A Nonparametric, Intersection-Union Test for Stochastic Order
by
Roger L. Berger
North Carolina State University
Institute of Statistics Mimeo Series No. 1685
May, 1986
I.
Introduction.
A distribution function (cdf) G(x) is said to be
stochastically larger than another cdf F(x) if G(x) " F(x) for all x with strict
inequality for some x.
~.1:
This relationship will be denoted by G
F.
stochastic ordering defines a situation in which, in a very strong sense,
observations from a population with cdf G tend to be larger than
observations from a population with cdf F.
To test if G
are
H~:
~.1:
G • F and
F, null and alternative hypotheses commonly considered
H~:
~.1:
G
F.
The Mann-Whitney-Wilcoxon rank sum test
(RST) is often recommended as a nonparametric test for testing
H~.
See for example Lehmann (1975).
H~
versus
Lee and Wolfe (1976) study a competitor
to the RST, based on a restricted maximum likelihood estimate, for testing
versus
H~
H~.
Using the asymptotic distribution of the RST statistic, it can be shown
that the power of the RST converges to one for any pair (F,G) for which
9
--=
= f-
Here 9
F(x)dG(x) ~ %.
The same is true of the Lee and Wolfe (1976) test.
P(X " Y) if X and Yare independent random variables with cdfs F and
G respectively.
If G
~.1:
F then 9
~
But there are mal1yother pairs (F,G)
%.
for which G is not stochastically larger than F and still 9
~
%.
If the model
includes such pairs, then rejection of the null hypothesis by the RST or Lee
and Wolfe test cannot be interepreted as convincing evidence that G
but only that 9
~
%.
~
more properly thought of as a test of
H~
F
Indeed, the power of the RST converges to 0 and oc
(the size of the test) according as 9
than as a test of
~.1:
versus
% or 9
H~ ':
= %.
Thus the RST might be
9 " % versus
H~ ':
9
~ %
rather
H~.
To develop a nonparametric test for which "rejection of the null
hypothesis" could be interpreted as convincing evidence that G
~.t.
F, one
might try to derive a size oc test of H~: G is not stochastically larger than F
1
versus H!: G ::...~ F.
Unfortunately, as we shall show in Section 4, this is not
The uniformly most powerful level a test of H~ versus H! is the
practical.
"no data" test which rejects H~ with probability a, regardless of the data.
The reason for this is that for any pair (F,G) satisfying H!, there is a
sequence of pairs (F,Gn ), n
= 1,2,...,
satisfying H! which converge to (F,G) in
such a way that the power of any test at (F,G) is the limit of the powers at
(F,G n ).
Since (F,G n ) satisfies H!, this limit must be at most a.
The proof of
this fact suggests that the difficulty may be that G(x) may be less than F(x)
for most x's but G(x) may be slightly greater than F(x) for some x in the tail
of the distributions.
Thus, although P(Y ::.. x)
=I
- G(x) ::.. I - F(x)
P(X ::.. x) for most x's, G is not stochastically larger than F.
=
In this paper we
consider null and alternative hypotheses which are similar to H! and H! but
which circumvent this troublesome tail behavior and allow the development of
a reasonable level a test.
Specifically, in this paper we consider testing
Ho :
F(x)
= G(x)
or
for all x
£
[a,b]
F(x) " G(x) for at least one x
£
[a,b]
versus
H.:
F(x)
III
G(x) for all x
£
[a,b] with strict inequality for
at least one x
£
[a,b] •
Here -. " a " b " • are fixed constants which define the interval of x values
on which F(x) and G(x) are to be compared.
The relationship described in H.
is the same as that which defines stochastic ordering except that the
inequality is required to hold only for x
£
[a,b] rather than for all x.
If the
interval [a,b] is reasonably wide, Ho and H. provide practical approximations
to H~ and H!.
Rejection of H o by a level a test (a small) will provide
convincing evidence that F(x)
III
G(x) for all x
2
£
[a,b].
In many situations Ho
and H. provide more meaningful and easily understood approximations to H~
and H~ than do the hypotheses H~' and H~' associated with RST.
In this
paper we consider the one sample problem in which F(x) is a known standard
and a random sample is available from G(x).
Research is currently being
completed on the two sample problem.
In Section 2, a level ex intersection-union test of H o versus H. is
described.
A table of small sample critical values and a large sample
approximation for the critical values are given.
of the test is provided.
prescribed value.
Also a graphical description
In Section 3, the size of the test is shown to be the
The test is shown to be the uniformly most powerful level
ex test among all monotone, nonrandomized, permutation invariant tests.
consistency class of the test is described.
The
The problem of testing H~: G is
not stochastically larger than F versus H~: G ~.~ F is considered in
Section 4.
2.
Derivation of the test.
Let F be a known cdf.
random sample from an unknown cdf G.
Let Y u"',Y n be a
Note that neither F nor G are
required to be continuous.
We will note in later results when continuity
simplifies something.
~
Let 0
~
ex
I be the desired level of the test.
the inverse of F in the usual way, namely, F- l (p)
~
F(x)
I for all x, define F- l (1)
F-l(l - ex l /
n )
"
= •• )
a " b ~ F- l (1).
= inf{x:
F(x)
~
Define
pl.
Let a and b be fixed values satisfying
Then a level ex test of H o versus H. is given
by the following.
First define constants Po'''''Pn- l by the relationship
(2.1)
P (B" i)
Pi
= ex,
(If
i
= O, ... ,n-l
,
where B is a binomial random variable with parameters Pi and n.
3
The
constant p
also depends on n and ex but that dependence will be suppressed
I
in the notation.
and ex
ci
= .01,
Table 1 provides values of p I for n
.05, and .10.
