Indian Statistical Institute

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
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Indian Journal of Statistics, Series A.
http://www.jstor.org
: 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
<
distribution
from
samples
We
LetX'
of Y-observations
asymptotic
hypothesis
such
tests
identical.
line.
the null
real-valued
relative
asymptotic
under
for
the
random
independent
on the real
in the
that they
performance
from the hypothesis.
this class
Among
run test and the Dixon
it
known
test,
distinguish
test
theory
be
, Yn
USA
sense
the well
instance
includes
...
respectively
are
populations
of the
statistics
Barbara,
O
studies
the null
under
Santa
by Sjt the number
statistics
be
Ylt
and
parent
paper
.. ,
m}
have
poor asymptotic
at a "distance"
of n"1^
test
an
This
m
2
hjc(Sjc) which
fc-1
of the form
natives
,m.
for
F
Denote
theory
Asymptotic
two
these
and
,Xm-i
functions
X-observations.
k =
1,...
X'.),
[X'v
of
families
efficiencies
. )k=
. ) and
h(
{h*(
tions.
...
Xl9
distribution
continuous
of California,
random
samples from
(d.f.s.) F(x) and G(y)
are identical,
i.e., the
A simple probability
us to assume
that the
Contract
No,
DAAG29-75-C-0024.
integral
support
20
LARS
HOLST
J.
AND
S.
RAO
of both
the probability
is the unit interval
distributions
[0, 1] and that the
first of them is the uniform d.f. on [0, 1], For the purposes
of this discussion,
as
can
this probability
be
done
of
loss
transformation
without
any generality
soon.
be apparent
Thus from now on, we will assume that this reduction
and that the first sample
has been effected
is from the uniform
distribution
= G o p-1
after the
Let
G*
the
the
second
denote
of
d.f.
?7(0, 1).
sample
The null hypothesis
to be tested,
transformation.
probability
specifies
will
=
HQ:G*(y)
0 < Xi
Let
The
sample
... <
<
Dk
=
=
Dkm
statistics
,
?
= 0 and
1. Tests based
put X0
X'm
in the literature
been
for the
considered
have
See for instance
Define
k =
for
sample.
by
...
on these
sample
(1.2)
spacings
goodnesss-of-fit
problems.
and Rao and Sethuraman
(1975).
..., m
1,
Sk
(1965)
(1953), Pyke
Darling
from the first
...,m
k=\,
we
where
(1.1)
are defined
for the X-values
X??X?_!
...
1.
y<
1 be the order
X'm_1 <
(Dv ..., Dm)
spacings
0<
y,
?
number
of
in the
%'s
interval
...
[X^_l5 X'k).
(1.3)
....
is to study various
test statistics
based on these numbers
{Sly
Sm}
be
for testing H0.
These
called
may
quantities
(since
"spacing-frequencies"
the frequencies
of y's in the sample
of the x's) or the
spacings
they denote
Our
aim
to the gaps in the ranks of the #'s in
they correspond
as well as the statistics
Since the numbers
based
sample).
{Sjc}
remain invariant under probability
no
there
is
loss
of
transformations,
(since
"rank-spaeings"
the combined
on them
in making
generality
It may be remarked
such
a transformation
here
that we
in the first
vations
We
and
aim
will
do
mv
Note
more
is to study
this through
assume
{nx] and
??
oo,
instead
as was
done
of the usual
earlier.
m
obser
(m?1)
this yields m numbers
...,
instead
of
{Sv
Sm}
on
notation.
Tests based
simpler
{St} have been
since
sample
(m+1),
leading to slightly
for the two-sample
considered
Rao
(1976).
Our
on the data,
take
the asymptotic
a nondecreasing
throughout,
nv ?>
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 <
p <
oo.
and
to infinity.
integers
...
{mv}
(1.4)
in (1.2) depend on mv the number of X-values
and it is
{Djc} defined
to label them as {Dkv}.
the numbers
appropriate
Similarly
{$#} defined
that
TWO-SAMPLE
NONPARAMETRIC
21
STATISTICS
on both mv and nv and should therefore
in (1.3) depend
be denoted
by {Skv}.
=
Thus we are dealing with
variables
random
of
1,
arrays
triangular
{Dky, k
v
the
to
v-th
1.
..., mv} and {Skv, k ?
1, ..., mv} for
>
1)
(v >
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 <?(X).
mean
covariance
matrix
S will
be
with
?i and
variable
distribution
while N(0,
0) stands
by N(/i, S) throughout
zero.
at the point
For 0 < x < oo, p(x)
with mean x and
distribution
represented
distribution
Poisson
Thus
m
7T^)
=
e-*.^/j!,
j
=
0,1,2,...
for
will
...
the
degenerate
the
represent
(1.7)
For p =
(n, p) will denote
(pv ...,pm), mult
n
distribution
with
trials and cell probabilities
variable
random
(r.v.) with density
exponential
of j.
probability
multinomial
the m-dimensional
the Poisson
(Pv
>Pm)'
A
negative
e~w for w ^
o and
zero
elsewhere
(1.8)
22
LARS
will
be
denoted
HOLST
AND
J.
RAO
stand
for an
by W while
{Wl7 W2, ...} will
of such r.v.'s.
The
distributed
(i.i.d.) sequence
throughout
and identically
independent
random variable
tj will have
a geometric
distribution
i
Pfo=j)=p/(1+P)'+1,
for 0 <
S.
=
with
p.d.f.
...
0, 1,2,
...
(1.9)
oo.
p<
is useful
conditional
between
these distributions
following
relationship
a
as
r.v.
a
Let
Let r? denote
above.
later on.
IF be
exponential
negative
r.v. which
Then the (unconditional)
for given W = w, is 7?(w\p).
distribution
of 7j is
The
?
=
P(v
j)
=
=
Err}(Wlp)
f-W/P
J (w?p)S -^r-
c~" dw =
p/(l+p)>+1,
j
=
0, 1, 2, ...
...
same as (1.9) above.
Thus t? has a geometric
on W = w, it has a ^(w/p)
distribution.
distribution
the
for any random variable
and we write Xn =
in probability
Also
such that P{ XJg(n)
|
the
denote
We
largest
shall
>
\ Ke} <
=
Xn
if for each
Op(g(n))
a sequence
consider
we write
=
if conditional
?> 0
op(g(n)) if XJg(n)
e > 0, there is a Ks < oo
s for all n sufficiently large. Finally
[x] will
in x.
contained
integer
GUV)
Xn,
(1.10)
of alternatives
specified
0< y <
y+(Lm(y))lm*,
by
the d.f.'s
...
1
(1.11)
= 0 and S >
=
. In terms of the
J
original d.f.'s F and
Lm(l)
Lm(0)
=
there is
P, while under the alternatives
C?, the null hypothesis
specifies G
as
to
F
the
size
a sequence of d.f.s.
increases.
Guthat
converge
sample
is
of
Indeed Lm(
)
(1.11)
given by
where
Lm(y)
=
m*{Gm{F-\y))-y).
is a function
...
(1.12)
on
(0, 1) to which Lm(y) converges.
on Lm(
For further conditions
(B) and (B*).
) and L(
) refer to Assumptions
a
sense
in
certain
and
has been
is
smooth
of alternatives
This sequence
(1.11)
We
assume
considered
that
before.
there
See
for
instance
L(y)
Rao
and
Sethuraman
(1975)
or Hoist
(1972).
: In Section
of this paper is as follows
organization
are established.
2.1 gives asymptotic
Theorem
results
The
nary
2, some prelimi
distribution
of
TWO-SAMPLE ttONfARAMEtRlC
of multinomial
functions
on
limit
the
while
frequencies
of non-symmetric
distributions
n
STATISTICS
2.2
Theorem
is of
which
statistics,
spacings
in Theorem
combined
3.1 to obtain
independent
It is
S = |.
the limit distribution
of Ty under
the alternatives
(1.11) with
distribution
clear that putting Lm(y) == 0 in this theorem, gives the asymptotic
an asymptotically
test for
of jTv under HQ.
The problem
of finding
optimal
These
interest.
are
a result
establishes
results
a given
3.2.
in Theorem
is considered
Some specific
of alternatives
sequence
are discussed
4 deals with
the
at the end of this section.
