shall consider the following types of tests:
,r'(E
'!' Nn ) -
,r'l(~n)
'!'
._
1,
U > c ,
n n
0,
U
=
1,
(2.19)
n
< c n'.
Iun -80 I >-
c ,
n
HE ) = 1V (E ) =
2 ""n
"'n
where
(2.20)
E{\jr(E",n )jF
... n E:w*}
n
<~.
_
(1)
If we have a vector of U-statistics i.e., ~n
U - (Un
we define the corresponding vector of 8 's by 8
o
"'0
_
- (8
(p)
, ... , Un
), p >
- 1.,
(1)
(p)
, ... ,8
),. the true
o
0
covariance matrix of U by E (F ) and the estimated covariance matrix by V .
run
... n '\In
""n
If U is strictly distribution-free for F E:w*,. we have,. E (F )
""n
"'n n
",n Nn
= ""n
EO
(independent
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of F EW*), and by an assumption similar to (2.12), we have
.... n
n
nL° is positive definite; nV is positive definite, in probability.
"'n
un
(2.21)
We consider a test-statistic
s =
(2.22 )
n
where ",n
A
.
L: 0 or
~s
"'n
free for F EW*.
"'n
n
[U -8 ] A -1 [U -8
n
o"'n
""n
0
J'
V according as U is strictly or asymptotically distributionn
Nn
Then, the third type of test is
1,
sn >
c
n
(2.23)
where c
n
satisfies (2.20).
THEOREM 2.1.
F E.W
U (i) ,
"'n
~
n
The main theorem of the paper is the following.
If (i) (2.11) hOlds, (ii) 8(!n) is stationary of order zero for all
( ~~~
... )
W.~s a reso 1·
v~ng su b set
n
0
f nn
d (. )
an~v
U ·~s
n
.
1
str~ct
.
y d·~str~-
bution-free for !nEW~, then (a) for all 0 < a < If ~2(!n) and ~3(§n) are robust
for F Ew, and (b) for 0 < a < -k..
--",n n
- c
~l(E
"'n
) is robust for F Ew.
"'n n
I f (i), (ii) and
(iv) remain same, while (iii)' W i.s an affine resolving subset of nn then for
n
all 0 < a < 1, ~l(E ) is robust for F EW.
"'n
~n
n
bution-free for
5
>
!nEW~,
we need
to
If Un is only asymptotically distri-
change (i) by (i)' that (2.11) holds for some
2.
The proof of this theorem is postponed to section 4.
3.
~
Let us define 8(F) and
respectively, F(n) as in (2.15), and let
~l(F)
as in (2.5) and (2.8),
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(3.1)
Also, we define 8(F ;F ) as in (2.6) and let
1 2
(3.2)
O(n
LEMMA 3.1.
PROOF.
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From (2.1) and (2.3) we have after some simplifications
8 (F )
(3.3)
Nn
n
1
= ---:---:'7"
n ... (n-m+l)
m
E
..1.
1
T···f 1 m
.::J:.
l
=
f . .. fHx l ,
... , x ) 7T" dF. (x.).
m 1
1 .
J
J
Also from (2.5) and (2.15), we have
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), where 8(E ) is defined by
(2.3).
1
I
-1
(3.4)
=
8(F(n»
n
n
n -m
L.
i =1
1
m
L. f ... fHx , ... , x ) 7T" dF. (x.).
l
i =1
m 1
1.
J
m
J
Comparing (3.3) and (3.4), it is easily seen that under (2.11), 18<'~'n)-8(F(n»)1
=
O(n
-1
).
LEMMA 3.2.
PROOF.
Hence, the lemma.
Under (2.11),. /nV{Un }-
We define 8.
as in (2.2) and let
.
11' ••• , 1
[m2(~1(F(n»-6~)]1 = 0(n- 1 ).
m
(3.5)
=
E{Hx. , ... ,x. ,X. , ... ,X.
)} - 8.
..
.;
1
1
J1
J m- c
1 , ... , 1 ' J , ... , J C
C
1
1
1
m c
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=
S(.
k
k
c ~ l' ... , ~'c) 'J l' ... , J'
m-c ; 1"'" m-c
(3.6)
E{'IT (.
c
~ l'
. ).
.
... , ~ c J l' ... , J m-c
(X. , •.. ,X. ) 'IT (.
~1
~
c
c
~ l'
. )k
•.. , ~ c
for c=l, ... ,m and all possible distinct its, jts and kts.
,.
