ON THE ASYMPTOTIC OPTIMALITY OF UI-LMP RANK TESTS FOR RESTRICTED ALTERNATIVES
*
MING-TAN M. TSAI AND PRANAB KUMAR SEN
University of North Carolina at Chapel Hill
SUMMARY. For testing against restricted alternatives, the optimality of UI-LMP
rank tests is characterized in the light of the asymptotic most stringency and
somewhere most powerful character. Under additional regularity conditions, this
characterization is also extended to the multivariate case. Some allied asymptotic optimality results on the UI-LMP rank tests are considered in the same vein.
l. INTRODUCTION
Host of nonparametric tests against ordered, orthant and other forms of
restricted alternatives have mostly been considered on ad hoc basis. The unionintersection (UI-) principle of Roy (1953) has also been effectively employed by
Chatterjee and De(1972,l974) and Chinchilli and Sen(198la,b), among others. These
authors have studied the (asymptotic) power superiority of the UI-rank tests to
their global versions. However, these were done mostly under rather restrictive
conditions on the associated covariance matrix and noncentrality vector, leaving
open the possibility fdr relaxation of some of these conditions. Sen (1982) incorporated the theory of locally most powerful (LMP-) rank tests in the UI-setup •
However, the resulting tests may not have the asymptotic optimality in a conventional sense. This naturally raises the question on the asymptotic optimality of
UI-LMP (and other related UI-) rank tests in a well defined manner. Our main
contention is to provide an affirmative answer in the light of asymptotic most
stringency and somewhere most powerful character of these tests.
AMS Subject Classification Nos : 62GlO, 62G20, 62G99, 62H15.
Key Words and Phrases: Kuhn-Tucker-Lagrange formula; likelihood ratio test; locally
most powerful; most stringent somewhere most powerful; multi-sample problem; rankpermutation principle; uniformly most powerful; union-intersection principle.
* Work partially supported by the Office of Naval Research, Contract No. N0001483-K0387.
RUNNING TITLE:
Optimality of UI-rank test for restricted alternatives.
-2-
Section 2 deals with the basic regularity conditions, preliminary notions
and a general formulation of such restricted alternatives where UI-rank tests
have been considered. Asymptotically optimal rank tests for restricted alternatives
in the univariate case are derived in Section 3. This characterization is extended
to the multivariate case in Section 4. Some further remarks on the asymptotic
optimality of UI-rank tests are appended in the concluding section.
2. PRELIMINARY NOTIONS
Xl""'~
Let
be independent random variables (r.v.) with continuous distri-
bution functions (d.f.) F , ... ,F , respectively, all defined on E = (_00,00). Suppose
l
N
that F, has an absolutely continuous probability density function (pdf) f., where
~
~
f.(x)
= f(x;sN')
where sN'=
(BlcN'l"
.. ,B q c N~q
' ), i=l, ... ,I\I,
~
~J. ~
~
~
~
(2.1)
,
the c . = (cN'l"" ,c , )
~ N~
~
N~q
,
6 = (6 , ... ,6)
~
q
1
are q-vectors of known constants, not all equal, and
is a q-vector of unknown parameters; q
~
1. [ To be more precise,
we should have started with a triangular scheme of r.v. 's and d.f. 'so However, for
the sake of notational simplicity, this refinement will be suppressed.]
In a nonparametric testing problem, one is usually interested in the null
hypothesis (H ) of the homogeneity of the F , i.e.,
i
O
H : FI= ... =F = F (unknown)
O
N
or,in (2.1),
(2.2)
6 = 0 .
In testing for H against a global alternative, we allow (3 to be arbitrary. On the
O
other hand, in testing for H against a restricted alternative, we need to pose
O
the parameter space
(of
a)
a positively homogeneous set
in a more structured way. We assume that there is
r ,
such that under the alternative
(2.3)
A 6 > 0 , where A
is of order a Xq and
o
is of full rank a (l<a <q) }.
o
The simplest form of
r
-
0-
--
relates to the orthant alternative:6 >0 i.e.,
positive orthant in Eq . Other notable forms of
~
r
r
=E+
q
, the
relate to the ordered al ternative:
6 <... < 6 , with at least one strict inequality, umbrella alternatives, tree
1 -
-
q
alternatives, loop alternatives etc. These alternatives are all special cases of
r ,
defined by (2.3), and often,in practice, they appear to be more realistic than
-3-
S~
the global alternative that
of restricted alternatives
r,
0 . Hence, we confine ourselves to a general form
and study the asymptotic optimality of UI-LMPR tests
for this problem.
Generally, for the nonparametric hypothesis testing problem, rank tests are
distribution-free (and hence, valid for a broad class of d.f. 's). However, they
are based on the use of some scores , suitably chosen to maintain the high power
property for an important subclass of such d.f. 'so In testing for H against a
O
global alternative, a convenient way to choose such scores is to maximize the local
(asymptotic) power of the test for a specific family of d.f. 'so In the literature,
this is referred to as the locally most powerful rank (LMPR-) test. A very elegant
treatment of LMPR tests is given in H{jek and Sidak (1967). We shall find it
convenient to adapt their notations for the restricted alternative case too.
First, note that for the o~thant alternative problem, we have
For any given Y
H
*
s=
=
*E
Y
=
element~),
{H
>0 }.
we denote by
(2.4)
O.
yy r H~
Also, for every
H
(y , ... ,y ) (with nonnegative
q
l
oy , for some arbitrary and positive
Y
Then, we have
H*
=
r = {6: 6
(2.5)
E > 0 , we let
*
Y
;O<O<E
}
*E
H
and
u
=
YE r
*E
Y
(2.6)
H
The basic idea is to incorporate the LMPR test theory for a given Y , and taen to
use the UI-principle to have the test constructed for the entire class in (2.5).