= F- 1 (PI)
simply requires that a
~
CI-1
1lI
a
Y(1)
1lI... 1lI
1Il
co.
CI and C,T-1 " b
i
= O,... ,n-l,
define critical values
Note that the restriction F- 1 (1 - ex 1/ n )
= ~1 (1).
and let Cn
= O,... ,n-l,
Next, for i
= 1,...,30,
a
1lI
Define integers I and J by the relationships
~
C,T'
Note that I
1lI
J since a
1lI
b.
Let
Y(n) denote the order statistics from the sample YU... ,Y n.
Then the
proposed test, that we will call test T, is defined as follows
(2.2)
o if and only if Y(i)
Reject H
Note that if 1
=J
= 1,... ,J-l
then the set i
rejects H o if and only if Y(J')
~
b.
c. , i
1
1Il
= I, •.. ,J-l
and Y(J)
~
b.
is empty and test T simply
But this corresponds to the atypical
situation in which the interval [a,b] is very short and C,T-1
1lI
a
1lI
b
~
C,T'
Typically the interval is wide so that H o and H. are similar to the stochastic
ordering hypotheses H~ and H! from Section 1.
discussion we will assume that I
to handle the 1
=J
~
J.
In all the remaining
The arguments can be easily modified
case.
= F- 1 (PI)'
Values for PI' that are used to define the critical values c i
= 1,... ,30.
are given in Table 1 for n
and Pn- 1
= (1
For larger values of n, Po
- ex) 1/n are easily calculated.
=1 -
ex 1/ n
For other values of i, a normal
approximation to the binomial distribution (with continuity correction) gives
reasonable approximations of PI as follows.
p
i + % - np.
B - np.
1
p. [ Jnp. ( 1-p. )
111
The approximation is based on
1lI
Jnp. (i-p.)
1
p
1
1
B - np.
1
[
p. Jnp.(l-p.)
1
1
1.
1lI Z
1
ex
where Zex is the lower (100ex)th percentile of a standard normal distribution.
So setting Zex
= (i+%-np, )/(nPI (1-Pi »%
and solving for Pi yields
4
=
p.
1
n + Z
2
a
Table 2 gives the actual and approximate values of Pi for n = 30 and a = .01,
.05, and .10.
The approximation is quite good, especially for i near %n.
approximation is within .005 of the actual value for 1
~
i
~
The
28 for a = .10.
The rejection region for test T can be related to the empirical cdf,
Gn(x) = n- l 1:'i'=l I(Y(d 1Ii x), of the sample Y1, ... ,Y n •
function.)
L
ci
•
The condition Y( i)
Similarly Y(.7)
~
~ Ci
(1(') is the indicator
is equivalent to Gn(x) 1Ii (i - l)/n for all x
b is equivalent to Gn(x)
~
(J-l)/n for all x " C.7'
These conditions define a step-shaped region as shown in Figures la and lb.
Test T rejects H o if and only if the graph of the empirical cdf lies entirely
The rejection region extends to
in the shaded, step-shaped region.
Figure la, if a
~. C 1
and, hence, I
no restriction on Y( 1 ).
But if c 0
~
2.
~
a
-CD,
as in
In this case the rejection region puts
L
c l' then the rejection region has a
finite lower endpoint, as in Figure lb.
Test T was originally derived using the intersection-union (IU) method of
test construction.
This method was used as early as 1952 by Lehmann.
More
recently the method has been used by a variety of authors including Gleser
(1973), Sasabuchi (1980), Cohen and Marden (1983), Berger (1984a) and Berger
(1984b).
The method has not always been identified as the IU method.
The IU method of test construction can be used if the null hypothesis
can be conveniently expressed as a union of sets.
In our problem, we
express H o as (U .1IiX~bGX)UG= where, for each x £ [a,b], G x = {G: G(x)
and G: = {G: G(x) = F(x) for all x
I:
[a,b]}.
~
F(x)}
The IU method prescribes that
each of the hypotheses H ox : G£G x as well as H o =: G£G: be tested with a level
5
ex test and H o is rejected if and only if each of the individual tests rejects
its hypothesis.
That is the rejection region for the test of H o is the
intersection of the rejection regions for each of the tests of Ho x and Ho:'
The usual test of Hox is based on the statistic B x : the number of sample
values Yi that are less than or equal to x.
B x is sufficiently small.
Pi-l " F(x)
.L.
Pi'
In particular, if
equivalent to Y( i)
all x
~
[C i -
1 '
"
x
.L.
c i , then
This is equivalent to Y(i)
~
c i " b, the requirement that Y( i)
.L.
1
Thus, from (2.1) we see that the level ex test of Hox rejects
Hox if and only if B x " i-I.
a
C i-
The hypothesis Ho x is rejected if
~
Cp
If Ci -
1
"
b
.L.
x for all x
C
~
~
x.
Now, if
[Ci-lIC i )n[a,b] is
n the requirement that Y( i)
c i )n[a,b] is equivalent to Y( i)
~
b.
~
x for
By the definition of I and
J, the only intervals of the form [Ci-uC i ) which intersect [a,b] are
[CI-l ,cI)"",[CJ'-l ,cJ')'
Thus the intersection over all x
~
[a,b] of the rejection
regions of the tests of the hypotheses H ox is the rejection region described
in (2.2).
test.
According to the IU method, H o: must also be tested with a level ex
But note that for any x .: [a,b], the test which rejects if Y(i)
-
-
also a level ex test for Box: G.:G x where Gx : (G: G(x)
~
F(x)}.
~
x is
Since Go:
C
Gx'
this test which was already included as the test for H ox is also a level ex test
of Ho:'
Thus, no additional test for Ho: need be included and, indeed, the
inclusion of a different test for Ho: would only serve to reduce the power of
the resulting test.