{( n )(2m-c)(2m-2c)}-1
~c , n =
2m-c
c
m-c
(3. 7)
where the summation
kts.
r.*c
k
l' ... , m-c
(X. , ... ,X.
~1
Also, let
"'* ,.
W
c ~c(il, .. ·,i c )jl, .. ·,j m-c ;kl, .. ·,km-c '
extends over all possible (distinct) sets of its, jts and
Then the variance of U
is given by [cf. Hoeffding (1948)]
n
(3.8)
(3.9)
n
~l
,n
may also be written as
m-l
-[2m-l]
'L.**l {f ... !¢(x 1, .. ·,x )¢(x , ... ,.x2 l)dF.(x) 7f
m
m
m~l m
1
(3. 10)
- e..
~ l'
where p
1 ::; i l
[ q]
f
= p
jl
f
.
J l' ... ,. J m - l
(p-q+l) and the summation
f
jm-l
f
kl
f ... f
e.
k
k}'
~l' 1"'" m-l
r.!*
km_l ::; n.
lemma 3.1, it can be shown that under (2.11)
c
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Thus, it suffices to prove that under (2.11)
Now, using (3.6) and (3. 7),
~
extends over all possible
Now, by the same method as in
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where 8(F ;F(n»
i
is defined by (2.6) and (2.15).
n
'£ 8 2 (F. ;F( »
n i=l
1
n
(3.11) !n-[2m-1] '£**1 8 . .
.
8. k
k
1 1, J 1,"" J m- 1 1 1, 1"'"
m-1
I = O(n
-1
),
Also, it is seen from (2.5),
(2.6) and (2.15) that
(3. 12)
Finally, using (2.8) and (2.15), we obtain that
~l(F(n»
2
+ 8 (F(n»
=
(3. 13)
n
-(2m-I)
L:
m-1
0
! ... !¢(x 1, ... ,xm)<l>(xm, ... ,x2m- l)dF.1 (x)
m
1
1
where the sunnnation '£ ~ extends over all possible 1 ::; i
for £=1, ... ,m-I.
(3. 13).
1
7r
1
::; n, 1::; j£ ::; n, 1 ::; k£ < n
(3.9) readily follows from (2.11), (3.10, (3.11), (3.12) and
Hence, the 1ennna.
1
LEMMA 3.3.
_
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1
Under (2.11) and (2.12), n2[Un-8(F(n»]/m{~1(F(n» - 6~}2 converges
in law to a standardized normal distribution for all F €W .
"'n
PROOF.
n
The proof of this lemma will follow readily from theorems 8.1 and 8.2
of Hoeffding (1948), and our 1ennnas 3.1 and 3.2, provided we can show that (in
addition to (3.9», ~l(F(n»
- 6~ > 0, uniformly in En€W n .
Now, by virtue of
(2.7) and (2.12), we have
(3.14)
for all i=l, ... ,n and uniformly in ",n
F €W.
n
(3.14), we obtain
Hence, from (2.8), (3.2), (3.12) and
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- 6 2 > 0 uniformly in F Ew .
n
Nn n
(3. 15)
ea
Hence, the lemma.
[Incidentally, the condition (2.12) may be easier to verify than the parallel
Also the third moment condition
condition by Hoeffding (1948, p. 313; (8.14».
implicit in his theorems may be replaced by (2.11)].
4.
n
As w is a resolving subset of Q, from (2.4) and
Proof of theorem 2.1.
n
(2.. 16), we have
for all F Ew .
(4.1)
.... n
n
Hence, using the three lemmas in section 3, we obtain under the conditions (i)
and (ii) of theorem 2.1 that
('
1
_
1
~(n2[U -8 ]/m{~l(F( »_62 }2) ~ N(O,l) for all F Ew .
(4.2)
non
Now, i f U
n
n
... n
n
is strictly distribution-free for ....Fn EW*,
by (2.14) and (2.16) we
n
have
~10
(4.3)
Hence~
> 0 for all ",n
F Ew n •
from (4.2) and (4.3)
(4.4)
where 6 2 , defined by (3.2), is zero either when F EW* or when w is an affine
n
""n n
n
n
resolving subset of Q .
normal distribution.
Let now t ex, be the
Then from (4.4),
100(1~)%
(2.l8~
point of a standardized
(2.19) and (2.20), we obtain that
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IC n -8 a -n
(4.5)
_1
2 mt
_1
Ic -n 2 mt
(4.6)
n
-10.
12
ex,
The robustness of tl(!n) (for 0 <
from (4.4), (4.5) and (4.6).