Towards the formulation of such UI-LMPR tests, we make the following assumptions:
[ Al ] For every N, we d e f ine
~N = «
CNkk
,»
=
~N
=
and let
N -l~N
~i=l~Ni
L~=l( ~Ni
-
~N
)(
~Ni
-
~N)'
(2.7)
.
We assume that there exists a positive definite (p.d.) matrix C , such that as
N increases, N-1C
-N
converges to C , and further
.
{max
l~m~
l<i<N
- '-1}
(~Ni - ~N) ~N ( :Ni - ~N)
[A2] Let us define f.(x;Oy) as in (2.1) with
~
i( ~l). f.(x;Oy)
_
~
-
S
=
/
(2.8)·
0 .
= oy, i > 1. Then, (i) for every
is absolutely continuous for almost all x and
Y E
A..
r.so that i f
-4-
we let ~ie(x;~) = (a/a~)fi(x;~) and f. (x;oy) = (a/a6)f.(x;oy), then, for
.
-,.
f. (x;cy) = y f.e(x;Oy), y
~y
-
. ~y
-
f. (x;O) =
-~
.
-
--1
UT
~
hm.t'LO 0
-
If.
lim.t'IO;oo
UT
-00
~y
-
€
r, x
~-
E. (ii) For almost all x
€
and
~_
-
If.
~y
,
€
r,
(x;O)
_ Idx is finite. (iv) The largest character-
-,
(a/a~)logf(X;~)(a/a~ )logf(X;~)l
= E [
e
e =6y
y, the limit
[f.(x;oy) - f.(x;O)l exists. (iii) For every y
(x;oy)
_ Idx =;00
_00
~(f)
istic root of
-
~y
is finite.
Consider now the score generating function (vector) :
~
*(x)
and let
=
{f(x;~)}
~(.)
= ( a
*
-1 .
~e(x;~)
Nl
*
= ( flex) , ... ,fq(x) ) , x
(.) , ... ,a
,
(.»
Nq
€
E,
(2.9)
with
aN.(i) = E{f~(UN*')}, i=l, ... ,N , j =l, ... ,q
J
J
(2.10)
~
where U* , ..• ,U * are the ordered random variables of a sample of size N from the
Nl
NN
d.f. F ( for which the p.d.f. is f(x;O». Finally, let ~_.
be the rank of X1 among
-~1
Xl' ...
for i=l, ... ,N. Then, we define the vector of linear rank statistics
'~'
TNj =
N -~
"N
:N
V'
,
= ( T
, .•. , TNq )
nl
For every (fixed)
"
*
N
~i=l c Nij aNj(~i) = ri=lcNijaNj(~i)' j=l, ... ,q;
y
€
r ,
(2.11)
•
on letting TN(y) = y':N' and proceeding as in
/.
Hajek and Sidak (1967) and Sen (1982), it follows that the test with the critical
region WN(y) =
{(\'." ,~) : TN(y)
~
kN(a)} ( with suitable critical value kN(a»
is the LMPR test for H against H* , at the level of significance a ( a < a < 1).
y
a
,
Note that for every
~ =
«
vNjj,N
V
Njj
-1
r
€
f , E[TN(X IHal = 0 and Var[TN(X) IHOl = X ~y , where
=«
CNjj ,»
*.
w1th
VNjj'CNjj'»
~N =
«
V
Njj ,»
' = (N_l)-l r~=l aNj(i)aNj,(i) , for j,j' =l, ... ,q.
defined by
(2.12)
Thus, if we let
(2.13)
then, in accordance with the VI-principle of Roy (1953), as in Sen (1982), we
•
may consider the following UI-LMP rank statistic
= sup{
*
TN~)
: J
€
f
(2.14 )
}.
As has been mentioned earlier, we consider the case of fof the form in (2.3),
so that for the c.omputationof Q
n
~ = ~
Y~ ~
and
Y'~Y
= 1.
in (2.1t.), VIe
If we let h(y) =
npen to
,
-r :N
ml'lximizE? ~''E", ~I!r;~,..t t--
' hl(y) = - A Y and h 2
(r)
-5-
=
Y
~Y
- 1, then incorporating the Kuhn-Tucker-Lagrange (K.T.L.) point formula
theorem, we arrive at the following solution. Let
U = A M-IT
=
and
-::"'~-N
-N
-1
A M
-
:.~
=
and also let J be any subset of P
A
I
(2.15)
{l, ... ,a }
o
a
of the 2
and J
I
be its complement. For each
set J , we partition (following reorganization, if necessary) U and 6
0
~N (~N(J~
k(J')
)
Also, fo reach J
U
and
)
=
~N(J
~N = (~N(JJ) ~N(JJI»)
k(J)
P
~
J
..s
6
I
-N(J J)
!1
I
I
_N(J J )
, we let
P
(2.17)
I
-N(J:J )
6
(2~:6)-'
as
(2.18)
I
-N(JJ:J )
Then [viz., Sen (1982)J, we have
'{M
-1 - M-1AI 6-1A M
-l}T
T
-N ::...~
::"'N - -N - -N
-N
+
PS~SP {~N(J:J')~~~JJ:J')~N(J:J,)}I(~N(J:;'» O)I(~~~JIJI)~N(JI)
<0), (2.19)
where I(B) stands for the indicator function of the set B. Our main interest is
to study the asymptotic optimality of this UI-LMPR test.