Section
examples
of
4.1 gives
statistics
distribution
the asymptotic
T*v. Theorem
symmetric
4.2
while
5=
Theorem
of
with
the
under
alternatives
T\
\
sequence
(1.1)
to note
tests.
It is interesting
can
alternatives
only distinguish
symmetric
4
unlike
the non-symmetric
to the hypothesis
at the slow rate of n
converging
at the more usual
can discriminate
alternatives
which
statistics
converging
_i
2 .
on sample spacings depend
rate of n
Similar results hold for tests based
See for instance
or not one considers
statistics.
ing on whether
symmetric
finds
the
optimal
that
Rao
and
some
test
among the symmetric
classes of test statistics
T\
further
remarks
2.
The
following
as well
functions
of this
results
and
Some
preliminary
5 contains
Section
the
limit
which
growth
be needed
will
properties,
functions
real-valued
(1980).
results
conditions
regularity
as supply
smoothness
the next section.
(A) : The
Condition
and Hoist
(1975) and Rao
and discussion.
Sethuraman
)} defined
{hk(
on
of
the
for the
{0, 1, 2,
...}
satisfy Condition (A) if they are of the form
*=l,...,w
hk(j)^h(kl(m+l),j),
for
some
function
=
0,1,2,.
=
0<
u <
1, j
in u except
if any, does not
for
finitely
for
defined
h(u, j)
?
(2.1)
0, 1, 2,
... with
the
many
u
the
properties
(i)
h(u,j)
is continuous
set
discontinuity
depend
and
onj.
(ii) h(u, j) is not of the form c >j+h(u) for some function A on [0, 1] and
a
(iii)
For
real
c.
number
some
d>
0, there
exist
IMn,j)| < cx [n(l-ii)]-i+S.
for all
0<
u <
constants
c1 and
c2 such
that
(/2+l)
1 and j =
0, 1, 2, ...
...
(2.2)
24
AND
HOLST
LARS
S.
J.
RA?
real-valued
functions
(A') ; The
Condition
(A') if they are of the form
satisfy
gic{x)
for
some
?
function
g(u, x)
k=
x),
g(kl(m+l),
1, ..., m
u <
0<
for
defined
on
{gjc(*)} defined
Condition
x<
0<
and
oo
0 <; x <
1 and
[0, oo)
oo with
the
u
the
properties,
g(u,
(i)
in u
is continuous
x)
set if any,
discontinuity
for finitely
except
on x,
not depend
does
g(u, x) is not of the form c
and a real number
c, and
(ii)
for some
(iii)
?>
0, there
\g(u9x)\ < c^l-^f
We
2.1
Lemma
g on
function
[0,
1]
that
c2 such
cx and
and
?+^(a2+l)
u <
simple
lemma,
: Let
h(u) defined for 0 <
in absolute
be bounded
x <
1 and 0 <
for all 0 <
following
u and
many
finitely
Then
the
require
for some
x-\-g(u)
constants
exist
many
is stated
which
...
oo.
without
(2.3)
proof.
u <
1, be continuous
except for
value by an integrable function.
i
m
2 A(Jfc/(m+l))-> J h(u) du as ra-> oo. D
(1/ra)*=i
o
(2-4)
of Tv defined
problem, we will obtain the distribution
we
the
statistic
consider
in two steps.
First
in (1.5), essentially
Tv for given
Since the numbers
D = {Dv ..., Dm}.
values
of the X-spacings
{Sv ..., Sm}
on
a
we
multinomial
a
the
need
result
multinomial
distribution,
given D have
to the main
Turning
sums.
for
We
formulate
the
given
on the
The expressions
2.1.
this part of the result in Theorem
of Tv
distribution
mean
of this conditional
and variance
asymptotic
D, are functions
of D.
limit distributions
of functions
for
in particular,
these
expressions
3.1 of the next
section
Theorem
lemmas
It
there,
given
is clear
frequencies
S =
thus giving
that
(Sv
the
of spacings,
the
combines
the required
conditional
..., Sm)
2.2, we
In Theorem
given
a general
result
us
to handle
allows
formulate
which
mean
and variance.
asymptotic
other
these
results along with
distribution
of
the
vector
spacings
of Ty.
distribution
asymptotic
the
vector
D =
of
(Dv
spacing
is
..., Dm)
TWO-?AMPLE N?NPARAMETRI?
25
STATISTICS
on D,
Therefore
the test statistic
..., Dm).
(n, Dv
Ty9 conditional
as
same
variable
null
the
random
the
the
distribution
hypothesis,
mult
under
Zv
where
(9^
variance
variables,
variables
=
?iv
taking
r.v.'s
Poisson
Theorem
be as defined
2.1
m
S hk(?kv),
(2.5)
...
cr2=
E(Av),
...
var(Av).
(2.7)
the asymptotic
of the multinomial
distribution
2 of Hoist
case of Theorem
as a special
(1979) by
theorem.
that
in
of
of
(?kv, hk(?kv))
place
(Xky, Ykv)
: Let
in (2.5),
(2.6)
k=l
be mult(n,pv
(<pl5 ...,9^)
(2.6) and (2.7). For 0 < q <
Av,=
thai there exists
Assume
...
?T=1
on
theorem
following
Zv can be derived
sum
mv
2 hk(9k)
mean
and
the asymptotic
is mult
Since
(n, Dv
...,Dm).
...,9^)
random
terms
Poisson
in
of
stated
of Zv can be more
simply
a triangular
we
random
Poisson
introduce
array of independent
v
1
is
and
set
where
%kv P(nvpkv)
{?lv, ..., gm v}, >
Av=
The
=
has
a q0 <
M
2
...,pm) and Zv, /iv, andav
=
1, set M
[mq] and
...
?*(&).
(2.8)
fc=i
1 such
that for
M
2 pk->Pq9
q>
qQ
...
0<PQ<1,
k=l
(2.9)
and
where AQ, BQ and Pq
are such
Aa -> Ax,
TAew
a?s
j^?>
BQ -> ^
1-0,
and Pa ->
...
1.
(2.11)
00,
^-^/^-?^OMx-u?.
A12-4
that as q-+
D
...
(2.12)
&6
From
?LARSHOLST AND j.
(2.7) an explicit
(2.6) and
fi(nD)
p?nD)
?7(0, 1).
m
?
S
S ?*(j)*r,(nD*).
?=l
Thus
have
we
=
pk
Dk,
consider
...
(2.14)
i=0
oo
Wi
k=i
2
A?(j) 7r?(a:/r).
i=o
on spacings have
(1965) and Rao
based
(2.13)
we
hypothesis,
=
This is of the form 2 gjc{mDk)where (/?(a;)
Statistics
by
i=0
the null
(1.7). Under
D are the spacings from
=
is given
...
2 hk{j) ttj (npjc)
fc=l
=
for the mean
expression
/?v= 2
the notation
using
==
k
1, ..., m where
S. tlA?
been
considered
earlier by Darling
LeCam
and Sethuraman
(1958), Pyke
(1975). Most
=
the
discuss
when
case, i.e.,
papers, however,
gk(x)
g(x)
symmetric
As Pyke
out (cf. Section
could
(1965) pointed
6.2), LeCam's method
=
to study
case.
the more
Let
general
non-symmetric
{gk{ ), k
=
0 < q<
be real-valued
measurable
functions.
For
1, let Mv
(1953),
of these
for all k.
be used
1, ..., m}
[mv
q].
Define
GQv=
M
2
gk(Wk)
...
(2.15)
fc=i
a sequence
of i.i.d.
Then
the
r.v.'s.
exponential
{Wv W2, ...} is
states
the
of
theorem
distribution
statistics
asymptotic
following
explicitly
of the type (2.14) and is easily established
by checking Assumption
(6.6) of
where
Pyke (1965).
Theorem
2.2
: Assume
0<
that
var (GQv) == o-2(Gqv) <
oo for all q and v,
...
(2.16)
and thatfor each q e (0, 1]
/ / 0 \ / At B9\\
{Gqv-EG9v)l<r{Gu)
\
... (2.17)
Mm
]^N[V \( 0 /), \( Bg q I
V S (F*-l)/m*
J
j
/
with
Aq
and Bq
such
that
Aq-*
Ax
Ba -> Bx
=
1 as tf->l-0
as q -
1-0.
...
...
