ASYMPTOTICALLY EFFICIENT RANK TESTS FOR RESTRICTED ALTERNATIVES IN A MANOVA
IN RANDOMIZED BLOCK DESIGNS *
MING-TAN M. TSAI
and
PRANAB KUMAR SEN
University of North Carolina at Chapel. HiU
Sunmary
In a randomized block design MANOVA model, for tests for the homogeneity
of treatments against restricted alternatives based on intra-block ran kings
and on ranking after alignment, asymptotic optimality is studied in the light
of asymptotically most stringency and somewhere most powerful character. For
ordered alternatives, the proposed tests are shown to be asymptotically most
powerful within the respective rank class. Tests based on the ranking after
alignment are preferred when the error distributions are homogeneous accross
the blocks.
1.
Introduction
We consider the usual randomized block design model consisting of n blocks
each containing p plots where p treatments are assigned at random; the response
(1)
lJ
(q)
is possibly multivariate. Let X.. = (X.. , .•. ,X..)
~lJ
lJ
I
be the response (vector) of
the jth treatment in the ith block, for j=l, ... ,p; i=l •... ,n. We assume that
(~i1'''''~ip) has a pq-variate continuous distribution function (d.f.) F~(x),
= (13·(1 ) , ... ,13·(q) ),
I
._
where letting S·
J-1, ... ,p and § = (..61'· .. '§p) ,
~J
J
J
pq
F?(x
- 13 •... , x~ p - ~Sp) • (x 1 '.·.,x ) E E ,
, .. ·,x_p ) = F.(x
1 ~1
1 _1 ~ 1
~
-p
and the d.f. F.1 is symmetric in its p arguments ( each a q-vector); although, the
( 1. 1)
form of the Fi may not remain the same for every i(=l , ...• n). Thus, we consider
independent blocks, and within each block, interchangeable errors. The case of
independent and identically distributed (i.i.d.) error vectors is thus included
as a particular one. In (1.1), the
S· stand for the vector of treatment effects
-J
* Work supported by the Office of Naval Resaerch, Contract No. N00014-83-K-0387.
Key words and phrases: Intra-block ranking; Kuhn-Tucker-Lagrange point formula~
locally most powerful rank tests; most stringent and somewhere most powerful test;
randomized blocks; ranking after alignment; union-intersection principle.
AMS Subject Classifications: 62E20, 62G10, 62G99.
-2while the block effects
~BY
not be additive and may even be stqchastic in nature.
Generally, in a MANOVA( multivariate analysis of variance), the null hypothesi~
relates to the homogeneity of the treatments i.e.,
HO : ~l =
= ~p ( = 0 , without any loss of generality).
A global alternative relates to the non-homogeneity of the S. . In this study, we
(1.2)
~J
shall be specifically interested in suitable restricted alternatives. For the univariate case ( i.e., q
= 1), the most common form of such a restricted alternative
is the so called ordered alternative
(1.3)
H<:
Sl ~ S2 ~ ... ~
Sp' with at least one strict inequality.
A general account of rank tests for ordered alternatives in randomized blocks is
given in Chapter 7 of Puri and Sen (1971). Later on, De (1976) extended the method
of Sen (1968) by effectively incorporating the union-intersection (UI-) principle of
Roy (1953) to form an aligned rank test for H against H< when the r.v. 's Xij have
O
i.i.d. error components. Boyd and Sen (1984) used the concept of locally most powerful
rank (LMPR) tests of Hoeffding (1950) and incorporated the UI-principle to
construc~
UI-LMPR tests based on intra-block rankings as well as on ranking after alignments.