3. ASYMPTOTIC OPTIMALITY OF THE UI-LMPR TEST
As
a first step, we shall study the asymptotic power equivalence of the UI-LMPR
test and the UI-LR (likelihood ratio) test for local alternatives. Towards this,
we define
for j =1, ... , q .
Note that the summands in (3.1) are
indep~ndent,
(3.1)
so that the central limit theorem
(in the multivariat2 case) can easily be invoked to conclude that for every y
* =
SN(Y)
where
EN
QO
N
=
=
(r
,-
~N
y) -~
* , CjJjj d)
« CNjj
with I( f)
*
: Y
SN(Y)
sup{
~)
Y ~N
E:
r
N(O,l)
=
«cp...
,».
JJ
[under HOl
,
P
(3.2)
If we let
},
then, by virtually repeating the proof of Sen(1982), we obtain that as N
-+
E:
a , under H as well as a sequence of contiguous
O
-!<:
alternatives { K : S = N 2 Y l
E: r} .
N
(3.3)
-+00,
(3.4)
r ,
-6o
Now, QN corresponds to the UI-version of the efficient scores statistic,
and is asymptotically (power-) equivalent to the UI-LR test statistic ( for
restricted alternatives) , under H as well as
O
{~}
. As such, we conclude that
the UI-LMPR test based on the specific scores ~N(') in (2.10) is asymptotically
power-equivalent ( under {K } ) to the UI-LR test, and thus the two tests share
N
the same asymptotic optimality properties. Note that
tildt there ex:'st p.d. matrices
~J:J'
As sucn, defining
elc.C for every y
and
E:
<
f
o .
and
I'[
and 6-<1,j
T
T
_,
: e}
6, such that N..
-~';
*
E
~~
~
:\1
l.
~
C (p.d.) as N ~
an d
"
u,
..... :"
p. uA ,as .,,.
~
-.co •
.....
as in (~.lJ) and (L.l~), we may conclude
} tilere is only one J : 0 ~ J ~ P , such that
Then the
so
00,
uI~L~~R
a
~J:J' >
test has asymptotically the best constant
power over the ellipsoid in the parameter space specified by y'{M-1 -M-1'
A 6 -1 AM -1 }y
.....
+ ~J:J'~JJ:J'~J:J'
= constant,
against the restricted domain
.....
.....
..........
--
.....
and is also the asymptotically most stringent test
f. However, it is generally not the asymptotically
most powerful test against the entire domain of alternatives in
f; in fact,
there may not be any such optimal test for restricted alternatives. Towards our
f for which the UI-L~R test has
main objective of locating a subspace of
asymptotically the best power. we consider the following.
THEOREM 3.1. For each J ( P SJS: P), let f
and
fa
= n~'::Js.p
fJ ,
J
={
~
= Ay
E:
+a
Eo, ~J:J' > 0 }
where E+ao is ao-dimensional positive orthant space. For
testing (2.2) against (2.3) ( under local alternatives in (3.4)), the UI-LMPR test
in (2. 19) is asymptotically most stringent for
powerful for
fo ,
and is asymptotically most
f
at the respective significance level
a.
Proof. Consider a r.v. W having a q-variate normal d. f. with mean vector My and
.~
~-
a (known p.d.) dispersion matrix M • Let Z
"""J
"'_
= AM-~
1: {ZJ'J'
f1=.J,=P - .
• and define
....,
....1
~J:J'~:J/} I(~J:J' > 0).
-1
I ( ~J ' J' ~J'
< 0).
(3.5)
where the partitioned matrices and vectors are defined as in (2.16)-(2.19). Then,
by virtue of (3.2)-(3.4). it suffices to show that for testing H : y = 0
O
H:~ >
1 - -
a , the test with the critical region Q* -> ka
against
is the most stringent one
-7-
for f and most powerful for f , where f and f
o
0
are defined in the theorem. For
this, it suffices to show that the shortcoming of Q* is equal to 0 for some
'(
inside the positive orthant f (excluding the null point). Towards this, we show
that Q* is power equivalent to the most stringent test for testing H :
O
against Hi:
~p.
~~,
where
~
is a given qXq matrix of real constants. Since 6
is p.d., there exists a upper triangular matrix B such that B6B'
a diagonal matrix. For each J (0 ~
so that D
-
=
/
DJ
-
\'
~JJ ' \
D.t
-J 'J :1
_J ' J
~JJ:J'
We also let Y
~JJ
D
=
J--= P),
=
(
we take
~J0J: J '
-
B
-
=
~)
6
B
-J
=(
D (say) is
-
-1
~JJ'~J'J')
E
0
I
• Then we have
/
-J ' J '
= ~JJ:J '
= ~JJ
r=0
(3.6)
BZ and partition Y as in (2.15), so that Y' =
'
(~J
~J'
).
Then
we may rewrite (3.5) as
Q* = W'(M- l _M- l A'6- l AM- l )W +
2:
fJ-=J<-.;P
Then the UI-efficient score test for H : Y = 0 against HJ:
O
~J~
> 0 is given by
(3.7). Thus, for each J( fJ.:;·J':=,P ), the most powerful property holds for Q*
whenever Y lies in
r
n fJ
, so that Q*is most powerful for the region
( which is) by definition,
f ), and hence, the desired
o
result follows. Q.E.D.
Note that in the characterization of the asymptotic optimality of the
UI-L~PR
test, the UI-efficient score statistic and (3.4) play the basic role. We shall
see in the next section that this feature remains true in the multivariate case also
Suppose that
~,
...
4.