(2.18)
(2.19)
TWO-SAMPLE NONPARAMETRIC
as
Then,
v-^
?
gk(mDk)-EGlv'jl<r(Glv))
..., Z>) are spacings
(Dv
corollary gives a simple sufficient
following
holds.
) in order that the above Theorem
: The
2.1
Corollary
asymptotic
...
1?J3J).
(2.20)
1). Q
U(0,
from
-> N(0,
The
gk(
27
oo,
??((
wAere
STATISTICS
normality
on the functions
condition
asserted
2.2
in Theorem
holds
for any set of functions {gk( )}which satisfy condition (A').
: To prove this
Proof
corollary, we need to check that the assumptions
It can be easily checked
condition
(2.16) to (2.19) hold when
(A') is satisfied.
that if g(u, x) satisfies
condition
then
(A'),
00 00 00
x)e~x dx as well
S 9?U> x)e~x dx, J g\u,
oo
satisfy
Lemma
of Lemma
conditions
2.1,
as
m
E(Gqv)lm
??
=
var (Gqx)jm=
2.1
in u.
Thus
as J g(u, x)(x?
l)e~x dx
o
from
the definition
of
Gqv
and
oo,
[mq] q
2 Eg(kl(m+1), Wk) -> JEg(u, W) du
(Urn) k=i
o
...
(2.21)
[mq] q
(1/m) 2 var
(g(kl(m+l),Wk)) -> J (var g(u, W)) du
o
k=i
...
(2.22)
and
cov (Gqv,2 Wk)\m =
M [mq]
2
(1/m)
l
k=l
~> J cov
o
from (2.3) of condition
Again
in q so that (2.18)
continuous
Finally
to
check
the
cov (g(kl(m+l), Wk), Wk)
(g(u, W), W)
fc=i
...
(2.23)
(A'), all these limits are finite.
and (2.19) are satisfied.
asymptotic
in (2.17)
normality
These
{a{gk{Wk)-Egk{Wic))+{Wle-\)}=
S
Jc=i
are
or equivalently
tm0l
[ma]
S
du.
g*k(Wk), say
...
also
of
(2.24)
LARS HOLST AND J. S. RAO
28
for all real a, we
identical
case.
have
only
is easily
It
for the non
the Lindeberg
condition
condition
if {gk( )} satisfy
(A'), so do
to verify
seen that
{gl( )} defined in (2.24). Let
=
(Wk) and sfm] * 2 erf.
k=l
=
(r? Eg\
...
(2.25)
Since {gl( )} satisfy condition (A'), we have as in (2.22) that
=
afmgI/?
to a finite
converges
non-zero
(l/m)2
o
[mq]
oo
k=i
jg\(w)tr"?w
o
from Lemma
constant
...
2.1.
(2.26)
Now
consider
[mQ] 2
j
(i/^)s
<
0;(*)6-*<fe
.
(mlsfmq])dim) 2
c1[(fc/(m+ i))(i_ t/(m+l)]-i+2'
J
(a;C2+l)2e-^da;
S
< {^/^?{(l/m)
J
{
#=i
[(?/(m+l))(l-i/(m+l))]-1+25}
(xe*+l)*e-*dx).
remain bounded
because
in the first two parentheses
As m ?> oo, the quantities
in
third
the
2.1 while
the integral
of (2.26) and Lemma
goes to
parenthesis
Thus the Linde berg
zero for any e> 0 since s[mq^ is of order (\/m) from (2.26).
condition
is satisfied
3.
for
(2.24) which
distribution
Asymptotic
define
for later use,
for
theory
statistics
non-symmetric
We
the assertion.
proves
the following
additional
functions
00
gx(u, x)
=
2
=
2
h(u, j) tt^x),
...
(3.1)
00
g2(u, x)
h2(u, j) tt} {x)
...
(3.2)
and
g3(u, x)
=
2
J-O
h(u,j)(j-x)
n} (x).
...
(3.3)
STATISTICS
TWO-SAMPLE NONPARAMETRIC
29
are well defined
(A), these functions
h(u, j) satisfies condition
Further
condition
For instance
condition
gx and gr3 satisfy
(A').
since
condition
of
for
gx
(A')
(iii)
implies
When
\gx(u,x)\
for x >
0.
(iii) of
(A)
I 00
=
i2 h(u,j)nj(x)
<
SCiMl-ttjf^a+^^W
;=o
_
<c;Ml_w))-*+s(1+/2)
(34)
To see the role of these
the moments
of the Poisson
distribution.
using
the representation
functions
Let
of r? given in (1.10).
gv g2 and g3, recall
over
the expectation
and variance
W while En?w Vn\W denote
Em Vw denote
over
the conditional
variance
and
Then from the
7} given W.
expectation
definitions
of gl9 g2, gz
ri)
Enh*(u,
7i)
some
after
And
Enh(u,
=
=
EwEn,
=
h(u,
r/)
h\u,
7i)
EwEnlw
w
=
...
Ewgi(u,
W\p)
Ewg2(u,
W/p).
(3.5)
...
(3.6)
calculations,
elementary
COV(h(u, 7?),7?)= EWEn]W [h(u, 7?)(7}-W/p)]
P^+P)-1
=
cov(gi(u,
W/p),
=
W)
Ew
gs(u, W/p).
...
(3.7)
Define
1,
a-2=
From
j var ?^
0
?u__
^
the Cauchy-Schwartz
1
,2
cov
\ jo
y) du
(h(Uj^
...
(t?).
/ /var
(3.8)
inequality
/
1
, 1
\2
( J cov (A(w,7/), ??)du <
J
i2
f J (var h(u, 7/))*(var?/)*dw 1
< var (t?) ( J*var A(w, 9/)dw ]
with equality if and only if h(u, j) =
number
o and
some
condition
satisfying
/i(x)
and observe
=
fiv(x)
that
for 0 < u <
c j+h(u)
a*2>
Thus
h(u).
=
For x
(xv ..., xm), we
function
(A).
=
2
g1(kl(m+l),
/?(wD) corresponds
xk)
=
*=i
?>vm
2
*=i
to the statistic
0 for
any
1 for some real
function
h(u,j)
define
co
2 hk(j)nj(xk)
;=o
in (2.14).
...
(3.9)
LARS HOLST AND J. S. RAO
30
we
Before
distribution
proceed
of Tv under
alternatives.
Consider
in (1.11),
given
state
to
theorem
the
a few words
the alternatives,
the
from
F-observations
S= i
i.e.,
(1.12), with
:GUV) = Gm{F-\y))
=
y+Lm(y)lmK
4?>
which
gives
about
the
0< y<
the
of
sequence
function
distribution
...
1.
in (3.10) with
S = |,
(B) ; For the alternatives
Assumption
there exists a continuous
function L(y) such that for 0 ^ y ^ 1,
Lm{y)
=
Also
that
the
suppose
some
continuous
outside
limits
right
derivatives
fixed
=
DI
and L\y) = %)
exist
in [0, 1] and that finite
on the open
interval
and
are
left
and
(0, 1).
Gm{F-\U'k))-Gm{F-\UU))
=
=
U'k, k
that
of a F-observation
inside
probability
falling
is given by the uniform
spacings
hypothesis
{Dk}.
is given by
the alternatives
(3.10), this probability
under
hand,
assume
the
the X-sample,
under the null
[Xk_v X?),
On the other
where
finite
exist
of the derivatives
Given
L'm(y)
subset
(3.10)
as m -> oo.
-> ?(y)
m*[Gm(-F-%))?y]
asymptotic
the
...
Z>*(l+A*/m*)
1, ..., m are order
statistics
from
(3.11)
?7(0, 1) with
U'0
=
0, U'm
=
1
and
Ak
random
variable.
We
proof
theorem
be completed
may be slightly
In any
the present
will
case,
Theorem
3.1
cr is defined
that for some small
(3.12)
one so that
A*? is a well-defined
probability
now state the main
theorem
whose
of this
section,
in Lemmas
to
3.7.
The
of this
3.1
conditions
weakened
but
conditions
at the expense
of added
complexity.
cases of statistical
interest.
cover most
: Let
Vv
where
...
[Lm(Uk)-Lm(U'k_1)]IDk.
0 with
that Dk >
Note
=
=
m
2
(hk(Sk)-Ehk(V))lm*
k=l
in
(3.8).