However, in either of these studies, no attempt was made to establish any possible
optimal properties of the proposed tests. For the ordered alternative problem. Araki
and Shirahata (1981) and Shirahata (1984) considered some rank tests ( for q
= 1)
which are analogues of the usual likelihood ratio tests. Interestingly, their proposed
tests may also be characterized as UI-LMPR tests, and hence, the question on their
asymptotic optimality properties remains open. The main objectives of the current
study are the following:
(i) To establish the asymptotic superiority (in terms of power) of the ranking
after alignment procedure to the intra-block ranking procedure for general forms of
restricted alternatives,
(ii) to show that the UI-LMPR tests are asymptotically UMP against the class of
ordered alternatives within the respective class of rank tests. and
(iii) to characterize that the UI-LMPR tests have the property of asymptotically
e
-3-
most stringency and somewhere most powerful character within the respective class of
rank tests for testing HO in (1.2) against a general form of restricted alternative
(1.4)
H*: S € r = { S€E Pq : AvecS > 0, A E '~a,pq) } ,
where vec~ denotes the pq-vector obtained by stacking the rows of f3 under each
- -
-
-
other and -da ,pq) is the set ofaxpq matri x of rank a : 12,a< pq ,
The basic regularity conditions and preliminary notions are presented in Section
2. UI-LMPR tests based on intra-block rankings and ranking after alignment ( for testing
HO against H* ) are considered in Sections 3 and 4. Asymptotic compariosons of the
power properties of these two procedures are made in the last section ..
2.
Preliminary notions
We make the following regularity assumptions:
.
.
[AI] Let 0 be an open set containing 0 and define
f.{x-6~)
1I'V"'"
'V
i
~
1.
(~
Then (i) for every i
fi{~-6~)
1),
as in (l.l) with
is absolutely continuous for almost
a
all x and
then for
~
~
,....,
€
=
a
0, so that f'R{x-~) = a
and f.I~ (x-6~)
= a~vf.(x-6~).
~I~ ~ ~
vec ~ f.{x-6~)
I ~ ~
~
~
I ~ ~
6~.
,....,
•
f.
1"Y
(x-6~)
"'J
all x and ~. the limit
=
(vec~)
,....,
i.
• •
fiR{x-~).
,....,,..,
fJ ,...., "'"
~
"""
pq
€ O. x € E
~ ... ~
u-"V
~
~
I
~
~
~
~
~
I~
~
largest characteristic root of llf) =
[A2]
~O
(ii)
For almost
(x) = lim 6-l[f.{x-6~) - f.{x)] exists. (iii)
I~ ~
I~
.
""J
~
I
every ~ € 0, lim I~ If. {x-6~)ldx = I~ If. (x)ldx
finite. where
~=6~,
~
is finite.
~
(iv)
to{a :ec ~ log f{X) [a
a vec X]'
~
For
The
log
f(~)}
is
._
stands for the expectation under the null hypothesis.
For the pq-variate p.d.f.
f.{x-~),
I
~
we denote the conditional p.d.f. of
~
(je)th coordinate. given the others by f 'n(x (e)
..
J~
IJ
{ell x ) and let
J
~o
-~.
a
g[ j] e(x) =
(2. 1 )
Also, let
(2.2)
f[jJ~
ax
log f j e(x) , j = 1 , ... , p;
denote the marginal p.d.f. of the
f[jJ~(X) = (Cl/dx)log f[jJ~(X)
(j,~)th
, j =l, ...• p;
Q, = 1 ,
... , q.
coordinate, and let
~ =l, ..... q .
-4-
We denote the pq-vectors with the elements in (2.1) and (2.2) by
~(~)
and
!* (~),
respectively, and assume that there exists a p.d. matrix H , such that
= H f*(x)
(2.3)
-- -
, for almost all x .
*
is di fferen(=1, ... ,q), f[j]i(x)
*1
• We denote by f[j]i(x) =
[A3] We assume that for each j( =1, ... ,p) and
tiable on the support of the p.d.f. f[j]i
~
and assume that there exists a positive r : a < r < 1/2 ,such that
sup sup sup
*1
(i)
r
(2 . 4)
l:.i<11 l<j<p 12.i:' q If[j]i (X ij )1 = 0(11 ) a.s., as 11 -+
00
3.
UI-LMPR tests based on intra-block rankings
Intra-block rank tests for MANOVA against global alternatives were considered
by Gerig (1969), and we shall follow the same notations along with general scores.
(i) be the rank of X..
(i) among X'(~) , ... ,X.(~) , for J. -1,
_ ... ,p,. i = l,oo.,n
.Let R..
1J
1J
1l
1P
and ~ = l, ... ,q. By virtue of the assumed continuity of the d.f.1s, ties among the
observations may be neglected, with probability 1.
R~l)
1P
R(l)
R.