MULTIVARIATE GENERALIZATIONS
,~
are independent r.v. 's with continuous (p-variate)
d.f. 's Fl, ... ,F , respectively, where p
N
Fi(~) = F(~;~Ni)':: E: EP ; ~Ni =
~
«
where S = «Sj~»
1 and, as in (2.1), we assume that
Sj~CNij~»l~J:'P;l~~.:.q'
is a pxq matrix of unknown parameters and ~Ni
i=l, ... ,N,
=
«cNij~
(4.1)
», i=
1, ... ,N are pxq matrices of known constants. The general multivariate linear
model, treated in detail in Chinchilli and Sen (198la,b) , is a special case of
-8-
~N' = 8c N·• the c N· are q-vectors and F(x;~N') = F(x.- ~N'). For the
1
-- 1
- 1
~
_ 1
~
-l 1
(4.1) where
-
general model in (4.1). the null hypothesis H relates to the homogeneity of the
O
F .• i.e .•
1
: F
H
. . .
=
l
O
= F = F
N
(or
= 0
8
)
.
(4.2)
and we consider a restricted alternative of the form
*
S
H :
where a
0
E:
{ 8
f
E:
EPq :
fl* = A(vecS
is a positive integer «
>
)
0
}
(4.3)
pq) and A is of oder a Xpq
o'
the pq-vector obtained by stacking the rows of
vec8
denotes
S under each other.
For this generalized orthant alternative problem. the test procedure is similar
to the one considered in the previous sections. First. we consider a simple null
hypothesis H against a simple(local) alternative for which for a fixed
O
8 = oy, where 0
>0 •
To obtain
LMPR
y
E:
f.
test for this specific case and to incorporate
the same in the formulation of UI-LMPR tests. we make the following assumptions.
[Al] F is absolutely continuous with continuous density function f(x) = f(x;8 ) •
8 = (e ..... e) E: f • which satisfies the conditions
(a) For every i( 12i2N).
-l
-p
f.(x;oy) is absolutely continuous for almost all x and y E:f. so that if we let
1
-
-
!i8(~ ;~)
=
= (a/a8)f.(x;8) and
-
1
-
(vecy) 'vec ii8(~;or).
f.
(x;oy) = (a/ao)f.(x;oy). then for 8 = oy.
1Y -
\J
x
E:
-
1
EP and y
E:
r .
-
-
_
f.
(x;oy)
1y -
-
(b) Conditions (ii) and (iii) in [A2]
of Section 2 hold. (c) Let
(4.4)
where EO denotes the expectation under the null hypothesis. Then. lCf) is p.d.
[A2] For the p-variate pdf f(x;0). we denote the conditional pdf of the jth coordinate • given the others. by f.Cx.; 8 Ix) • and let
J
J
-
-
g.(x.; 0 Ix) = (a/aO.)log f(x;0) = (a/ae.)10gf.Cx.;8 Ix_).
-J
J
-
-
-J
- -
-J
J
J-
(4.5)
for j=l •...• p. Also, let f[j] denote the marginal pdf of the jth coordinate. and
* ](x.;e J = (a/ae.)logf[j](x ; e,) • j=l, ... ,p.
fee
(4.6)
j -J
.
J -J
-J
-J
*
Note that both the ~j and ~[e.] are q-vectors. We denote the corresponding pq-vectors
-
*
*
-J
by g(x;8) and f (x;8). respectively, and assume that there exists a p.d. matrix L.
such that
g(x;O) = L* f*(x;O) , for almost all x
--
E:
EP .
(4.7)
~
•
-9-
For multivariate linear models, (4.7) holds for elliptically symmetric d.f. '5.
[A3]
For the c ., [AI] in Section 2 holds (where we use the vec c"., if
~ N1
~"l
needed) and assume
sup
1~ i~
[A4]
For each j
sup
sup
N 1~ j~ P
1~e ~
q
ICN'1 J'e I
l
J lOgNlog
= 0[
N ]
(4.8)
(j=l •.... p) and e (l'=l .... ,q), f[j]e(x:~) is differentiable
> 0
(a.,b.).
J J
r E
(0.
(ii)
on
let
its
suPPOt't
some
1
2)
a.s.
Fur ther,
(4.9)
let
E {f*(X.;9)(f*(X.;9)'} = 1* = ((CPm*m'»)
o
~l ~
~
~l ~
1~m;
m
,
(4.10 )
~
pq
and define the bolck diagonal matrix of 1* by
oi ag [ E0 { : [ 1 ] (X 1 ; ~ 1 ) ( : [ 1 ] ( X1 ; ~ 1 ) ) , } .
8(1*)
... , EO U*[
~
THEOREM!.1
p
] (X p ; 9~p ) (f*[
] (X ; 9 ))'}]
~ p
p ~p
For the family of distributions f(x;9).
~O
Let!N = ((9
))
Nje
j=1, ... ,p;e=I, ... ,q
estimator (M.L.E.) of 9
~*O
~*O
under the assumptions
8(1*)1*-1.
[AI] and [A2]. then L
~o
(4.11)
~*O
be
the
maximum
likelihood
(parameter of multivariate density function f) and
I
9". = (9 N . 1 •.. ·.9 N . ) ,
~.'J
'J
,Jq
(j=l, ... ,p) be the M.L.E.
of 9.
~J
(parameter of
jth coordinate density function f[j]). Under some regularity conditions,
and
~:a
may then be solved by the likelihood equations
A*O
INi=l _f*(X.
',8) = 0, respectively. Note that
_~ _
8
-N
is assumed to be nonsingular, IN
g(X. ;8)
i=l - -~ -
a
A*a'
I~=l ~(~t;~)
A*a'
= (8-Nl , ... , -eN
l P
iff
I~=l f*(~i;~)
=
~
the
AO'
~N
and
). Since
. -L
=
a.