?>
0,
In
addition
- cr
to Assumptions
\Lm(t)-Lm(s) | < c3(ia-5a) for 0 < s < t<
and for (1?e) < s <
t <; 1
...
(A) and
(3.13)
(B) assume
6
...
(3.14)
TWO-SAMPLE NONPARAMETRIC
where
7/8 <
a <
1.
Then
the alternatives
under
31
STATISTICS
(3.10),
...
?(Vy)->N(b,l),
(3.15)
where
i
b=
Proof
rewritten,
: Observe
using
c.
Jo cov (h(u>7?),7j) l (u) du pl(l+p)
first
relation
the
that
in
constant
centering
(3.13) may
be
(1.10)
?
m
m
2 Ehk(7?)= 2
&=1
fc=l
=
2 hk(3)E7?i(W?p)
j=0
...
Efiv(W?p)
(3.16)
and W =
(Wl9 ..., Wm) are i.i.d. exponential
vector
the
2,
(Sl9 ..., Sm) given D* is mult
in (3.11).
D* has the components
(n, D*) where the m-vector
Using
D\ given
we
write
conditional
may
expectations,
in (3.9)
?iv(x) is defined
in Section
As explained
where
r.v.'s.
E(e^vv)
= E
E(e^Vv\D*)
=
...
E(Jv(iD*)Ky(D*))
(3.17)
where
JV(D*)
= exp
(itm-i \ji {nD*)-E/i
(W?p)]) ...
(3.18)
and
KV(D*)=
Now
e(
from Lemma
exp
(?im-i[
S hk(8k)-/i(n
A*)] )|l>').
... (3.19)
that
3.4, it follows
E{Jy(D*)) -> exp {ibt-ct2?2)
with 6 and c defined in (3.38) and (3.39) respectively.
?(m-i[pv(nD*)-Eii(Wlp)])->
so that
in distribution.
one,
random
JV(D*) converges
i.e., for almost every
vector
KV(D*)
By
Hence
N(b,c)
Lemma
3.5, with
...
(3.20)
probability
D*,
-> e-***'*
(3.21)
LARS HOLST AND J. S. RAO
32
with d as defined in (3.43). Combining (3.20) and (3.21), with probability
the
one,
in distribution.
since
But
converges
JV(D*)KX(D*)
so
moments
that
this
also
the
of
the
1,
convergence
implies
product
|<
|JV(D*)KV(D*)
E(JV{D*)KV(D*)) -> exp (ibt-(c+d)t*?2).
theorem
the continuity
Using
of the
the assertion
Lemma
: If
3.1
m-* 2
Ar=l
theorem
for
...
and Lemma
functions
characteristic
(3.22)
the conditions
of Theorem
3.1 hold,
then
2 h(kl(m+l), j)Midfy-n?nDk)\
;=o
oo
m
= m-1 2 A* 2
h(kl(m+l),JXJ-nD^inD^+o^l)
where
Ak is as defined
*=i
2
\hfc(j)\
? 1
-(j-nDk)
n~*2
[
the difference
Ak/m* \7Tj{nDk)
2 h%j)7Tj
2
{nDktf [
|exp {jlog(l+Aklmi)-nDkAklm*}
...
bracket
some
elementary
of
\exp{jlog(l+Aklmi)-nDkAklm*
^l^j^^^A^/m^l2^^)]*.
After
(3.23)
A*/m*}]
j=0
ifc=i j=o
on
inequality
2 hU^inDD-n^nD^l+U-nDk)
< m-* 2
...
in (3.12).
the Cauchy-Schwartz
Applying
Proof:
the two sides in (3.23), we have
m~* 2
3.7,
follows.
calculations,
we
see that
the term
in the second
(3.24)
square
is
exp (DkA?lr)-1 -D*A
j/r.
...
(3.25)
TWO-SAMPLE NONPARAMETRIC
Since
that
2.2 and
condition
h(u, x) satisfies
(A), using Theorem
the right hand side of (3.24) can be estimated
m-* 2 ^.^/(m
=
we
show
Observe
that
by
max(Af/m*)
k
this max
that
33
it is clear
(3.25),
by
+ l^^Kl+im^)02^^^
?=i
Now
STATISTICS
...
Op(l).
(Af/m*) goes
to zero
(3.26)
in probability
when
a>
7/8.
(3.14)
*
|
Lm(Ui)-Lm(UU)
I/ Dk
|
A*/m*
|=
<(U?-U?L1)lm*-Dk
...
<D*klm*-Dk
since
(1953)
s<
for 0 <
(t?s)a
(t*?sa) <
e
we
have
for any
> 0,
1/ min
Therefore
from
oc>
probability.
Dk)
(m2+e
=
a <
1.
Also
from
Darling
0P(1).
: If
3.2
...
lA^/m^l <O??(m(2+8>(1"a)-i).
choosing
7/8, by
the
This proves
Lemma
1 and
(3.27)
max
Since
t<
(3.27)
0<
e<
(4(1?a))-1?2,
max
(3.28)
|&k?m*\
-? 0
in
lemma.
the conditions
of Theorem
3.1 are satisfied
then, for
any
?>0,
[me]
oo
...
Urn sup?m-1 2 A* 2Hkl(m+l)9j)(j-nDk)n,(nI>k)\<Kta+?
m
k=l
?>oo
with
(3.29)
y=0
probability
one.
: In terms of gs(u, x) defined
Proof
in (3.29) is
(A'), the expression
in (3.4) and which
satisfies
condition
[mc]
m-i
2
k=>i
=
A12-5
&kgz(kl(m+l),nDk)
[me]
2
[Lm(Uk)~Lm(U'k^)]gz(kl(m+l),
nDk)(rnDk)~\
...
(3.30)
?4 LARS ?OLST
condition
Using
(3.14),
ANt> J. ?.
M
and writing
=
JEtA?
[mg], this
is
M
2
0<
it/)*) I(mD*)-1
g8 (*/(m+l),
i [LM)-Lm{U^x)]
< clCs 2 (J/?-?/?LiXfc/m^l+?AD*)
i
now make
We
...
2].
use of the representation
of the
(3.31)
spacings
r.v.'s
exponential
Wv
... with
W2,
mean
1. Writing
Wk
of i.i.d.
in terms
?
k
=
2 Wjjk,
the
in (3.31) is
RHS
?7.A
M _
cOH0
__
S [WtHk/M)?(fl$+ W*)
W%^{{lc-l)IMYlklM)<>.
1
Wa+C2
rr m
= G -6?+'.
f^"*'2'
if-1!
if? ^5)
?Ff-i(*/Jf)?+'-i(?P?+1+
...
'[{?-ii-wtik.wirymkwt)].
as k ?> 00, ?Fj;?> 1 a.s. and Wk?
Now
{1-(1-
?> 0 a.s.
a
WklkWtf}HWtlkWt)->
(3.32)
so that
...
a.s.
(3.33)
Using the Holder inequality,
~
|
Wr< (W+<->W?+' Rl-(1?WtlkWt)-)HWtllcW,)]
1/JP,
<[i?wr"p[?fwJo'**']
J U,
Uf!
Using
the
fact
that
in (3.34) converges
if ak ?
\
H
WkiikWje)
n
1 as k -? oo, w12
a.s. to the finite
limit
i
a? -?
1 as n -?
oo, the
RHS
TWO-SAMPLE NONPARAMETRIC STATISTICS
term
the other
Similarly
we get the desired
Lemma
can be handled
so
that
-Q
: Under
3.3
Wk* W*? in (3.32)
involving
result.
35
the conditions
of Theorem
3.1,
m-1 2 &kgz(kl(m+l),nDk)
k~i
l
?> J l(u) cov (h(u: 7j), 7j) du
o
: For
Proof
any fixed
?e]
ing of 3 parts
, 2
fc-l
and
?-[me]
2
can be used
In view
nDk)
A*gf3(?/(m+l),
t
k-[tnt]
The
in probability.
. Lemma
2
1-*
[m(i-a)]
will
proof
the sum in (3.35) as Consist
consider
shows
3.2
that
the
that
the third
first
jfe-[m(l-e)j
sum is negligible.
A similar analysis
a.s. by Ke*+?.
term is also bounded
m-1
(3.35)
m
[m(l-e)]
2
viz.,
0, we may
e>
...
in probability.