Some properties of the test T that we have developed using the IU
methodology will be discussed in the next section.
3.
Power properties of test.
In this section we will discuss three
properties of the power function of test T.
First we will show that test T is
a level ex test and give conditions under which the size is exactly ex.
Next we
will show that test T is a uniformly most powerful (UMP) level ex test in the
6
class of nonrandomized, monotone, permutation invariant tests.
Finally we
will describe the consistency class for test T.
3.1
Size of test.
Test T is a level
01
test of H o versus H..
In Theorem
3.1, a condition is given under which the size of the test is exactly the
specified value
level
01
This property is important because sometimes the use of
01.
tests in constructing a test using the IU method will yield an overall
test with size much less than
01.
Thus the constructed test will be overly
conservative.
Note, in particular, that the condition in Theorem 3.1 that
ensures a size
01
enough that I
~
test is satisfied if F is a continuous cdf and [a,b] is wide
J.
THEOREM 3.1.
Test T is a level
01
test of H o '
That is,
(3.1)
If in addition, for some j satisfying I
F(x)
= PJ
then test T has size exactly
PROOF.
Fix G
£
Ho '
Either G
£
~
j
01,
~
J, there exists an x such that
that is (3.1) is true with equality.
Gx for some x
£
[a,b] or G
£
G:.
G
£
Gx then, as explained in Section 2, the test will reject only if Y( I)
G
£
G=,
then, as also explained in Section 2, for any x
£
If
:10.
X
[a,b] the test for
(3.1) is verified.
To prove the equality in (3.1) note that the definition of c J as F- a (PJ),
the right continuity of F, and the existence of an x with F(x)
imply that F(c J
a ~ c J ~ b.
)
= PJ
Fix an x*
and c J + a
:10.
:10.
cJ •
= PJ
together
The definitions of I and J imply that
b and for any v satisfying 0 ~ v ~ l-p J define Gv(x)
7
as the cdf of the probability measure which puts mass p J + v on the point c J
and mass 1 - PJ - v on the point x*.
For every v, 0 ~
inequalities which define the rejection region in (2.2).
V
L.
I-PJ' G v
I:
Ho
If G v is the true
distribution, the inequalities corresponding to subscripts i satisfying
I
~
i
~
j are true with probability one.
satisfying j + 1
to
i
to J
The inequalities for subscripts i
are true if and only if Y(J+l)
that is, if at most j of the Y i'S equal c J •
= x*
(recall CJ + 1
::..
c J ),
But B, the random variable that
counts the number of Yi'S equal to c J ' has a binomial distribution with
success probability PJ + v.
Combining (2.1), (3.1), and the fact the Gv
I:
Ho
we have
P
Pj
(B to j)
= ex
.
Thus the equality in (3.1) is proved. c
3.2
Optimality property.
Test T is a UMP level ex tesf Of H o in the class
of nonrandomized, monotone, permutation invariant tests.
A nonrandomized
test is called monotone if (y uuo,y n) is in rejection region of the test and
y~
ill
test.
Y lJ i
= l,... ,n,
imply that (Y~,uo,y~) is in the rejection region of the
Monotone tests have monotone power functions in that if G ::...~ G* then
PG(reject H o )
ill
PG*(reject H o ) for any monotone test.
Optimality properties
such as this for IU tests iIi other problems have been discussed by Lehmann
(1952), Berger (1982), and Cohen and Marden (1983).
It is interesting to note
that the following results does not require test T to have size exactly ex.
The distribution F may be discrete and the test may have size less than ex
and still the test will be UMP in this class of tests.
8
THEOREM 3.2.
Test T is a UMP level ex test of H a in the class of
nonrandomized, monotone, permutation invariant tests.
PROOF:
We will show that any monotone, permutation invariant, level ex
rejection region is a subset of the rejection region for test T defined in
(2.2).
Hence test T is UMP.
We will show this by showing that any
monotone, permutation invariant rejection region which is not a subset of the
region in (2.2) has size greater than ex.
Consider a monotone, permutation invariant rejection region which is not
a subset of the region in (2.2).
Let (y 1I."'y n) be a point in this rejection
region which does not satisfy all the inequalities in (2.2).
Fix j,
I " j " J such that the inequality in (2.2) involving y (J) is not satisfied.
That is y(J)
that a
L
c J ' if j
L
and b
L
C:r
considering, F(y*)
L
PJ.
L
Cx
max(y(n),b).
= J.
J, or y(J) " b if j
Set y*
= max(y(J),a).
so that, regardless of which subscript j we are
Choose v ~ 0 so that PJ - v ~ F(y*).
Note that x* ~ y*.
£
Fix x* ~
Let G v denote the cdf of the probability
distribution that puts mass PJ - v on y* and 1 - PJ + v on x*.
Gv
Recall
H a since a " y* " band Gv(y*)
= PJ
- v ~ F(y*).
variable which counts the number of Yi's equal to y*.
binomial distribution with success probability p J -
V
L
Then
Let:B be the random
Under Gv ' B has a
P J'
Now consider sample points (n-vectors) consisting only of x*'s and y*'s.
Since x* ~ max(y(n),b) .. y* .. y(J)' by the monotonicity and permutation
invariance of the rejection region, all such sample points with at most j y*'s
are in this rejection region.
sup
GeH
PG(reject H )
0
0
Thus for this rejection region, by (2.1) we have
.. PG (reject H )
v
= Pp.-v (B " j)
0
~
J
.. P (B " j)
G
v
P
p.
(B " j) = ex.
J
This rejection region has size greater than ex as was to be shown. []
9
A nonparametric upper confidence bound for the cdf G might be used to
construct a level IX test of Ho •
A test which rejected Ho only if the 100(1-IX)%
upper confidence bound was less than or equal to F(x) for all x l: [a,b] would
be a level IX test of H o•
But the usual upper confidence bounds, such as
those described by Sandford (1985), would produce a nonrandomized,
monotone, permutation invariant test.
