Indian Statistical Institute
Asymptotic Theory for Some Families of Two-Sample Nonparametric Statistics Author(s): Lars Holst and J. S. Rao Source: Sankhyā: The Indian Journal of Statistics, Series A, Vol. 42, No. 1/2 (Apr., 1980), pp. 19-52 Published by: Indian Statistical Institute Stable URL: http://www.jstor.org/stable/25050211 Accessed: 18/06/2010 09:10 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=indstatinst. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact
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: The Indian Journal Sankhy? of Statistics 1 & 2, pp. 1980, Volume 42, Series A, Pts.
19-52.
ASYMPTOTIC THEORY FOR SOME FAMILIES OF TWO-SAMPLE NONPARAMETRIC
STATISTICS
By LARS HOLST* and Uppsala
Madison
of Wisconsin,
University
Sweden
University,
and S. RAO
J. University
SUMMARY. two the
null
the
ordered
Let
that
hypothesis
is studied
natives,
trie
type
of test 1,.
test
are not
for
the maximum
has
metric
type
can
Wilcoxon-Mann-Whitney asymptotic
test
against
"optimal"
Connections
with
rank
as well
as under
m 2 h(Sk), fc-1
form
based
is an example
specific
are
1.
briefly
On
are
methods
from
alternative, explored
Introduction
the
here
the United
States
interval
and
limiting
and
Army
under
only
illustrative
and
of the
those
symme alter
distinguish
of symmetric tests, which the Dixon is shown that tests
of the rate
nonsym of
n'112.
investigating one to allow
select
standard After
which tests
among
tests
that
can
of alter
sequence on $*'?
hand,
suggested
some
and
that the two d.f.s. are the same. hypothesis transformation z->F(z) would permit carrying by
a suitable
other
of
the
examples
type
the
m 2 hj?(Sjc). *-=l
provided.
notations
Xl9
Sponsored
be
in the
theory
at the more converging to this class. which belongs
and Yl9 ..., Yn be independent ...,!,?_! continuous distribution two populations with functions to test if two these wish We populations respectively. Let
test
... ^ X'~
falling
symmetrically
It is shown
in {Sjc}.
efficiency.
alternatives
any
to
on these numbers based hypothesis, {Sjt}. Let some functions condi satisfying simple regularity
symmetric
alternatives,
wish
in Dixon
problem
oo
and
that
sequence as v ?> oo, rv ?>
Godambe
(1961)
as m and n tend
theory
= mv/nv
(1940),
p,
of positive
0
Corresponding = let functions and real-valued k be ..., 1, array, satisfying Av( ) {A?v( ), rav} certain regularity conditions 2). Define (see Condition (A) of Section mv Z A*V(S*V)
?*v=
...
(1.5)
*=i
and
s
r;=
...
av(^v)
fc-i
(1.6)
on the (mv~l) X-values and the nv 7-values. Though T* is a special case of Tv when on k, we will distinguish these two cases {hkv( )} do not depend since their asymptotic is quite different in the non-null situation. behaviour
based
It may
be noted
that
here
the Wald-Wolfowitz
(1940)
iun test and
the Dixon
are of the form
test is of the Wilcoxon-Mann-Whitney (1940) T*v while in the combined the form Tv. In fact, any linear function based on the X-ranks can a as case be of also Section 5.) sample, (cf. expressed special Tv. test
A few words
the
m
Tv
=
law A
), {hk(
A(
v is suppressed
suffix
: Though
the notations
about
as the functions
as well
except
the quantities m, n, r, Dk, Sk on v, for notational convenience
)} depend it is essential. where
2
hk(Sk), T*?
fr=i
a
of
random
normal
=
S *=?i
and r will
h(Sk)
stand
for (m?n) etc.
for
instance,
The probability
(or random vector) X will be denoted by