(pjl+p)
then
-
be
to demonstrate
of (3.7),
it is enough
to show that
? l(u) E(gs(u, W/p) du
since
complete
...
(3.36)
e is arbitrary.
=
except possibly
lm(y) exists and is continuous
By our assumption
L'm(y)
If lm(y) is continuous,
then bounded
for a finite number of points on (?, 1?e).
condition
ness of lm(y) along with
the fact g3(u, x) satisfies
(A') allows us to
: from the Glivenko-Cantelli
2.2 as follows
theorem,
apply Theorem
max
k
Theorem
Also from
A*;?lm(k?m+l)
|
\?> 0 with probability
2.2,
[m(i-?)l
2
m-1
gs(k!(m+l), nDk)
k-[me]
Hence
the
sum
1.
in (3.36) has
[ma-?)]
m-1
2
the
-
same probability
0^(1).
limit
as
lm(kl(m+1)) g3(kl(m+1), nDk)
k=[me]
which
from Theoren
Now
will
not
2.2 is the required
if lm( ) has a finite
create any problem
limit given
set of discontinuity
since the function
in (3.36).
inside (6, 1? e), this
points
is bounded
in this intervajt
36
LARS
on m.
S>
Take
J.
S.
RAO
in (0, 1) except at y = yQ. By our assump
right limits at this point and the point does
our
0 so that 0 < y0?8 < y0-}-S <
1. From
that lm(y) is continuous
Suppose
tions lm(y) has finite
left and
not depend
AND
HOLST
it follows that with probability
theorem
and the Glivenko-Cantelli
assumptions
and m is sufficiently
A
<8
one,
\
large. From
|k\m?y?
| a: | is bounded whenever
seen
to the sum
that
the
contribution
this it is easily
arguments
by analogous
(3.36)
of y0 can be made arbitrarily
in the neighborhood
small
a
It is obvious
of
finite
that the situation
8 sufficiently
small.
from
such
terms
by choosing
set of discontinuities
3.4
kind
can
be
on m.
depend
handled
This
same way,
the proof
the
completes
if
of
: Let
JV(D*)
be as defined
first
not
set does
the discontinuity
Lemma
3.3.
Lemma
the
of
in (3.18).
=
exp (Um-* [?y(nD*)-Efi{W?p)])
Then
under
->
E(JV(D*))
the conditions
of Theorem
3.1,
...
exp (ibt~ct2/2)
(3.37)
where
i
b=
...
cov {h{u, r?)9r?)l(u)du p/(l+p)
oI
(3.38)
and
c == ?varg^u,
0
JV(D*)
In Lemmas
?fiv(nD)]
=
exp
established
already
in probability
converges
to b.
Thus
(?m-*[p,(nD)?E/iv(Wlp)]))
g-^u, x) satisfies
asymptotic
gx{u, W?p))du)
.
...
/
...
(itm-^v(nD*)-/iv(nD)]+[j^v(nD)~E^v(Wlp)]).
3.1 to 3.3, we
?/(exp
Since
cov(W,
\ J0
(3.39)
can write
We
Proof:
W?p)du?(
normality
condition
(A'), Corollary
that
we
the first part m~*[/iv(nD*)
need only show that
-> exp
2V g^k/im+l),
k=i
nDk)
...
(-ct2/2).
2.1 of Section
of
fiv(nD)=
(3.40)
2 holds
(3.41)
and
the
TWO-SAMPLE NONPARAMETRIC
STATISTICS
37
var (gx(u ,W/p))
and cov (W, gx(u, Wjp))
2.2.
Further
by Theorem
as functions
of Lemma
in u, satisfy the conditions
2.1, so that as v ?> oo
is assured
. m
-War
A;=l
1 /
-> I v^r(g1(u,
Wlp))du~(
0
\
,
mm
v
2 gi(kl(m+l), Wk?p) 7-m~2
\
Nil /
1
cov2 ( 2 Wk, 2 g(k?(m+l), Wk\p)
v2
J0 cov (If, gx(u9W?p))du) /
...
-c.D
Lemma
i.e., for
: Under
3.5
almost
the assumptions
(3.42)
3.1, with probability
of Theorem
one,
every D*
KV(D*)= El exp [itm-*\ 2 hk(Sk)-/i(nD*))\
D*J
j
->exp(-dt2l2)
where
d = i E[g2(u, WIP)-9l(u, Wlp)fdu-p
0
: The
Proof
Theorem
2.1 hold
theorem
Cantelli
lemma will
be proved
by
and showing d = Ax?B\.
one
that with probability
M
2 Dl =
fc=i
U'M+m-*
...
( ) Eg3(u,
'
v 0
W?p)du)".
that
verifying
we
First
have
Lm(U'M) ^q^Pq
the
(3.43)
of
conditions
by the
Glivenko
...
(3.44)
=
from
[mq] and Uk is the ?-th order statistic
?7(0, 1). Clearly
conditions
and
since PQ = q?? 1 as g ?> 1?,
of
(2.9)
(2.11) of Theorem
part
a
For real numbers
and b, consider
2.1 hold.
where
M
h(u>3)
It is easy
to verify
that
if h(u,j)
=
ah(u,j)+bj.
satisfies
condition
...
(A), then
(3.45)
so does
hx(u,j).
Consider
M
= m-*2
W%+l),6)-^iW(w+l),6))
?
k=i
...
(3.46)
HOLST
LARS
38
fl9 ..., ?m are independent
it follows
that for some positive
AND
J.
S. RAO
and ?k is p(nD*k).
From
...
we
constants
have
cl9 c2,
where
the assumptions,
M
V(Q
= m-i 2 var
(^(?/(m+1), &))
M
2
<m?V1
[(kl(m+l))(l-kl(m+l))Y(nDl)e*+ct
k-i
Im
2
<c;
by the Holder
inequality
nDl
=
6
2 P/im+lJKl-^m+l))]^!):)2^!)
*=*i
<mr^c1
e
c
\
...
(nDl)*?m)5 +c8
and Lemma
From
2.1.
(3.47)
the assumption
(3.14),
-*
nDic+niLmiUi??LmiU't^))!
< nDk+KxDk m*+K2 D%rr*
...
< K3(mDk)+K2(mDk)*m*-*.
=
the
mDk
representation
c
numbers
that
for
>
0,
large
Using
lim
is finite with
probability
one.
m^1 2
?o
m ?>
it follows
(WkIWm),
1
As a >
(3.48)
by
the
i
with
\, we have,
using
{mDlp
one.
< m^X
2
i
Thus
with
(X3mZ>*+X2(mZ>*)a7n*-"-fl)C6+1
we will
verify
(Q <
oo.
...
(3.49)
>
0.
...
(3.50)
that
liminf
assumption
-> X4
one
probability
lim sup var
By
theorem
the binomial
i
probability
Now
law of
[mDjcY
mm
m-1 2
strong
(A), it follows
that
var(Q
there
exists
an interval
[a, b]C
(0, 1) and
a
u < 6, Again from the
integers jx ^ j2 such that hx(u,jx) ^ hx(u,j2) for <
TWO-SAMPLE NONPARAMETRIC
STATISTICS
and our assumptions,
strong law of large numbers
0 < C < D < oo, with probability
one
{k: a < k?(m+l) <b,
=H=
for n
Therefore
sufficiently
var >(Q
with
similar
one.
probability
fashion
for any
that
0.
large,
< *<
?>(w+D
Hence
it follows
seen
C < nD\ < D}/m -> Kx >
2
a(m+l)
it is easily
$0
(3.50)
var (A1(A/(m+1),&))/m > K%> 0
is satisfied
with
probability
one.
In a
that
m
?k)\%+'?m< oo.
lim sup 2 Elh^kKm+1),
*?i
the Liapunov
Therefore
condition
M
*
!a+e/( var (Q)
2 E |h1(k?(m+1), h)
i
M
= m-'2 2 ^
|A1(Jfc/(w+1),(t) !a+8/(var (Q)1+4/2 m
i
-> 0 as m?>
is satisfied
with
one.
probability
one.
probability
var
By
(Q
(?))*) -> #(0,
the next
->
(3.61)
Thus
??Ovar
with
...
oo,
Lemma
1)
3.6, we
have
that
in probability
(azAq+2abBqp^+b2qp-1)
?.