Thus, by Theorem 3.2, test T is a more
powerful test than a test constructed in this way.
Test T is not UMP among all level IX tests.
PG(reject H o )
~
IX if G(x)
even if G l: H..
= F(x)
Test T is biased in that
for a nondegenerate interval of x values,
On the other hand, the randomized "no data" test than
simply rejects H o with probabibility IX, regardless of the data, has
PG(reject H o )
= IX
for all G.
Thus the "no data" test has higher power than
test T for some G l: H•.
3.3
Consistency.
described.
Let "'I
In this section, consistency properties of test Tare
= inf.,"x,"b[F(x)
- G(x)J.
We show that the asymptotic value
of PG(reject H o ) depends on the value of "'I.
An interpretation of the
parameter "'I is given at the end of this section.
The following two theorems
relate the power of the test to "'I.
THEOREM 3.3.
For test T, if "'I " 0 then PG(reject H o ) " IX, for every
sample size n, and hence, lim n -+CD P G(reject Ho ) " IX.
THEOREM 3.4.
For test T, if "'I
PROOF OF THEOREM 3.3.
:10.
0 then lim n -+CD P G(reject Ho )
If "'I ~ 0, G(x)
:10.
= 1.
F(x) for some x l: [a,b].
G l: Ho' and by Theorem 3.1, PG(reject H o ) "IX.
If "'I
=0
and G(x)
Thus
= F(x)
for
some x l: [a,b], then as explained in Section 2, the test which rejects if
Y(i)
:10.
x (where C i -
1
"
x ~ c i ) is a level IX test of Hox:Gl:G x
10
= {G:G(x)o.F(x)}.
So also for this type of G we have PG(reject H o ) 6P G(Y( I)
following argument handles the remaining case, 7
X
6
ex.
The
and G(x)
L.
F(x) for all
[a,b].
£
Since [a,b] is compact, there exist
xm
=0
x)
It..
[a,b] such that
£
F(x o )
It..
limm~_ X m
= Xo
Xo
and
£
[a,b] and a sequence of values
lim~_
[F(x m) - G(x m)]
G(x o )' the right continuity of F and G imply that
Xm
= O.
Since
must approach
from below and
G(x-)
o
(3.2)
= lim
xlx
= lim
G(x)
xlx
o
= F(x-)
o
F(x)
o
Note that
o
(3.3)
L.
F(a)
6
F(x-)
o
= G(x-)
0
6
G(x )
x
~
0
L.
F(x ) 6 F(b)
0
L.
1.
Define a cdf H(x) by
G(X)/G(x:)
(3.4)
H(x) =
[
For any v
£
1
(0,1) define Gv(x)
= (1
x
o
- v)G(x) + v H(x).
First we will show that for any v
£
(0,1), Gv
Ho and hence, by
£
Theorem 3.1,
PG (reject Ho )
(3.5)
6 ex •
v
Recall that F(x;)
= G(x;)
lim [F(x ) - G (x )]
Dr"+-
m
vm
L.
1
= H(x;).
= F(x-)
0
Thus
- (1 - v)G(x-)
0
= v (G(x-) - 1)
o
Thus Gv(x m )
It..
L.
0 .
F(x m) for some x m, verifying that Gv
11
£
Ho and (3.5).
Xo
Now we will show that
(3.6)
lim P (reject H )
Gv
0
~
PG(reject H )
v.,j,O
0
which with (3.5) will complete the proof that PG(reject H o )
= (Yu...,Y n )
Y
from H.
Fix v
=
l;
(0,1).
= 1) = v.
with P(I t
Y*
be a random sample from G.
01
Let Zll...,Zn be a random sample
All the V's, Z's, and I's are mutually independent.
(Y~, ... ,Y~) is a random sample from the cdf G v •
= Y*
= o.
if B
~ P
= 1 +... +1 n•
1
Let A
= the
Define
rejection
Note that Band Yare independent and
Using Theorem 3.1 and the fact that G v
= P(Y*
I:
A) ~ P(y*
= 0) = P(Y
I:
A)P(B
(reject Ho )
G
Let
Let I 1 , . . . ,I n be independent Bernoulli random variables
region for test T and B
Y
~ 01.
I:
A, B
I:
H o we have
= 0)
v
= P(Y
I:
A, B
Taking the limit as v
.,j,
0 yields (3.6).
= 0) = PG(reject
Ho)·(l - v)n •
0
SKETCH OF PROOF OF THEOREM 3.4.
A proof of Theorem 3.4 can be based
on the graphical representation of the rejection region described in Section 2.
Let [a] denote the greatest integer less than or equal to a.
Weak Law of Large Numbers can be used to show that for any v
lim n-+_ P[nv]
= v.
This convergence is uniform for v
I:
Then the
I:
[F(a), F(b)].
(0,1),
These
facts can be used to show that the step-shaped boundary, with vertices at
the points (F- 1 (p t), i/n), converges to F(x) uniformly on the interval [a,b].
On the other hand, by the Glivenko-Cantelli Theorem, the empirical cdf
converges to G(x) uniformly on [a,b] with probability one.
"I
= inf."x"b[F(x)
F(x) - "1/2
~
- G(x)]
~
Thus, if
0, for large n the boundary is greater than
G(x) + "1/2 for all x
I:
[a,b].
12
Test T will reject H o if
Gn{x)
L
G{X) + "1/2 for all x
approaching one as n -+
OD.
I:
[a,b] and this event has probability
C
Ht:
Theorems 3.3 and 3.4 say that test T is a level ex test of
H:: "I ~ 0 and the test is consistent against all points in H:.