(3. 1)
-1
i1
Let then
, fo r i = 1 ,. . . ,n,
=
and define R.* to be the matrix derived from R. by permuting the columns in such a
-1
-1
* i=l •... ,n
way that the top row is in the natural order. Let S(R.),
_1
be the set of
[(p!)n ] matrices which are permutational1y equivalent to the _1
R~, i=l, ... ,n. Since
under HO' X' l , .... ,X. are interchangeable random vectors, their joint distribution
-1
-1 P
remains invariant under any permutation of these p-vectors among themselves. This
implies that under HO' given a particular realization of -1
R.* , the (conditional)
* That
distribution of R. will be uniform over the p! possible realizations in S(R.).
-1
is, for any r.
_1
E:
-1
* we have
S(R.),
_1
* H } = (p!) -1 , for all F. , i= l, ...• n .
S(R.),
O
-1
1
Moreover, the rankings are made independently for the different blocks. Hence
= -1
r.
(3.2)
P{ R.
(3.3)
P{ -1
R. = ,.1
r., i =1, ... , n
-1
for every r.
-1
E:
* ), i=1, . oo , n, H } = (p!) - n ,
S(R.
-1
O
*
S(R.). i=l, ... ,n , whatever be the F. , i=l, ... ,n.
-1
I
1
-5-
We denote by ~~l) the probability measure over the (p!)n conditionally equally
likely realizations in (3.3), and define the matrix of linear rank statistics by
(3.4)
where
~j2(r) = '{-f[j]2(UNi )},
i=l, ... ,n; j=l, ... ,p; 2=1, ... ,q.
(3.5)
with UNl",.,U
being the ordered random variables of a sample of size n from
Nn
(0,1) uniform d.f.
It is thus easily seen that
(3.6)
Also, it is easy to verify that
o
0
I (1)
-1
Cov(TNj2 , TNj '2' ~n ) = (Ojj' - p )U n22 ,
(3.7)
for j,j' = l, ... ,p; 2,e' = 1, ... ,q, where 0jj' is the usual Kronecker delta and
(3.8)
is defined,by
(3.9)
Thus, if we write
0'
~1 = vec ~
(3.10)
then
g{TN1TN11~(1)}
= ,Jl
V 1 ~ (I _! 1 1')
....' ....'
n
",p
p ",p",p
where
~
=~
L· .
1
(3.11)
say.
denotes the Kronecker product.
Now we consider a sequence {K ;
n
~
=n
_.1-
2~, ~
is fixed} of alternative
hyPOthesis where K specifies that the random vectors X.. - a.
n
",1 J
",1
j
= 1, ... ,p
are interchangeable for i
= 1, ... ,n
and
~1'
'"
...
'~
",p
- n
-~
~
..
",J
are p real q
-6p
vectors with
!~.
. I~J
J=
= O. Thus under {K }
n
-"i (i)
~j
Fi[j]l(x) = Fi[e]l(x-n
(3.12)
V j=I •.... p; l=I ..... q.
)
_
(3.13)
Moreover. we assume that the following limits exist
-1 n
= lim n
Jl-lIID
(3.14)
! Fi[e]l(x).
i=1
(3.1S)
and let
Then. under assumption [AI] and following Lemma 7.3.10 of Puri and Sen
have. as n ...
(1971).~
co.
(3.17)
and
(3.18)
where
~1
--
Based on
~1'
=
~1
--
(3.19)
1 I')
-p-p
First we note that the set r in (1.4) is positively
homogeneous in the sense that for every
> O.
-p
we then can derive a suitable test statistic for testing HO
against H* defined in (1.4).
~
_!p
8 (1
~
€
r and 0 > O. 0
~
€
r.
So for a given
under the assumptions [AI] through [A3]. by mimicking the proof of Theorem
4.2 of Tsai and Sen (1987). the LMPR test for testing HO against
-I
~'~(~1)~1~1'
H~
is based
where ~(~1) denotes the block diagonal matrix of ~1.
o~
Namely,
-7-
B{~·I)
~
= Diag
'"
_!p
VI' (I",p
,.,n
-1
~1
where Diag !nl is the diagonal matrix of !nl' and
inverse matrix of
~1.
Furthermore, for every
~
1 1')
",p",p
€
(3.20)
stands for the generalized
r, we may write
'"
(3.21)
By noting that H*
=
U H~ and making the use of UI-principle, then the overall
~€r
'"
'"
test statistic for H versus H* is granted as
O
(3.22)
For the computation of GNI in (3.22), we need to maximize
~
-1
~'~{~I)~I!Nl
-1
subject to ~ ~ ~ and ~'~{:Nl):Nl~{:N1)~ = 1.