This verifies
that the assumptions
of
Aq ->AV Bq -J>B1 as q ?> 1
one.
2.1 are satisfied with
From
Theorem
the definition
(3.43)
probability
of d as well as the expressions
and
for
and
(3.55)
(3.54)
Aq
Bq, it follows
where
d=
which
proves
Lemma
Under
the
3.6
^1-?2
...
(3.52)
lemma.
: Given D*,
let (?l9 ..., i;m) be independent
3.1
Theorem
of
and
%k be ^(nDj).
the assumptions
m"1
m
2 var ^(?/(m-J-1),
i
?*)) -> a^Aq+2abBqp^+b^f-1
...
(3.53)
40
?. s. rao
where
in probability
=
Aq
and
and
holst
Lars
q
=
Bq
p-i
? E(g3(u,
o
Wjp))
...
du.
to those
in Lemma
3.1,
that,
2 A2(?/(m+l),j)[7r^Z>*)-^(nD^)]^0
fc=i
i=o
2.2, we
Theorem
Using
get
M
q
m-1 2 A2(?/(m+l),j)7T^Z)^->
o
j (Eg2(u,Wjp)) du
fc=i
Therefore
in probability.
M
q
?*) -> oJ (%2(^,Wjp)) du.
m-1 2 EWikKm+l),
i
other
The
terms
can be handled
analogously
=
c+d
where
(3.5),
: From
c+d =
(3.39) and
the definitions
(3.7), we
(3.6) and
...
(3.43) of c and d and from
identities
get
/
1
J var (g^t*,-W?)
du?
o
W
v2
1
J cov (If, ^(w, Wjp)) du)/
\ /0
Eg3(u,Wjp)?S /
1
1
\
v))) d^
J Fw(???|w(H% y))) du+ 0J Ew{Vn w(h(u,
0
-(!+/>)
==
(3.56)
(3.8) respectively.
+ 0} E\g?u,Wlp)-gt(u, Wlp)fdu-p(
=
the assertion.
proves
(T*
(3.43) and
in (3.39),
c, d, tr2 are defined
Proof
which
:
3.7
Lemma
By calculations
for instance
?
M
m-1 2
in probability.
it follows
(3.54)
(3.55)
=
Proof : Recall from (3.45) that hx(u,j)
ah(u,j)+bj.
similar
...
Wjp)]* du
f? EMu> Wlp)-gi(u,
[
? cov {h(u,y), y) pj(l+p)
1 r
J var (A(w,9/))dw?
0
i
du^
y?
cov
(A(w,?/), 9/)du
L 0J*
J p2j(l+p)
= <r2.
...
(3.57)
STATISTICS
TWO-SAM?LE NONPARAMETRIC
lemmas
These
lemma
following
Lemma
gives
: A
3.8
the origin
to
3.7
: We
condition
sufficient
have
0<
=
Lm(0)
0 < L'Ju)
<
for 0 <
s<
3.1.
of Theorem
proof
condition
for (3.14) to hold.
to hold
(3.14)
for
: Under
3.1
c -u*-1 for
The
in a neighborhood
of
u <
0<
(3.58)
c(t?-s?)lot-(Lm(t)-Lm(s)).
!> 0, the assertion
follows.
the null
(1.1),
of Vy defined in (3.13) is N(0,
...
e.
?
t<
du =
s| (cu^-L'Ju))
0 and L'm(u)
Corollary
This
the
complete
sufficient
a simple
is that
Proof
Since
3.1
4l
hypothesis
the asymptotic
distribution
1).
is a direct
of Theorem
3.1 and is obtained
consequence
by
1
u
0
in
This corollary
the null distri
<
<
l(u) ^ 0,
(3.15).
regarding
of Vv can also be reformulated
in the following
form using
interesting
taking
bution
Lemma
result
2.1.
Corollary
variables with
random
7?x,r\2, ... be a sequence of i.i.d. geometric
the asymptotic
Then
null distribution
given in (1.9).
of
: Let
3.1'
p d.f.
m
im
S hk(Sk) is N(E[
i
var
2
x i hk(7ik) ' ,
regression
See
finding
(3.10),
given
coefficient
?
also Holst
the optimal
i.e., a given
\
v
m
2
2
v i hk(7ik)-~? i 7?k' where ?
is
the
by
, m
=
/ m
cov
|
m
2 hk(7jk),2
v i
7/fcj
,m
>
Ivar ? 2
t/^J.
now
2. We
of
the problem
consider
(1979), example
choice of the function h(u, j) for a given alternative
sequence
sequence of functions Lm(u) with the property
Lm(u)
=
m*(Gm(F-l(u))?u)
as m->
-> L(w)
oo.
...
(3.59)
: If the sequence of alternatives
is such that the assumptions
of
test
3.1 are fulfilled,
most powerful
the
Theorem
then an asymptotically
(AMP)
of
the simple alternative
hypothesis
against
(3.10) is to reject H0 when
Theorem
3.2
m
2 l(k?m+l)Sk >
?;=i
a
12-6
c
...
(3.60)
LARS HOLST AND J. S. RAO
42
I is the derivative
where
statistic
of this optimal
l(kj(m+l))(Sk-ljp))
under
I
Arnr*
\
the alternatives
test which
I
?l\u)
(3.10)
.
/ ->X(/>-1
rejects HQ when
1
.
i
the asymptotic
c is determined
/r
it follows
that
(r?)
...
.
that
...
l(u)-j.
the AMP
by
(1972), we have
3.1 of Hoist
follow
distributions
asymptotic
3.1 for the above special case.
this result,
of the
-i*
the
3.1 and Corollary
power
(3.63)
1
var
(A(w, ?/), 7/) dw > /
=
...
J*(var h(u, t?))du
1 ,2
J cov
h(u,j)
on
(3.62)
J Z2(^) dw,-cra / ,
o
j cov (?(^, 7/), ?/) l(u) du /
as in Lemma
argument
when
is maximized
From
(3.61)
...
?=l
the same
results
,
that
3.1, it follows
m
2 h(kj(m-j-l),
Sk) >
Theorem
- Jf
The
...
(3.59)
satisfying
2
l(kj(m+l))(Sk~ljp))
k=i
Ph =
quantity
<r2)
du)(l+p)jp2
m
: From
Proof
Using
->X(0,
distribution
with
or*=
while
asymptotic
\m
i
cQlm-t 2
under H0
The
in (3.59).
of L, mentioned
is given by
(3.64)
this
(3.65)
directly
from
Theorem
test of level a is explicitly
given
by :Reject HQ if
r
rn-
r
, i
i
ni
J l*(u)duj(l+p)p^ ]> Aa
j V_l{kj{m+\){Sk-ljp) j j \m (
Aa is the
3.2 we
Theorem
where
standard
normal
of the iV(0,
upper a-percentile
find that the asymptotic
power
c.d.f. is given by the expression
Also
from
1) distribution.
of this test in terms of the
...
*(-*?+(
... (3.66)
Sl2Wduj(l+p)f).
(3.67)
TWO-SAMPLE
Relative
seen from Theorem
in using h(u, j) =
it is easily
Furthermore
(ARE)
Efficiency
NONPARAMETRIC
[0, 1] instead of the optimal h(u,j) =
e=
43
STATISTICS
3.1 that the Pitman
>j for
d(u)
some
Asymptotic
function
d on
l(u) j is
... (3.68)
/ f d(u)l(u)du
\2j { J d2(u)du-( } d(u)du\2}j J Z2(^)cfel
: translation
some appli
now consider
alternatives.
We
Example
on
we
cations of the above results
tests.
shall look at
First
non-symmetric
the translation
alternatives.
Let Xv
be absolutely
continuous
..., Xm_1
i.i.d. random
Let Yl9 ..., Yn be i.i.d. with
variables
with
distribution
F.
3.A.
d.f.
G.
to test
wish
We
:G(x) = F(x)
#0
against
the
of translation
sequence
:G(x) =
A&>
Let
f(x)
=
Lm(u)
=
and
iff'(x) exists
at the continuity
Uu)
tically
now
illustrate
optimal
test
Example
normal
L(u),
-> l(u)= -
6f\F-\u))lf{F-\v)).
based
3.2 may
on
...
to obtain
(3.70)
#'s then,
many
finitely
be used
(3.71)
the asympto
{Sk}.
or normal
der Waerden
...
say.
score
type
:
test)
For
the
d.f.