The parameter "I
= inf.6 x "b
0 versus
Thus the
parameter "I plays the same role for test T that the parameter 8
plays for the RST.
"I ..
= P{X
.. Y)
[F{x) - G{x)] may be
interpreted as follows:
"I ~
0 if and only if P{X " x)
and P{X
"I "
L
~
x)
P{Y " x) for all x
~
P{Y
L
[a,b]
I:
x) for all x
0 if and only if P{X " x) " P{Y " x) for some x
or P{X
L
x) " P{Y
L
L
[a,b]
I:
x) for some x
If F is continuous, then each of the inequalities involving
in the above interpretation.
[a,b]
I:
[a,b] •
I:
can be dropped
As mentioned in Section 1, we believe that in
many situations the hypothesis
m:
"I ~
0 provides a more meaningful
approximation to H!: G ~ _'to F than does H~': 8 ~ %, the alternative hypothesis
associated with the RST.
4.
UMP test of H~ versus H~.
There is no satisfactory level ex test for
testing H~: G is not stochastically larger than F versus H!: G ~_'t F.
level ex test of H!, PG{reject H~) " ex for every cdf G including all G
For any
I:
H!.
This means that the randomized "no data" test that rejects H~ with
probability ex, regardless of the data, and has PG{reject H!)
the UMP level ex test of H! versus H!.
= ex
for all G is
This fact is proved in the following
theorem.
THEOREM 4.1.
Consider testing H~ versus H!.
there exists a sequence of cdf's Gm , m
= 1,2,... ,
13
Let G be any cdf.
such that Gm
I:
Then
H! for every m
and, for any (possibly randomized) level
(4.1)
~
IX
lim P
lD""k>
PROOF:
m
= 1,2,... ,
Hm(x)
fix a value
= 1 for
x
~
Xm.
= (Yu...,Yn)
for any level
IX
y is observed.
Zm
~
test of H!,
PG(reject H:) .
The proof is similar to the proof of Theorem 3.3.
(l - lIm)G(Xm) + 11m
Let y
(reject H:)
G
m
IX
= (Zlm,,,,,Znm)
Xm
such that F(xm)
Let Gm
~
= (l
11m :!o. F(Xm).
- lIm)G
L.
be a cdf such that
+ (l/m)Hm• Then Gm(xm)
=
So Gm is not stochastically larger than F.
denote a sample point and let cjl(y) denote the test function
test.
That is cjl(y) is the probability H! is rejected given that
Let Y
= (Y
1 , ...
,Yn) be a random sample from G, let
be a random sample from Hm and let 11 m, ...,l nm be Bernoulli
random variables with success probability 11m.
l's are mutually independent.
Y f (l - 1 f m)'
Hm
11m and let
For each
Let y~
Assume all the V's, Z's and
= (Y~m""'Y!m)
where Y~m
Note that Y~ is a random sample from Gm•
and note that Y and B m are independent and Y
conditional distribution of y~ given B m
=0
= Y~
Let B m
if B m
= O.
= ZfmIfm +
= 1 1m
+...+ I nm
Thus the
is the same as the conditional
distribution of Y given B m = 0 and this is just the uncond.itional distribution
of Y.
Thus we have for every m
= 1,2,... ,
P (reject H*) = E cjl(Y*) = E(cjl(Y*)/B =O)P(B =0) + E(cjl(Y*)/B :!o.O)P(B :!o.O)
G
o
m
m
m
m
m
m
m
m
(4.2)
~ E(cjl(Y*)IB =O)P(B =0) = E+(Y)(l-l/m)n = PG(reject H*)(l-l/m)n.
m
Since Gm
£
m
m
o
H! for every m = 1,2, ... , P G (reject H!)
taking limits as m
m
~
• in (4.2) yields (4.1). c
14
6
IX
for every m.
Thus
In the above proof, since F(xm )
inf{x: F(x)
~
OJ.
That is, the
Xm
~
11m, we must have lim,..-+...
are in the left tail of F.
Xm
~
This suggests why
we were able to develop a reasonable test of Ho versus H..
These
hypotheses ignore the tails of F.
REFERENCES
Berger, R. L.
sampling.
(1982). Multiparameter hypothesis testing and acceptance
Technometrics 24, 295-300.
Berger, R. L. (1984a). Testing for the same ordering in several groups of
means. DesiPJ of Experiments: Ranking and Selection 241-249. Ed.
T. J. Santner and A. C. Tamhane. Marcel Dekker, New York.
Berger, R. L. (1984b). Testing whether one regression function is larger
than another. COJD1B. Statist. A - Theory Methods 13, 1793-1810.
Cohen, A. and Marden, J. (1983). Hypothesis tests and optimality properties
in discrete multivariate analysis. Studies in Econometrics, Time Series,
and MUltivariate Statistics 379-405. Academic Press, New York.
GIeser, L. J. (1973). On a theory of intersection-union tests.
Bulletin 2, 233. Abstract.
IMS
Lee, Y. J. and Wolfe, D. A. (1976). A distribution-free test for stochastic
ordering. J. Amer. Statist. Assoc. 71, 722-727.
Lehmann, E. L. (1952). Testing multiparameter hypotheses.
Statist. 23, 541-552.
Lehmann, E. L. (1975). NonparBl//etrics:
Holden-Day, Oakland.
Ann. Math.
Statistical Methods Based on Ranks.
Sandford, M. D. (1985). Nonparametric one-sided confidence intervals for an
unknown distribution function using censored data. Technometrics 27,
41-48.
Sasabuchi, S. (1980). A test of a multivariate normal mean with composite
hypotheses determined by linear inequalities. Biometrika 67, 429-439.