Hence
-10_
"0
"*
eN = vec e = e =
-N
-N
A*O
vec ~N .
(4.12)
A
O is a
consistent estimator,
~N
Under the regularity conditions of Wald (1943),
so that the usual expansion of the scores LN g(L' e"0 )
i=l- :;'1.' -N
around e
under HOl and the use of the weak laws of large numbers lead to
A
!<:
N 2( e - 0 ) {l + 0 (l)}
f) { N-~ N
(4.13)
Li=l~(~i;~) }
-N
-p
A*
!<:
N2( e - 0 ){l + o (l)} = B-l(I*){ N-~
(4.14)
L~=l!*(~i;~)}
-N
-p
1-\
and these two equations lead us to
LB (1*)
=
I (f) = LI*L'
(4.15)
This completes the proof of the theorem.
Let Rij be the rank of Xij among the set X ,X , ... 'X
, for i=l, ... ,N,
lj 2j
Nj
and j = 1, ... ,p, and let
~
be the pXN rank collection matrix and
pxN matrix formed by permuting the columns of
~
*
~
be the
in such a way that the top
row is in the natural order; it is termed the reduced rank collection matrix.
For any given
*
~
, let
*
S(~)
be the set of N! possible rank collection matrices
which can be reduced to ~* by column permutations. Then, by virture of the
Chatterjee-Sen (1964) rank permutation principle, we have
p{ ~
We denote by P
=: I
N
(Nl)-l ,
vi :
* •
E. S(~)
(4.16)
the permutational probability measure generated by (4.16).
*
Also, for every j (=l, ... ,p), working with the f[e.](';~)'
we define the scores
-J
b (i), i=l, ... ,N,as in (2.10)[ with F being replaced by F[jl ]; note that
-N
~Nj(k) = (bNjl(k), •.. ,bNjq(k»',· for k=l, •.. ,N and j=l, ... ,p.
Consider then the pXq matrix
!~
=
« T~j£»
(4.17)
of linear rank statistics
o _ -~ N
N
*
()
1
TNj £- N Li=l ~ij bNj£(R ij ) = Li=lcNij£bNj£ Rij , j= , ... ,p;
n 1
N=
, •••
,q.
(4.18)
For later use, we write
T_
o
N
T
= vec -N
* -1 C =
and C
-N = N -N
« ci:Nmm ,) ) m,m '1
= , ... ,pq '
~Ni = (bNll(Ril) , ... ,bNlq(Ril),···,bNP1(Rip) , ... ,bNpq(Rip»',
(4.19)
(4.20)
for i =1, ... ,N, and let
V
-N
=
(N-I)
-1
N
'
Li=l ~Ni~Ni
= «
v Nmm ' »m,m' =l, ... ,pq ,
(4.21)
11
We assume that
~N
is p.d., in probability. Then, as in Puri and Sen(1969), we
have
E(
~N
~N
* ,v Nmm ' » , and we let
= « c Nmm
I
and
E( T T'
-N-N
(4.22)
where
= li~ ~ E(~NIHO).
L
(4.23)
Further, under the Assumptions [Al] - [A4]
and
TJ
-N i. ~}
N
X
(L\, l: ) , A= vee
pq --
(4.24)
where {KN}is a sequence of contiguous alternatives, defined as in Section 3.
THEOREM 4.2.
For the family of distributions in (4.1), under the assumptions
[AI] through [A4], the test with the critical region (vecr)
,
~!N ~
kN(a) is
*
locally most powerful rank (LMPR) test for testing H in (4.2) against HI:
O
S = oy ,y
E r
and fixed, at the significance level
a .
Proof. Note that
*
P{R N = rIH.,} =
N
N
_
tv
f .... f 11 f(x';~N') dX 1 d~2 ... d~N
R =r i =1
1
1
NN
N
N<
N
-
-
N
N
n f(x. ;O)dx ... dX
1
N
i=l
N1
N
+ f.,. ,f
R =r
NN
[ni=l
f(x
N
N
N
N
;~N')
i
N
N
-
1
n
i=l
f(x, ;O)]dX
N1
N
N
1
(4.25)
... dX
NlN
N
Then, virtually repeating the elegant proof (for the uniparameter case) of
Hajek and ~ida1k (1967,pp.71-72) and extending it to the multiparameter case,
we obtain that under [Al] through [A4] , as
0
+ 0,
N
f .... f
R-_=r
:..~
N
p
N
+
~
q
Z c
Nije ., j e
i=l j=l e=1
Z
Z
N
11
f (x. ; 0)
,;Olx)
f .... f g'e(x.
J
1J
i=l
R =r
NN
N
N
N1
N
N
d~ 1 .. ,d~N +
0
(~ )
(4,26)
Noting that the gradients in (4.5) have all null expectations, we obtain on
summing over all r E
*
S(~)
that as
o +
0,
12
N
N!
J .... ~
II f ( x . ; 0 ) dx 1 ... dX
i=l ~1 ~N
R =R
~~
~~
+
(4.27)
(6 )
0
From (4.26) and (4.27), we have
*
P{~~.;=:IH1V[
E * P{RN=rIH*}]
rES (R)
~
l.
~
N
1 ;.
N!
[
E
Ji=l
P
q
E
X
N
j=11=1
J .... J eN i .e
R =r
R =R
where (N!)
J
"f.
J
e g.J e (x.
,; 0 Ix)
~lJ ~ ~
II
i=l
dF (x
.0
~i'~)
_N
-1
~N
j
l
°l----~~N.:-=-~- : : N : - - - - - - - - - - - - - J
J., .. *J
.