If we let h{~) = -~'~{:Nl):Nl!N1'
programming problem, the Kuhn-Tucker-Lagrange (K.T.L.) point formula theorem
be used to arrive at the" following result:
can
Let
-1
~1 = ~ ~ (~I)!Nl
{3.23}
and
O = {1,2, ... ,a} and J' be its complement.
also, let J be any subset of A
For each
a
of the 2 set J, we partition ~1 and ~1 as
~1
= r~I{J) ]k{J)
l~I{J') k{J')
and
~1
where k{J) denotes the cardinality of set J.
_
[~I{JJ) ~1{JJ')]
(3.25)
~1{J'J) ~1{J'J')
Also, for each J{~ ~ J ~ A )' we
O
let
~1{J:J')
~1{JJ:J')
Then, we have
-1
= ~1{J)
- ~1{JJ') ~1{J' J') ~1{J' )
= ~1{JJ)
- ~1{JJ') ~1{J' J') ~1{J'J)
-1
(3.26)
(3.27)
-8-
~1 = 5n ~~~1
-
~1 ~~ ~1
+ qJCA
-_~ 0
{~I(J:J') ~~(JJ:J') ~1(J:J')}
...
1{~I(J:J') > ~, ~1(J'J') ~1(J') ~ ~}
(3.~
where I(B) stands for the indicator function of the set B.
-1
Let ~1 = ~
Theorem 3.1.
(1)
tf\
J (~~ J ~ AO)' assume f J
and f~l)
family
=
~
= {~1
+a
€
E
(1)
; ~:J'
= ~IJ
For each
-1
- ~1(JJ')~I(J'J')~IJ' ~ ~}
f}I), where E+a is a-dimensional positive orthant space. If the
n
qg~o
f.(x-~)
1 '" '"
H* :
I
.
(~1 )~1~ and ~1 = ;:; g(~1 HO)'
satisfy the assumptions [AI] through [A3], then for testing HO vs.
(under {K }) ~1 in (3.28) is asymptotically most stringent for f and
n
€ f
asymptotically UMP for r~l) within the class of intrablock rank tests at the
respective level of significance a.
Proof. The proof follows directly from Theorem 4.4 of Tsai and Sen (1987) and
hence is omitted.
Corollary 3.2. For q
the~
= 1.
under {Kn} and Assumptions [Al] and [A2J. within
class of intra-block rank tests. the UI-LMPR test based on (3.28) is asymptotically UMP at the significance level a .
Outline of Proof. The testing (1.2) against (1.3) can be written as for testing
(1.2) against (1.4), where A = ",q
I 8 L with
L
-1
1
0
o
-1
1
1'/
=
0*
'Y
_p
wi th
-
'Y
*
",p-l
0
0
=
(3.29)
000
Then we have
o
o
-1
1
-9-
~~ = (Diag y-II)Y I ~ .. j=I.2 ....• p
--J
(3.31)
-J
-1
And
*
-1
~I = (Diag ~I ) ~I(Diag ~I ) 8 ~
(3.32)
where
*
~
Let
~
*
= «uij»i.j=I ..... p-I
with
= {q.2q •.... (p-I)q} and J be any subset
for any J
($
a~j = t1
of~.
if i=j
I=1
li-j I > 1
if li-j
if
(3.33)
J' be its complement. then
~ J ~ ~)
(1)
~:J' ~ 0
(3.34)
if ~I ~ ~ .
By noting that if q=I. then ~*j = ~ .. V j=l •...• p. hence the corollary is proved.
-
-J
.
Note that for q = 1 and i.i.d. errors, the condition (2.3) is not needed
as there H is an identity matrix); however, for q
> 1,
(2.3) is a sufficient
condition to ensure the stated optimal ity properties i,n Theorem 3.1.