15
TABLE 1
Critical values Pi for « = .10
n
i
3
4
6
2
1
~
0 .9000 .6838 .~358 .4377 .3690 .3187
.9487 .8042 .6795 .5839 .~103
1
.965~ .8574 .7534 .6668
2
3
.9740 .8878 .7991
4
.9791 .9074
~
.9826
6
7
8
9
10
11
12
13
14
7
8
.2803 .2501
.4~26 .4062
.5962 .5382
.7214 .6554
.8304 .7603
.9212 .8531
.9851 .9314
.9869
9
.2257
.3684
.4901
.5994
.6990
.7896
10
.2057
.3368
.4496
.5517
.6458
.7327
.870~ .8124
.9392 .8842
.9884 .9455
.9895
11
.1889
.3102
.4152
.5108
.5995
.6823
12
.1746
.2875
.3855
.4753
.5590
.6377
.759~ .7118
.8308 .7813
.8952 .8458
.9505 .9043
.9905 .9~48
.9913
13
.1623
.2678
.3598
.4443
.5234
.5982
.6691
.7363
.7995
.8584
.9120
.9583
.9919
24
.0915
.1526
.2069
.2575
.3059
.3525
.3976
.4416
26
.0848
.1415
.1920
.2392
.2842
.3277
.3700
.4111
.4513
.4907
.5293
.5671
.6043
.6407
.6764
.7114
.7456
.7791
.8117
.8434
.8740
.9034
.9312
.9568
.9794
.9960
27
.0817
.1366
.1853
.2309
.2745
.3166
.3575
-; 39:74
.4364
.4746
.5120
.5488
.5849
.6204
.6552
.6894
.7229
.7557
.7878
.8191
.8495
.8789
.9071
.9338
.9585
.9801
.9961
28
.0789
.1319
.1791
.2232
.2655
.3062
.3459
.3845
.4224
14
.1517
.2507
.3372
.4170
.4920
.5631
.6309
.1423
.2356
.3173
.3928
.4640
.5317
.5965
.69~4
.658~
1~
.7568 .7178
.8149 .7744
.8691 .8280
.918~ .8782
.9613 .9241
.992~ .9640
.9930
n
i
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
16
.1340
.2222
.2996
.3712
.4389
.5035
.56~4
.62~0
.6822
.7371
.7896
.8394
.8862
.9290
.9663
.9934
17
.1267
.2102
.2837
.• 3519
.4164
.4781
.5374
.5945
.6496
.7027
.7539
.8028
.8494
.8932
.9333
.9683
.9938
18
.1201
.1995
.2694
.3344
.3960
.4550
.5118
.5667
.6198
.6712
.7208
.7686
.814~
.8582
.8994
.9371
.9701
.9942
19
.1141
.1898
.2565
.3186
.3775
.4340
.4886
.5413
20
.1087
.1810
.2448
.3042
.3607
.4149
.4673
.5180
.592~
.~673
.6421
.6902
.7367
.7817
.8249
.8661
.9049
.9405
.9717
.9945
.6152
.6618
.7071
.7509
.7933
.8341
.8731
.9098
.9436
.9731
.9947
21
.1038
.1729
.2340
.2910
.3452
.3973
.4477
.4966
.5442
.590~
.6356
.6795
.7222
.7637
.8038
.8425
.8794
.9142
.9463
.9744
.9950
22
.0994
.1656
.2242
.2789
.3310
.3812
.4297
.4768
.5228
.5675
.6112
.6538
.6954
.7358
.7752
.8133
.8500
.8851
.9183
.9488
.9756
.9952
23
.0953
.1588
.2152
.2678
.3180
.3663
.4131
.4586
.5029
.5462
.5885
.6299
.6703
.7097
.7482
.7856
.8218
.8568
.8903
.9219
.9511
.9766
.9954
16
.484~
.5264
.5674
.6076
.6468
.6852
.7228
.7~94
.7951
.8297
.8631
.8950
.9253
.9532
.9776
.9956
2~
.0880
.1469
.1991
.2480
.2947
.3397
.3833
.4258
.4673
.5080
.5477
.5867
.6249
.6623
.6989
.7347
.7697
.8038
.8368
.8688
.8994
.9283
.9551
.9785
.9958
.4~94
.4958
.5316
.5667
.6013
.6352
.6686
.7013
.7335
.7650
.7958
.8259
.8551
.8834
.9105
.9362
.9600
.9808
.9962
29
.0763
.1276
.1733
.2160
.2570
.2965
.3349
.3725
.4092
.4452
.4806
.5154
.5496
.5832
.6163
.6489
.6809
.7124
.7433
.7736
.8032
.8322
.8603
.8875
.9137
.9385
.9614
.9815
.9964
30
.0739
.1236
.1678
.2093
.2490
.2874
.3247
.3611
.3968
.4319
.4663
.5001
.5334
.5662
.5985
.6303
.6616
.6924
.7227
.7524
.7816
.8101
.8380
.8652
.8914
.9166
.9406
.9627
.9821
.9965
TABLE 1 - Continued
Critical values Pi for ex = .05
n
i
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
1
.9~00
4
6
2
3
~
.7764 .6316 .~271 .4~07 .3930
.9747 .8647 .7~14 .6~74 .~818
.9830 .9024 .8107 .7287
.9873 .9236 .8468
.9898 .9372
.9915
7
S
.3482 .3123
.~207 .4707
.6587 .~997
.7747 .7108
.8712 .8071
.9466 .8889
.9927 .9~36
.9936
10
9
11
.2831 .2589 .2384
.4291 .3942 .3644
.~496 .~069 .4701
.6~51 .6066 .~644
.7486 .6965 .6502
.8313 .7776 .7288
.9023 .8500 .8004
.9590 .9127 .8649