(~.28)
~N
;.
0
(0 )
II dF(x. ;0)
i=l
~l _
represents the probability ratio under H ' Hence, using the
O
Neyman-Pearson Lemma on this conditional probability setup, we obtain that a
* is given by the coefficient of
LMPR test statistic for testing H vs. HI
O
in the right hand side of (4.28); using (4.7), (4.16) and (4.17). we may
further simplify this as
N
(vee
N
JR =rf
L X
"f)'
~
i=1
~N
~
[:*(~i;~) x ~Ni]
l,ll
__ l
e·
dF(~i;~)
(4.29)
N
II
dF (x. ; 0)
i=l
_1
~
where for every k(=l, ... ,p) and i(=l, ...• N).
*
:*(~i;~) x ~Ni = ((:[l](xil;~) x :Nil)'·
*
f[k](xik;~) x :Nik
( f *[ ] (x , ; 0) x
P
1P
~
*
*
= (f[k]l(xik;~)eNikl'
.... f[k]q(xik;~)cNikq)
e..T •
~.,
1P
) I ) ;
(4.30)
Now. for every k(=l, ••.• p), let Uik.i=l, .••• N be i.i.d.r.v. 's having the
uniform (O,l)d.f., and define
-1
•
F[kJ(t) =
inf{
x: F[kJ(x;~) > t}
, for t
E (0,1)
•
(4.31)
Also let
-1 N
GNk(t) = N ~'-l
l(U'~k < t)
~-
(0,1), (4.32)
L
-1
where the quantile function G (.)
Nk
(N+l)
-1
R
ik
yields that
<
13
*
f[k)t(Xik;~)
*
-1
= f[k)t(F[k)(t);O) +[N
-~
*'-1
ZNk(t)][f[kJt(F[kJ(~);~)]
1
for some;
such that It-sl~ N-~IZNk(t) I·
N -
-
-1
If *'
[k] t (F [k] (;) ;~)
N
(4.33)
= (25 loglog N)/N = (25 1og2N)/~.
~),
Then, by [A4J, we have for some r S (0,
sup
S <; <1-s
Let SN
,
a.s.
-1
=
/ f [k] (F [kJ (;) ;~)
0
r
(N )
(4.33a)
•
Further, by Theorem 4.5.5 of Csorgo and Revesz (1981), the first factor of the second
term on the right hand side of (4.33) satisfies
sup
lim
(4.33b)
S <t<l-s
N~
N- -
N
Thus, combining (4.33) through (4.33b) and letting Wkt(R
*
-1
f[k]t(F[k] (R ik /(N+l»;9) , we obtain that for every
a.s.
(N+l)-lR
su~
~ (SN,1-S )
ik
N
IWk,Q,(R
We verify (4.35) only for R
ik
tail. Let
~
0 < t
k(l<k~p)
ik )
~
and
~
(1 < t
I = o«N r-l logZN) ~
~q),
~t ~
as N
q,as N
l-r
+
~
».
00
+
00
(4.34 )
,
) a.S..
SN' For N sufficiently large, if U
ik
S O(Nr)t S O(Nr)~N
N
o [lOg2 ]
N
~
*
= f[k]~(Xik;~)
-
(4.35)
(N+l)£N ' as a similar proof holds for the upper
t: ik
If Uik
t: 1
)
o«(1og N)/N
2
===
Next, we claim that for every
ik
~
t , then
a.s.
(4.35.a)
l-r
t. then
=
l-r)
o(log2N / N
a.s.
(4.35.b)
This proves (4.35). By (4.34) and (4.35), we obtain that for every t:1 <t <q,
14
N
a.s. o(Og2 ]
N1- r
0)
(4.36)
(vecr)'~~N
Using (4.8) and (4.36), we may reduce (4.29) (a.s.) to
' and this
completes the proof of Theorem 4.2.
We may remark that for the coordinatewise independendence case, we do not need
.'
~
.../
(4.27)-(~29).
(4.9) as the Hajek-Sidak (1967) arguments directly apply to
the LMPR test statistic based on
(vecr)'~~N
depends not only on
In general,
y but also on the
unknown matrix L. For testing (4.2) against (4.3), according to Roy's (1953) UIprinciple, the overall test statistic is given by
Q = sup{\' L T
~
A'LE
Thus we let h(A) = -A'LT
, h (A)
.., ..,_N ..,1..
= -/-I
N
/
"':' • .N
L'A , A = vec
• ..N • •
..
= -tv.
_..,
and
~, ~ E
h 2 (~)
r}
= ~
I
(4.37)
:-:N: ' ~ -1, then
for this non-linear programming problem, the K. T. L. point formula theorem
can be used again.
For the lagrange function
* * ) is a K,T.L. point if it satisfies the system
the point (A * ':1,t
2
•
(b)
A A
( C)
A
*
I
*
~
: ~
0
N:'~ * -
1
o
(4.38)
=0
(e)
From (4.38)-(e), we have
2
t; ~*
=
(L
~N
:,)-1 : :N
+
(:
~N
L,)-1
~':~
(4.39)
After some manipulations, consequently we arrive
* *'
2 t 2 :1{~(: ~N :')
-1
::N
1
+
*
~(: ~N :')- ~':1}
o
(4.40)
Define
*
~N
(4.41)
15
and
(4.42)
where
~(:N)
is defined as in (4.11), Then by Theorem 4.1 and (4.23) we have
* - ~o ...P 0
~
N ...
as
(4.43)
oo
Further let
U
~N
~ ~
=
-1
and
(:N):N
~~-1(:N)
~N
Z
~N
1
8- (Z
~
~N
)A'
~
(4.44)
Then from (4.40) and (4.43) we have
*'
:1
(~N + ~N
*
:1) = 0
If we denote J be any subset of A
O
in probability as N ...