6
n
o(n 2) for tl(E
);
",n
ex,
1
S2
10
ex,
I = o(n _10.2)
<!) and
Further, if
W
n
for t 2 (E ).
"'n
t2(~n)
(for 0 <
ex,
< 1) then follows
is an affine resolving subset of nn,
; 0 for all F EW and hence from (4.4) and (4.5), the robustness of tl(E )
",n n
"'n
(for all 0 <
ex,
< 1) follows.
Now, if Un is asymptotically distribution-free for .....n
F EW*,
n its estimated
variance (Vn ) is such that nVn estimates m2~1(F) for all IUn
F EW*.
n
~l(F)
is a regular functional of degree 2m, and hence, the corresponding U-statistic
is an optimum estimator of it [cf. Fraser (1957, p. l42)J.
~l(F)
Now, by (2.8),
If this estimator of
is used then nV is an optimum unbiased estimator of m2~1(F) for all F EW*.
n
"'n n
Now, if (2.11) holds for some 5
~
2, it follows from our lemmas 3.1 and 3.2 that
under (2.12)
(4.7)
1
1
Now, for asymptotically distribution-free U , we need substitute (nV )2 for m!;2
n
n
10
in (4.5) and (4.6).
By virtue of (4.7), the desired result will again follow
from (2.18),_ (2.19), (4.4), (4.5) and (4.6).
It remains only to show that t (E ) in (2.23) is also robust.
3 IVn
By a
straightforward extension of lemma 3.3, we have
t(S ) ~ X2 for all F EW*,
(4.8)
n
where S
n
p
Nn
n
is defined by (2.22) and X2 has a X2 distribution with p degrees of
freedom (d. f.).
p
Hence, from (2.23), we have
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IC n -X 2p,a., I =
(4.9)
0(1),
er
where P{X 2 > X2 } = a.,
p - p,a.,
Let us furst consider the case when U is strictly distribution-free for F €w*.
n
"'n n
On replacing the two ¢'s in (2.8) by ¢
we obtain
~l(r,s)(F)
r
and ¢
for all r,s=l, ... ,p.
s
(with m=m
r
and m respectively,)
s
We consider then the following matrix
(4.10)
Similarly, replacing the sum of squares in (3.2) by an analogous sum of products,
we define /::, (
) for all r, s= 1, ... ,. p, and let
n r, s
(4. ll)
n /::, = «m m /::, (
)) ) .
Nn
r s n r,s
By assumption (2.21), n
~
'" n
(F) is positive definite for all F EW*, and n /::, is
-n n
"'n
positive semi-definite for all F Ew.
Nn
n
As U
n
is strictly distribution-free for
F EW*,. by (2.21) and (2.22),
.... n n
(4.12)
n An = n ",n
~ (F) = n ~ (F( )) = a known matrix, for all F €W .
.... n
n
Nn n
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Let now for Nn
F Ew n ,
(4.13)
B
",n
=
~ (F( n )) - /::', S*n
'" n
~n
= [U -8 ] B-
Nn .... o ... n
1
[U -8 ] t
"'n"'o
•
Then,. by a straightforward extension of (4.4), we have
(4.14)\
2
1.(S*)
F €w n .
n ~ X for all Nn
Our desired result will follow then from (4.9), (4.13) and (4.14), provided we
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S > S* for all E and all F €w .
n
n
"In
""n n
(4.15)
Upon noting that B = A - 6, nA positive definite r
"'n
.... n
....n
"'n
(4.15) readily follows from
(2.22), (4.13) and the following simple lemma (whose proof is omitted).
LEMMA
4.1
If
~1
(and
~2)
are two positive definite (and semidefinite) symmetric
matrices of the order p x p and
-1
~ ~1
~
is any p-vector with real elements,
x' .
'"
This completes the proof for distribution-free U . When U
n
n
is asymptotically
distribution-free (for F €w*), and V is its estimated covariance matrix, as in (4. 7),
"'n
we have under (2.11) (for 6
In
(4.16)
n
"n
~
2),
V - n ....
L: n (F( n »
"'n
o
I ~ ""pxp
for all F Ew .
.... n n
The rest of the proof will follow from (2.22), (4.13), (4.14) and (4.15).
Hence
the theorem.
5.
Illustrations.
~
Let us first consider the hypothesis of matching-invariance.
(5.1)
for all i=l, ... ,n.