.9943 .9632 .9212
.9949 .9667
.9953
13
12
.2209 .20~8
.3387 .3163
.4381 .4101
.~273 .4947
.6091 .5726
.6848 .6452
.7~47 .7130
.8190 .7760
.8772 .8343
.9281 .8873
.9695 .9340
.99~7 .9719
.9961
15
14
.1926 .1810
.2967 .2794
.38~4 .3634
.46~7 .4398
.5400 .5108
.6096 .5774
.6750 .6404
.7364 .7000
.7939 .7563
.8473 .8091
.8960 .8~83
.9389 .9033
.9740 .9432
.9963 .9758
.9966
n
i
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
1~
16
17
18
19
20
21
22
23
24
25
26
27
28
29
16
.1707
.2640
.3438
.4166
.4844
.5483
.6090·
.6666
.7214
.7733
.8222
.8679
.9097
.9469
.9773
.9968
17
.1616
.2501
.3262
.3956
.4605
.5219
.5803
.6360
.6892
.7399
.7881
.8336
.8762
18
.1533
.2377
.3103
.3767
.4389
.4978
.5540
.6078
.6~94
.7088
.7~60
.8010
.8437
.91~4 .8836
.9501 .9203
.9787 .9530
.9970 .9799
.9972
19
.1459
.2264
.2958
.3594
.4191
.4758
.5300
.5819
.6319
.6799
.7261
.7703
.8125
.8525
.8901
.9247
.9555
.9810
.9973
20
.1391
.2161
.2826
.3437
.4010
.4556
.5078
.5580
.6064
.6~31
.6980
.7414
.7829
.8227
.8604
.89~9
.9286
.9578
.9819
.9974
21
.1329
.2067
.2706
.3292
.3844
.4370
.4874
.5359
.5828
.6281
.6719
.7142
.7550
.7943
.8318
.8676
.9012
.9322
.9599
.9828
.9976
22
.1273
.1981
.2595
.3159
.3691
.4198
.4685
.~1~~
23
.1221
.1902
.2493
.3036
.3~49
.4039
.4510
.4964
.~609
.~405
.6048
.6475
.6887
.7287
.7673
.8044
.8401
.8740
.9059
.~832
.6246
.6649
.7039
.7418
.7784
.8137
.8475
.8798
.93~4 .9102
.9618 .9383
.9836 .9635
.9977 .9843
.9978
17
24
.1173
.1829
.2398
.2923
.3418
.3891
.4347
.4787
.5214
.5629
.6032
.6424
.6806
.7176
.7536
.7884
.8220
.8543
.8851
.9141
.9410
.9650
.9850
.9979
25
.1129
.1761
.2310
.2817
.3296
.3754
.4195
.4622
.5036
26
.1088
.1698
.2229
.2719
.3182
.3626
.4054
.4468
.4870
.~439
.~262
.5832
.6214
.6586
.6949
.7301
.7644
.7976
.8297
.8605
.8899
.9177
.9434
.~643
.6016
.6379
.6734
.7079
.7416
.7743
.8060
.8367
.8662
.8944
.9210
.966~ .9457
.9856 .9678
.9980 .9862
.9980
29
27
28
.1050 .1015 .0981
.1640 .1585 .1534
.21~3 .2082 .2016
.2627 .2542 .2461
.3076 .2977 .2884
.3506 .3394 .3289
.3921 .3797 .3680
-.4323 .4187 .4060
.4714 .4~67 .4429
.~095 .4938 .4790
.5466 .5300 .5143
.~829 .5654 .~488
.6184 .6000 .~82~
.6530 .6338 .6156
.6869 .6669 .6480
.7199 .6993 .6797
.7521 .7309 .7107
.7834 .7617 .7411
.8138 .7918 .7707
.8432 .8209 .7995
.8715 .8492 .8275
.8985 .8763 .8547
.9241 .9023 .8808
.9478 .9269 .9058
.9690 .9497 .9295
.9867 .9702 .9~15
.9981 .9872 .9712
.9982 .9876
.9982
30
.0950
.1486
.1953
.2386
.2796
.3190
.3570
.3939
.4299
.4651
.4994
.5331
.5661
.5984
.6301
.6611
.6915
.7213
.7505
.7789
.8067
.8337
.8598
.8850
.9091
.9319
.9~31
.9722
.9880
.9983
TABLE
1 -
Centinued
Critical value. Pi fer
¢(
= .01
n
i
4
6
8
9
10
1
2
3
~
7
11
0 .9900 .9000 .7846 .6838 .6019 .53~8 .4821 .4377 .400~ .3690 .3421
.9950 .9411 .8591 .7779 .7057 .6434 .5899 .~440 .5044 .4698
1
2
.9967 .9~80 .8944 .8269 .7637 .7068 .6563 .6117 .5723
.9975 .9673 .9153 .8577 .8018 .7~00 .7029 .6604
3
4
.9980 .9732 .9292 .8791 .8290 .7817 .1378
.9983 .9773 .9392 .8947 .8496 .8060
5
6
.9986 .9803 .9467 .9068 .8656
7
.9987 .9826 .9525 .9163
8
.9989 .9845 .9572
9
.9990 .9859
10
.9991
11
12
.3187
.4395
.5373
.6222
.6976
.7651
.8254
.8785
.9241
.9610
.9872
.9992
13
.2983
.4128
.5062
.5878
.6609
.7271
.7871
.8412
.8892
.9305
.9642
.9882
.9992
14
.2803
.3891
.4783
.5567
.6274
.6920
.7512
.8053
.8543
.8981
.9360
.9669
.9890
.9993
27
.1568
.2217
.2766
.3264
.3727
.4166
.4584
-.49:84
.5370
.5742
.6102
.6450
.6787
.7113
.7428
.7732
.8024
.8306
.8574
.8830
.9071
.9295
.9500
.9682
.9834
.9944
.9996
28