= {1,2, ... ,a o}
(4.45)
00
and J' be its complement,
thus (4.45) implies
*
-1
0, :1(JI)
(4.46)
-~N(J'J') ~N(J') ~ 0
and
(4.47)
~N(J:J') > ~
a
Moreover. Kudo (1963) showed that the collection of all 2
O
sets
aO
-1
RN(J) = {~N E E ; ~N(J:J') > ~, ~N(J'J') ~N(J') S O}
a
O
is a disjoint and exhaustive partition of E
(4.48)
Therefore we have
where J is the set such that conditions (4.46) and (4.47) hold.
Therefore
the UI-LMPR statistic in (4.37) is given by
2
QN
-1
-1
]
' [ -1
-1
= :N ~N - ~ (~N)~t~N ~ ~ (:N):N
-1
+
th-~A {~N(J:J') ~N(JJ:J' )~N(J:J')}
r: - 0
-1
1{~N(J:J') > O} 1{~N(JtJ') ~N(J') S O}
(4.50)
in probability, where the partitioned matrices and vectors are defined as in
(2.16)-(2.19).
THEOREM 4.3.
For testing (4.2) against (4.3)
{K }), i f (Al]- (A4] hold, then
N
the
UI-LMPR
(under the local alternative
test
statistic
and
the
corresponding UI-LR test statistic are asymptotically power-equivalent.
Define
£I..Q.2.!.
N
SO
_N
*0
~N
C
*
Nijt gjt
C
*
Nijt
[C:1
N
=
[C:1
(Xij;~I~»)]j=1,... ,p;t=l. ... ,q
(4.51)
f;j]t(Xij;~»)]j=1""'P;t=1, ... ,q
(4. 52)
,
and write
~N
= vec
~~
and
*
~N
*0
vec ~N
then it is easy to show that
(4.53)
where
with
I(f)=((~
-
mm
)),
•
and
€: = [[~:m C:mm ']]m;m =1, ...
l
~
Also,
~
-1 ~*
let
(~N)
"..
~N = ~~;
~N ~
-1
"'*
1
(~N)~')'
*
~N (:N = ~ ~
(4.54)
,pq
-1 A*
(~N)~N)
A 1
and
~*
~N = ~~;~' (~N = A
Then the UI-score statistic for testing (4.2) vs.
(4.3) turns out to be otthe form
~N(J:
J' )}.
..
(4.55)
We also define
(4.
+
~
ep:J<:A O
f *'
A*_1
l~N (J:J') ~N (JJ:J')
> O}
56)
A_1
*
l{~N (J ' J') ~N (J') ~
O}.
17
A*
Note that ~N = ~~N~' (as ~N
*
~~N
), and hence, by Theorem 4.1 and the implications
of contiguity, we have
P
r - B(~*)r*-lB(~*) + 0 (as N + 00), under H as well as {K }
(4.
O
-N
- -N -N - -N
N
2
*2
Thus, by (4.48) and (4.12)-(4.14), QNO and Q are asymptotically equivalent in
N
sn
distribution under HO as well as {K } . Furthermore, under some regularity conditons
N
.
.
Q2
d
h
t h e UI -score stat~st~c NO an t e corresponding UI-LR statistic are asymptotically
equivalent in distribution under {K } .For this we may refer to Chapter 4 of Puri
N
and Sen (1985) where the asymptotic quadratic mean equivalence of T~~
*
N
*
~i=l cNik~ f[k]£(Xik;~)
test statistic
Q~
and
(for k=l, ... ,p; ~=l, ... ,q) has been established. Thus, the
and the corresponding UI-LR statistic both have the same asymptotic
power function under {~}. This completes the proof of the theorem.
THEOREM 4.4.
Let"
1
A 8- (X)1:>" and
..:l
+a
define r
J
For each J (¢~J=Ao),
- lim
- N-t'IO E ( ~N H a ).
I
-1
= {~ e E a ~J:J' = ~J - ~JJ' ~J'JI ~J' ~ a}
-rhen for testing (4.2) vs.
(4.3) (under {K }).
N
and r a -=
n
4x:J~AO
r J.
the UI-LMPR test statistic
with critical region W (:) = {~; Q~ ~ kN(a)} is asymptotically most stringent
N
for
r
and is asymptotically most powerful for
ro
at the respective level of
significance a.
Proof. The proof follows directly from Theorem 3.1 and Theorem 4.2, and hence
is omitted.
5. GENERAL REMARKS
We consider here some specific problems and append some useful discussions.
(I) Univariate k-sample location/scale alternative problems. Let X.. , j=l, ... ,n,
~J
F .. (x) ,x
i=l, ... ,k be independent r.v. 's with continuous d.f. 's
F~J'(x)
...
~J
1
R ,
where
(5.1)
= F.(x) = F«x- 8.)/0.), j=l, ... ,n;i=l, ... ,k.