=... sFnsF,
If F
1
(5.1) implies that the joint distribution
of (Xl' ... ,X ) remains invariant under arbitrary matching {(X ,X .), i=l, ... ,n},
n
l i 2R
1.
where (R , ... , R ) is any permu ta tion of (1, ... , n), and hence,. (5. 1) is termed the
1
n
hypothesis of matching-invariance.
tests for H
o
When F s .•. sF , genuinely distribution-free
n
1
in (5.1) based on U-statistics are due to Kendall (1938), Hote11ing
and Pabst (1936) and others [cf. Hoeffding (1948, pp. 316-321)J.
We shall show
that these tests are all robust for a wider class of distributions.
the following four cases:
We consider
I
I
-14-
I.
_I
II.
III.
IV.
Neither G.'s
nor H.'s
are all identical.
~
~
It is easy to verify that except in case IVT w, defined by the set of
n
equations in (5.1), is a resolving subset of an, Further, if in case II (or
1
1
case III) J H.;(x)dHj(x) = t (or J G.(x)dG.(x) = t) for all i,j=l, ... ,n, wn
0
~
J
o
.L
is an affine resolving subset of an (when Kendallts tau or Spearman's rho is
used).
Also, for Kendal1 t s tau or Spearmants rho, the kernels are bounded and
hence (2.11) holds for any finite 6, and it may be verified that (2.12) will hold
1
i f the integrals
1
J
o
H. (x)dH. (x) (in case III) or
~
J
J
o
G. (x)dG. (x) (in case II) for
~
J
i, j= 1, ... , n are bounded away from 0 and 1, i.e' T the marginal cdfts are all
mutually overlapping.
This clearly indicates that the nonparametric tests by Kendall (1938) and
Hotelling and Pabst (1936) are robust for heterogeneity of the marginal cdfts
of one of the two variates (i.e' T case II or case III).
Next, let us consider the problem of sign-invariance.
cdf F.(x) defined on the real p-space, (for some p
~
~
Let X. have the
l)T i=l, ... ,n.
~
The null
hypothesis states that X. and (-l)X. have the common cdf F. for all i=l, ... ,n,
~
~
~
i.e., F , ... ,F are all diagonally symmetric about
n
l
2.
[For p=l, this simplifies
to
(5.2)
F. (x) = l-F, (-x-a) for all i= 1, ... , n. ]
~
~
It is, again T easy to verify that if F , ... ,F are all diagonally symmetric, then
1
n
F(n)' defined by (2.15), is also so.
Thus, here also w
n
is a resolving subset
I
I
I
I
I
I
I
-.
I
I
I
I
I
I
--I
I
I
Ie
I
I
I
I
I
I
I
-15-
of On.
Moreover, w
n
can also be shown to be an affine resolving subset of On
if the signed-rank statistic of Tukey (1949) is used.
As the kernel of this
U-statistic is bounded, (2.11) holds for all F €w, if the marginal cdf's (of
"'n n
each variate) corresponding to F , ... ,F are mutually overlapping.
n
1
Thus, the
test based on the p-vector of signed-rank statistics corresponding to the p components of Xi (i=l, ... ,n) is robust for any heterogeneity of the cdf's Fl, ... ,F .
n
In the case of p=l, it has been shown by the author [cf. Sen (1967)J that when
Fl, ... ,F
n
are not all identical, the asymptotic relative efficiency of the signed
rank test with respect to the t-test (for shift alternatives) is usually greater
than that in the case when F
=... sFn .
1
By virtue of the results derived in this
paper, it can be shown by some routine analysis that the above results also hold
for the general case of p > 1.
Other examples of robustness can also be cited.
.e
REFERENCES
I
I
I
I
I
I
Ie
I
[lJ
FRASER, D. A. S. (1957).
John Wiley.
Nonparametric methods in statistics.
[2J
HOEFFDING, W. (1948). A class of statistics with asymptotically normal
distributions. Ann. Math. Statist. ~ 293-325.
[3J
HOTELLING, H. and PABST, M. R. (1936). Rank correlation and tests of
significance involving no assumptions of normality. Ann. Math.
Statist. !J 29-43.
[4J
KENDALL, M. G. (1938).
81-93.
[5]
SEN, P. K. (1967). On a further robustness property of the test and estimator based on Wilcoxon's signed rank statistic. Ann. Math.
Statist.
[6]
TUKEY, J. W. (1949). The simplest signed-rank tests. Mimeo report 17,
Statistics Research Group, Princeton University.
A new measure of rank correlation.
New York:
Biometrika, ~