F(x)
we
=
of fr(F~\u))
how Theorem
(3.69)
as m ?> oo
except for at most
we have
statistic
(A van
:
...
F(x-d\m?).
-> -df(F~\u))
is continuous
points
=
Then
m\Gm(F-\u))-u\
And
We
Gm(x)
continuous.
be
Ff(x)
alternatives
=
=
<J>(x)
(2n)~*
? J aoexp
(-
\t^)dt,
-
oo <
x<
oo
find
-f'(F-\u))lf(F-\u))
It is easy to check
are
3.1
Hence
satisfied.
that
the
the AMP
T=
=
4>-H?).
conditions
required
regularity
on
test is based
the statistic
m
S (D-^m+l))?*
...
of Theorem
(3.72)
44
from
Theorem
3.2.
LARS
AND
J.
under
m
J x2y(x)dx
=
1and 2 O-^Jk/im+l)) = 0,
fc=l
-ao 0
the null
that
hypothesis
(T/w*) -> N(0,
From Theorem
3.2,
?
0(
Aa+0(1+/?)-*),
To find
test based
(3.73)
test relative
of the Wilcoxon
efficiency
Since
(3.68).
only need to calculate
(3.72), we
.J (2u-l)Q)-1(u)du
=
y(x) dx
f 2(?(x)-l)x
0 ? oo? oo
1
and
...
(l+p)?p2).
test
power for a one-sided
asymptotic
same as that of the Student's
?-test.
the
the
the Pitman
on
RAO
S.
facts
?
=
J (O-1^))2^
have
the
Using
1
we
HOLST
= 2
f
1/3,
o
test versus
Wilcoxon
to the optimal
dx =
(<p(x))2
=
J (<S>-\u)Y du
1, using
o
scores
the normal
formula
(3.68)
test
statistic
has
(3.72)
: scale alternatives.
Example
random variables
under scale
..., Yn be
Yv
i.i.d.
Next
the
scale
Lm(u)
=
the
consider
continuous
absolutely
Let Xv
..., Xm_x be i.i.d.
= 0. We wish
to test
alternatives.
=
F(0)
0(0)
G(y) with
...
F(x),
(3.75)
alternatives
:G(x) =
H[">
If the density
we
(3.74)
as
asymptotic
properties
der Waerden's
rank tests.
H0:G(x)=
against
...
same
the
of
test
type
and van
Fisher-Yates-Terry-Hoeffding
positive
and
F(x)
77-*
the ARE
e = 3/tt.
3.B
is
1
=
J (2u-~l)2du
The
a
of level
f(x)
=
F'(x)
m^GUF-^uV-u)-*
Gm(x)
=
is continuous,
L(u)
...
F(x(l+6?m^)).
=
then
-
(3.76)
as m ?> oo,
df(F~\u))
.
F~\u).
...
(3.77)
TWO-SAMPLE
And
if f'(x)
exists
to
analogous
NONPARAMETRTO
is continuous
and
45
STATISTICS
for finitely
except
many
then
points,
(3.71),
F-\u)jf(F-\u))}
-> if*)= -d[l +f'(F-\u))
Uu)
where
statistics
/' exists.
Optimal
case of translation
alternatives.
:
Example
distribution
F(x)
on
based
score
or exponential
(Savage
=
(1?e~x) for x > 0 we
l(u)
=
-
{Sk}
...
can be derived
For
test).
(3.78)
just as in the
the
exponential
find
...
#(l+log(l-*?)).
of Theorem
3.1 can be verified
assumptions
is given from Theorem
3.2, by
The
and hence
(3.79)
an optimal
statistic
m
T=
...
2 log (l-kj(m+l))(Sk-ljp).
(3.80)
fc=i
i
Since
oj
=
(l+log(l?u))2du
1 we
that
get
c?{Tjm*)-> N(0,
and
the asymptotic
that
...
(l+p)jp2)
is
power
...
?(-Aa+d(l+p)-i).
The
ARE
statistic
of Wilcoxon
statistic
is an approximation
T
(3.81)
to T
relative
to
the
Savage
(3.82)
in (3.80) above
is 3/4.
statistic
(see Lehmann,
has
which
4.
This
i.e.,
the
test for the above
power
asymptotic
deals
with
the
(3.82) as the statistic
theory
distribution
Asymptotic
statistics
on 2 Xkj
is the test based
2
k=*l
same
section
situation
class
for
of statistics
symmetric
symmetric
T
1975
n
m?1
p. 103). The UMP
The
fc=l
Yk
in (3.80).
statistics
in {Sv
..., Sm],
of the form
T*=
?h(Sk)
...
(4.1)
46
LARS
AND
HOLST
J.
S.
RAO
tests are an important
subclass
Such symmetric
for some given function
h(j).
tests and hence are suited for testing the equality
invariant
of the rotationally
is also
statistics
of two circular populations.
Clearly this class of symmetric
if the
in the last section.
Indeed
by the asymptotic
theory discussed
not
the
section
the
function
last
with
of
does
function hk(j)
k, i.e.,
vary
h(u,j)
we
is
of
then
obtain the symmetry
of
and
is a function
u,
j
independent
only
i
But since J l(u) du = 0, it follows from Theorem
in the numbers
[Sl9 ..., Sm}.
o
covered
distribution
of?7* under the sequence
3.1 that the asymptotic
3.1 and Corollary
Thus
the null hypothesis.
that under
of alternatives
(3.10) coincides with
are
that
alternatives
cannot
the
of
statistics
type (4.1)
distinguish
symmetric
at a
of n~* and
'distance'
Therefore
more
zero
power
we
:G&y) = y+Lmiy)?mW,
4?
close
such
against
have
efficiency
comparisons,
Let
with 8 =
1/4 in (1.11).
alternatives
distant
have
to make
in order
to
alternatives.
consider
the
1
0< y <
with
Lm(u)
For
this
assumptions
=
we will make
situation,
symmetric
:
(B*)
Assumption
is a function L(u), 0 <
: Assume
u <
...
mV\Gm{F-\u))-u).
is twice
Lm
is twice
1, which
(4.2)
the following
slightly
stronger
on [0, 1] and
differentiable
differentiate
continuously
there
and
such that
?(0)
where
Notice
V
=
define
D?
We
observe
?(1)
I and
to
analogous
Dk(l+Aljmv*)
that
under
max
1^
k^m
0,
sup
V denote
smooth
\Lm(u)-L(u)\
=
-
L" =
for such
that
sup
We
=
first
the
sup
=o(l),
and
also hold
=
o(l).
(4.3)
of L.
derivatives
second
\L'm(u)-l{u)\
...
o(l)
:
...
(4.4)
(3.12)
=
A*c
with
the above
|AJ| <
and
the following
alternatives,
(3.11)
=
\L?(u)--L\u)\
conditions,
we
\lm(u)\ < K <
oo.
regularity
sup
0^
m^1
...
[Lm(Uh)-Lm(Uk_x)]IDk.
(4.5)
have
...
(4.6)
NONPARAMETRIC
TWO-SAMPLE
The
STATISTICS
: Suppose
4.1
Theorem
that there exist constants
c? and
satisfy
that
(4.7)
(h(Sk)-Eh(7i))?m^cr ...
(4.8)
and
Assumption^*)
c2 such
...
\Mj)\ <c1(/2+l)/oraZZj.
Let Lm(u)
of the symmetric
distribution
theorem gives the asymptotic
following
T* under the alternatives
(4.2).
statistics
47
let
m
P;=
2
fc=i
where
a2 =
t? is
and
the
alternatives
var (h(7?))?[cov
random
geometric
...
(h(r?), r?)]2?var (r?)
variable
defined
in
(4.9)
Then
(1.9).
under
the
(4.2)
1) as
<?(V*)->N(A,
...
v -> oo
(4.10)
where
to
: Following
Proof
show that
the method
m~*
We
...
l2(u)du\ cov(h(7?), 7i(7}-l)-47ilp)p2l2(l+p)2(T.