.1517
.2146
.2679
.3162
.3613
.4039
.4447
.4837
.5214
.5578
.5930
.6271
.6602
.6922
.7233
.7533
.7824
,8104
.8373
.8630
.8875
,9107
.9322
.9519
.9694
,9840
.9946
.9996
29
.1468
.2079
.2596
.3066
.3505
.3920
.4317
.4699
.5066
.5422
.5767
.6101
.6426
.6741
.7047
.7343
.7631
.7909
.8177
.8435
.8682
.8917
.9140
.9347
.9537
,9705
.9845
.9948
.9997
12
13
14
15
.2644
.3679
.4532
.~28~
.5969
.6~97
.7177
.7713
.8205
.86~4
.9056
.9406
.9693
.9898
.9993
n
i
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
16
.2501
.3488
.4305
.5029
.5690
.6299
.6866
.7393
.7883
.8335
.8749
.9122
.9446
.9713
.9905
.9994
17
.2373
.3316
.4099
.4796
.5434
.6025
.6577
.7094
.7578
.8029
.8448
.8832
.9178
.9481
.9731
.9910
.9994
18
.2257
.3160
.3912
.4583
.5199
.5772
.6309
.6814
.7290
.7737
.8156
.8546
.8904
.9228
.9512
.9746
.9915
.9994
19
.2152
.3018
.3741
.4387
.4983
.5538
.6060
.6553
.7020
.7460
.7876
.8267
.8632
.8968
.9272
.9539
.9760
.9920
.9995
20
.2057
.2888
.3583
.4207
.4783
.5321
.5829
.6309
.6766
.7199
.7610
.7999
.8366
.8708
.9025
.9312
.9564
.9773
.9924
.9995
21
.1969
.2768
.3439
.4041
.4598
.5120
.5613
.6082
.6528
.6953
.7358
.7743
.8109
.8454
.8777
.9075
.9347
.9586
.9784
.9928
.9995
22
.1889
.2658
.3305
.3887
.4426
.4933
.5412
.5868
.6304
.6721
.7119
.7499
.7862
.8207
.8532
.8838
.9121
.9379
.9606
.9794
.9931
.9995
23
.1815
.2557
.3181
.3745
.4267
.4758
.5224
.5669
.6094
.6502
.6892
.7267
.7626
.7969
.8295
.8603
.8893
.9162
.9408
.9624
.9804
.9934
,9996
18
24
.1746
.2462
.3066
.3612
.4118
.4595
.5048
.5482
.5897
.6295
.6678
.7047
.7401
.7740
.8065
.8375
.8668
.8944
.9200
.9434
.9640
.9812
.9937
.9996
25
.1682
.2375
.2959
.3488
.3979
.4443
.4884
.5306
.5711
.6100
.6476
.6837
.7186
.7521
.7844
.8152
.8447
.8727
.8990
.9235
.9458
.9655
.9820
.9940
.9996
26
.1623
.2293
.2859
.3372
.3849
.4300
.4729
.5140
.5535
.5916
.6284
.6639
.6982
.7312
.7631
.7938
.8232
.8513
.8780
.9032
.9266
,9480
,9669
.9827
.9942
.9996
30
.1423
.2016
.2519
.2976
.3403
.3808
.4195
.4567
.4927
.5274
.5612
.5939
.6258
.6568
.6868
.7161
.7445
.7720
.7987
.8245
.8492
.8730
.8956
,9170
.9370
.9553
.9715
,9851
.9950
,9997
TABLE 2
Comparison Ot actual and approximate
critical values Pi tor n = 30
i
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
«=. 10
approx.
actual
.0739
.1236
.1678
.2093
.2490
.2874
.3247
.3611
.3968
.4319
.4663
.5001
.5334
.5662
.5985
.6303
.6616
.6924
.7227
.7524
.7816
.8101
.8380
.8652
.8914
.9166
.9406
.9627
.9821
.9965
.0802
.1282
.1715
.2124
.2515
.2894
.3264
.3625
.3979
.4326
.4667
.5003
.5334
.5660
.5981
.6297
.6608
.6914
.7215
.7512
.7802
.8087
.8365
.9636
.8899
.9152
.9392
.9616
.9815
.9967
«=.05
approx.
actual
«=.01
approx.
actual
.0950
.1486
.1953
.2386
.2796
.3190
.3570
.3939
.4299
.4651
.4994
.5331
.5661
.5984
.6301
.6611
.6915
.7213
.7505
.7789
.8067
.8337
.8598
.8850
.9091
.9319
.9531
.9722
.9880
.9983
.1423
.2016
.2519
.2976
.3403
.3808
.4195
.4567
.4927
.5274
.5612
.5939
.6258
.6568
.6868
.7161
.7445
.7720
.7987
.8245
.8492
.9730
.8956
.9170
.9370
.9553
.9715
.9951
.9950
.9997
19
.1110
.1601
.2044
.2460
.2857
.3239
.3609
.3969
.4321
.4665
.5002
.5332
.5655
.5973
.6284
.6590
.6890
.7184
.7472
.7753
.8028
.8296
.8556
.9807
.9048
.9278
.9492
.9688
.9857
.9977
.1798
.2283
.2724
.3137
.3530
.3905
.4267
.4617
.4957
.5287
.5608
.5921
.6226
.6523
.6813
.7095
.7370
.7637
.7897
.8149
.8393
.9628
.8853
.9068
.9271
.9460
.9632
.9794
.9907
.9987
REJECTION REGION
IN TERMS OF EMPIRICAL CDF
1
.
F(x)~
/
FiguXl&e
~a
1
ij
F(x)~
FiguXl&e 1h
F(x)=l-exp(-x)
20
alpha=.l
n=2Q