1
E:
~
In the conventional case, we take the scale parameters 0l""'Ok to be all equal
(and equal to 1, without any loss of generality), so that (5.1) corresponds to a
special case of (2.1) where
c
-Nl
(0,1, ..• ,0)', ... , :N«k-l)n+l) =
I
= ... = ~Nn =(1,0, ... ,0) , £Nn+l = ... = ~N2n =
=
c
-NN
=
(0, ... ,0,1)
,
and N
=
kn. Here. we are
18
interested in testing for
61 = •.. = 13 k
(i.e., the homogeneity of the F.) ,
(5.2)
~
against the simple ordered alternative
A
=
(~~ L;.:::.~ . ~) }·
(5.3)
•
o
0
0 ... -1
1
Let R.. be the rank of X.. among the N observations of the combined sample,
1J
~J
for j=l, ... ,n, i=l, ... ,k. We also denote the corresponding order statistics by
~(l)
< ••• < ~(N)' respectively. Thus, as in (2.7), we have the scores
aN(r) = E {-f(~(r»/f(XN(r»
} , for r =l, ... ,N,
(5.4)
and the corresponding linear rank statistics are given by
(5.5)
TN'
• 1
We let
= (N_l)-l LN [
()
]2
-1 N
r=l aN r - aN
; aN = N L i =l a N(i)
2
ON
Then the dispersion matrix of
~N
2
= ON
.
= e~N l~'=ON 2 ~
(5.7)
•
O
-~
~
6 = N Y
*
e-
(under H ) is given by
( k~k - ~~' ) , 1 = (1, .•. ,1)' ,
Also, note that under ~
~N
~N
(5.6)
-1
= Ay and UN = ~ EN ' so that
*
t::. * = «Oij»
where
-{-~o :, I
i = j )
with
*
C••
1J
i-j
I
(5.8)
=1~
li-jl >2 •
Partitioning
~J:J'
and t::. * as in (2.16)
~
-
> 0
-
)
\J
J
:
o c.;
J
-
~ ~
(2.18) , it is easy to verify that here
= {I •... ,k-I}
, if
~
> 0
(5.9)
and hence, by Theorem 3.1. we may conclude that for testing (5.2) against (5.3)
(under local contiguous alternatives), the UI-LMPR test based on the score function
in (5.4) is asymptotically uniformly most powerful for all alternatives in the
order-restricted parameter space.
Consider next the scale problem ( with homogeneous locations), where
-1
Fi(x) = F( C x) , for
i
.
~
=l, •..• k , x
~
(5.10)
E.
Here the null hypothesis relates to the homogeneity of the o. while we have an
1
ordered alternative as in (5.3) with the vector 6
replaced by
,
0 = ( 0l, ... ,ok) .
19
For this k-sample scale ordered alternative problem, using the scores
•
bN(r) = E{ -XN(r)f(XN(r»/f(~(r»} , r = 1, ... ,N,
(5.11)
it can be shown that the corresponding UI-LMPR test is asymptotically UMP for the
entire (restricted) parameter space under the ordered alternative.
(II) Bivariate two-sample ordered location problem. Chatterjee and De (1972, 1974)
developed an UI-rank statistic for this problem and showed that their test statistic
is better than the unrestricted one . A generalization of this result to the general
multivariate case is still an open problem. However, instead of the maximization
of the Bahadur-efficiency [as has been considered in Chatterjee and De (1974)],
we may maximize the Pitman-efficiency to determine the UI-LMPR test for the
general restricted alternative problem under more stringent assumptions on the
underlying density function. For the particular problem considered by Chatterjee
and De (1972,1974), our UI-LMPR test is asymptotically uniformly most powerful for
the restricted parameter space under the alternative hypotheses.
(III) Other models. The asymptotic UMP property of the UI-LMPR test holds for
other problems as well : (i) a positive orthant alternative when the limiting
dispersion matrix is diagonal, (ii) an ordered alternative problem when the
limiting covariance matrix is of the form
~ ~ ~l +
(~~')~ ~2 ' and this is
typically the case with randomized block designs and with multivariate multi-sample
problems, (iii) profile analysis and (iv) in alternatives which put constraints
on the parameters in the form of lower dimensional hyperspaces. Chinchilli and Sen
(198lb) have considered the power superiority of the UI-rank test in the general
multivariate problem under certain restraints on the covariance matrix. Under the
same condition on the covariance matrix, by our Theorem 4.4, we are able to obtain
the asymptotic UMP character of the UI-LMPR test for a wider class of alternatives.
In passing, we may remark that the UI-LMPR test is invariant under any permutation
of the p-coordinates, while the corresponding rank test based on the step-down
procedure is not so, and the latter may not be asymptotically UMP.
(IV) A nonparametric version of Schaafsma and Smid (1966) procedure. For simplicity,
20
we consider only the case of the positive orthant alternative where A = I
and
-pq
pq
r = E+ . Under assumptions [Al] and [A2] in Section 4, the use of the NeymanPearson Lemma
a
size
(on the distribution of the ranks) leads us to the LMPR test of
which rejects H
O
for
(5.12)
~N
where
and :N are defined as in (4.19) and (4.22) respectively.
Assume t
is the half-line of points B
A,
B > 0 and
for any two such
vectors t and m. let 1'(t,m)
where <.>
:N
(5.13)
denotes the inner product with respect to :N and II.II
Euclidean norm with respect to :N'
.N
X
is the
Following Schaafsma and Smid (1966), the
test in (5.1 2 ) is asymptotically most strigent somewhere most powerful if
~o
e
r
satisfies
1'0
sup
mer
where the desired half-line
Abelson and Tukey (1963).
l' ( to'
m.)
• •
:0
inf
sup
1'(t
,m)
eer mer
can be determined by applying the method of
Namely
o
0
1'(t,e
1'(!':1) = 1'(:':2) =
o
•• pq
(5.15)
)
o
where the edges e. (j=l ..... pq) is defined by
-J
o
e. : A. > 0
-J
J
and
A
k
=
0
if k
~j
v
j, k=l, ...
,pq
(5.16 )
Since the maximum shortcoming of this test is an increasing function of
1'(~O'~)'
it becomes
unsatisfactory for large values of 1'0'
- 21 -
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