A=[\
used
[fly(nD*)~ju,v(nD)]
in the proof
?>A
of Theorem
3.1,
(4.11)
it suffices
in probability.
have
m-*[ju,v(nD*)-fiv(nD)]
oo
m
= m-* 2
2 hU^n^nDD-rr^nDfc)]
oo
m
2 h(j)n}(nDk)[( 1+ A*/??1'*)/ exp -( raZ>*A?t/m1'4)
= m-* S
fc=l
?=o
_l_j_WjDA;A*/m1/*-{j(j-l)-2>2)&+(?I>i)2}A|/m*
m
+w-?>4 2
fc=i
??
+WI-1 2
oo
2 h(j) TrunD?iJ-nDjk) A*
y=o
00
Vm)n}{nDk){j(3-l)-23nDk+{nDk)*}
?%\2.
...
(4.12)
LARS HOLST AND J. S. RA?
48
some
After
direct
it may
calculations,
two terms on the RHS
of
be verified
tedious
of the first
limits
probability
but
(4.12)
are
that
the
zero while
that of the third term is
under
-
J l\u)du)
(
the assumptions,
(cov (%), v(v-l)-mP)
thus
=
or putting
hk(j)
h(j) -y- k
null
result on the asymptotic
1 in Theorem
0, 0 < u <
3.1', we get the following
the symmetric
statistics.
for
)
the proof.
completing
=
l(u)
Taking
in Corollaries
3.1,
distribution
-p2j2(l+p)2
4.1
4.1 : Let V*v be as defined in (4.8). Then under the null hypothesis
Corollary
a N(0,
1) distribution
(1.1) F* has asymptotically
h(
) satisfies
if the function
condition
(4.7).
3.2, we now
in Theorem
As
function
) for the
h(
the conditions
the form
of Theorem
: Reject H0 when
on the optimal
choice
of the
case.
symmetric
: For
4.2
Theorem
a result
consider
the sequence of alternatives
given by (4.2), satisfying
test is of
most powerful
4.1, the asymptotically
(AMP)
m
2 Sk(8k-l)>c.
...
(4.13)
k=l
Proof
test of the
: From
Theorem
4.1,
it follows
(4.1) is a maximum
Observe
that
form
maximized.
when
that
the
asymptotic
A given
the quantity
cov (r?, r?(7j?l)??7ijp)
=
0
power of a
in (4.11) is
...
(4.14)
and
var
where
?
?
(h(r?))
is the usual
cov2 (h(y), 7j)j var
linear
?
Therefore
2p-2(\+p)2Aj
we
can
regression
=
cov
?
(r?)
var
...
(h(7j)??ri)
(4.15)
coefficient
...
(h(r?), 7/)/var (r?).
(4.16)
rewrite
=
Jo l2(u)du cov (h(ri)-?ri, r){<r?-l)-4?///>)/[var (%)-^)]*
=
cor (%)-/??/,
<
[var (ti(V-l)-4?/p)]*
7i(t)-1)-4:7iIp)
=
[var M?/-l)-4^//>)]*
2p~*(l+p)
...
(4.17)
TWO-SAMPLE NONPARAM?TRIC
49
STATISTICS
with equality in (4.17) if and only if
for some
a and
real numbers
b.
=
maxi
0
4.1, we
Theorem
further
have
off
( [ 2^ ?*(?*-l)-2m//|
and
that
the asymptotic
from
the
r?(ri?l)
and
(4-18)
under H0,
1) ... (4.19)
of level a is
}nu)du)j(i+p)].
proof
we
see
cor2(h(7?)?
?y}
t?(t?? 1)??7\\p).
above
=
im^pp-^l+p)])^^,
?[-k+(
Further,
that
for a test
power
by h(7?)
Q
J* Z2(w)du?(l+p).
?
Using
is maximized
A
Thus
that
the
in using
ARE
Hh(Sk)
instead of 2 Sk(Sk? 1) is
e=
m
...
(4.20)
m
The statistic 2 SI which is equivalent to 2 Sk(Sk? 1) was proposed by
*=1 k**l
Dixon
(1940). Blumenthal
test while
and Weiss
Blum
and Weiss
(1957) also
show
(1976) discuss the ARE of this
(1963) and Rao
the consistency
properties.
LMP
is asymptotically
(1957) consider
that the Dixon
test
"linear" alternatives with density {l+c(y?|)},
shown
that Dixon
For
test
a nonnegative
is indeed AMP
if we
r,
integer
1
f
alternatives
I0
against
but we have
0<y<l(|c|<2)
against
Blum
of the form
(4.2).
define
for
x=
r
h(x)=<
...
(4.21)
otherwise,
then
T*v
is the
r.
statistic
This
A12-7
=
Qw(r), the proporiton
statistic
has been discussed
m
2
fc=i
h(Sk)
of values
in Blum
are equal to
among
{Sk} which
and Weiss
(1957) from the point
50
LARS
AND
HOLST
J.
RAO
S.
of consistency.
the asymptotic
Our results establish
normality
of alternatives
(4.2). After
H0 as well as under the sequence
tions we find from Corollary
4.1 that under the null hypothesis
A
~> N(0,
nQm(r)-pl(I+p)r+1])
of Qm{r) under
some computa
...
or2)
(4.22)
where
^2 = W(l+P)r+1}[l-(W(l+P)r+1){l+(^-
-.
(4-23)
run test
Let U be the number
(1940) is related to Qm(0).
of runs of X's and 7's in the combined
The hypothesis
H0 is rejected
sample.
of Qm(r), it follows easily that
when Ujm is too small.
From the definition
The Wald-Wolfowitz
<
\(Ujm)-2(njm)(l-Qm(0))\
Thus
the asymptotic
thus have,
distribution
and we
under
the ARE
as shown
in Rao
(4.24)
of 2p(l?Qm(0))
H0,
...
-> N(0, 4p/(l+p)3).
t?(mK(Ujm)-2j(l+p)])
Therefore
as that
same
is the
of Ujm
...
1/m.
of the run-statistic
the Dixon's
against
statistic
(4.25)
is pj(l+p)
(1976).
5.
Further
remarks
and
discussion
in this paper gives
is interesting
to note that the theory
developed
on
are asymptotically
to
the corresponding
which
equivalent
{Sk}
tests in all the known
in Section
3. For a unified
discussed
examples
It
tests
rank
based
to the theory of rank tests see Chernoff and Savage
(1958) or Hajek
approach
and Sidak
that in general,
(1967). We
given any rank test, one
conjecture
can construct
a test of the form (3.60) which
the same
has asymptotically
null
distribution
here
seems
and
to lead
power.
to much
If this
is the
test
case,
statistics
then
which
the
are
theory
linear
presented
in {Sk} as
simpler
to the corresponding
rank tests based on score functions.
compared
optimal
one can derive
are linear in the ranks
the fact that tests linear in
Using
{Rk},
{Sk}
the asymptotic
of statistics
of the form (3.60) from rank theory.
distributions
3.1 nor the fact that tests
results of Theorem
general
are
seems to follow from rank theory.
such
(3.60)
asymptotically
optimal,
Further
these
two
between
groups of tests is under investigation.
relationships
It may also be remarked
Theorems
that the theory presented
here, especially
4.1 and 4.2, covers many other tests that are not based on ranks as, for instance,
But
neither
the more
as
the run test and the median
test and seems
to be more
general
to that
extent.
TWO-SAMPLE
The
theorems
statistics
also
be
to
applied
the
censored
samples
in the X-sample.
in the same way
[(m-i)ff]
T=
2
...
l(k\(m+l))(Sk-l\p).
?=i
Under
test
similar
study
are
that the
For
censored.
instance,
suppose
samples
order statistic
at the right by
the [(m?l)q]-th
X[{m_1)qj,
as in Theorem
Under
the same assumptions
3.2, we obtain
is given by
test statistic
that optimal
when
are
can
here
presented
51
STATISTICS
NONPARAMETRIC
(5.1)
H0,
?(T\m?) -? N(0, f/ l2(u)du-(f l(u)du))*}(p+l)/p2)
and
the
on
results
Sobel
in rank
censoring
or Johnson
(1960)
... (5.2)
J
l2(u)du(/ l(u)du^\ (l+p)}V2)).
( l2(u)duj{[/
0(-Aa+
For
is
power
asymptotic
The
Acknowledgement.
led to many
improvements
instance,
Rao,
and
Savage
(1972).
in the presentation
comments
the referee whose
to thank
wish
authors
for
see,
theory
and Mehrotra
of this
paper.
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metrika,
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