4. UI-LMPR tests and ranking after alignment
In intra-block ranking, because of the lack of information from the interblock comparisons. the tests are generally less efficient than the aligned
rank tests which incorporate the inter-block information through the alignment
procedure. For the standard MANOVA model in two-way layouts, Sen (1969) formulated
aligned rank tests based on general scores and studied their asymptotic efficiency
in a unified manner. Here, we extend the results to testing against general forms
of restricted alternatives [ as in (1.4)] and show that under fairly general
regularity conditions, aligned rankings lead to more efficient tests for such
alternatives too. To do this. we need to eliminate the (nuisance) block effects
by simple alignment procedures; namely, we subtract suitable estimates of the
block effects ( vectors) from the respective t.~) and on the residuals, we
lJ
make an overall ranking ( ignoring blocks) of all the treatments ( in a coordinatewise manner). We may use any trans1ation-equivariantestimator of the
block effects; for simplicity, we take them as the block averages.· Thus, we
-10-
define the aligned random vectors as
Y
~ij
=X
~ij
1
- P
P
! X.
j=l~lj
,
i
= 1, ... ,n;
j
= 1, ... ,p.
(4.1.
°
For convenience, we assume that !ij has a q-variate continuous c.d.f F ij ,
V i = 1, ... ,n; j = 1, ... ,p. Let S~~) be the rank of Y~~) among the N (= np)
(e) for j = 1, ... ,p, i = 1, ... ,n, e = 1, ... ,q. Thus,
observations Y(e) 'Y (e) , ... ,Ynp
11
12
corresponding to the aligned observation Y ., we have a rank vector
~i J
~ij
(q) ,
(1)
= (Sij ",.,Sij ) , i = 1, ... ,n, j = 1, ... ,p. We also define the rank
collection matrix &- by (Sll""'S ). Note that under HO' Y. 1 ,Y i2 , ... ,Y. are
~
~
-np
~l
~
~lP
interchangeable random vectors, so the joint distribution of
' , ... ,Y'
YM = (Y
'1""'Y
~1
~ 1p
-np ) remains invariant under the finite group
~,
transformation {gO} (which maps the swl~le space onto itself).
~
n of
Thus for any gO
n
€
n
~n' there exists ~ = ~ which is permutationaslly equivalent to!N'
If we
°
-
denote ~
* the rank
. collection matrix corresponding t0!N'
* then ~* = ~ and ill'
permutationally eqUivalent to
~.Thus,
under H ' the conditional distribution
O
° °
.
.
* = ~; ~ € ~n}
of ~ over the (p!) n reahzatlon
{~
realization haVing the conditional probability (p!)-n.
is uniform, each
Let us denote this
condi tional probabili ty measure by ~~2) and define the scores
as in (3.5) with F[j]2 being replaced by
~2' ~2
and
~2
the same as in
~1' ~1'
F~j]2'
!nl'
°
~(k),
Similarly define
~1' ~1
and
~1
k=l, ... ,N,
Q~2' ~2' ~2'
respectively with
~je(k) being replaced by ~je(k) for j=l ..... p; 2=1, .... q, k=l, .... N.
Furthermore, consider a restricted (contiguous) alternative
_.1.
{KN:
~
.-
= N 2~.
~ €
r.
p
!~.
. l~J
J=
= a}.
(4.2)
Then we have
(4.3)
and
e
-11-
o
0
-~ (~)
Fi[j]ii'(X,y) = Fi[.]ii'(X - N ~j
-~
(i')
y - N ~j
).
(4.4)
Finally. we define ~ and ~ the same as in ~1 and :1 with Fi[.]i(x) and
o
0
Fi[.]ii'(x,y) being replaced by Fi[.]i and Fi[.]ii'(x,y) respectively. Then under
parallel arguments as in the previous section we have
"
(4.5)
~2 {~}. I pq (.; :~.~).
Theorem 4.1 .. Let ~ = ~
and f (2) =
O
(~)~
(2)
Ih
J (~£ J £
-1
AO)' assume f J
n
~Ta
and
~ = lim g{~2IHO}'
For each
N~
+a
= {~€ E
(2)
; ~:J' = ~J
-1
- ~(JJ')~(J'J')~J' ~
O}
f (2). If the family F?(y-P)
satisfy the assumption [AI] through
1 __
~-O
J
[A3]. t~en for testing (1.2) against (1.4) (under {~} defined in (4.2». the
UI-LMPR test Q;2 is asymptotically most stringent for f and asymptotically liMP
for t~2) within the class of aligned rank tests at the respective level of
significance a.
Corollary 4.2. Under the same regularity assumptions as in Corollary 3.2. the
corresponding UI-LMPR test is asymptotically UMP (under {~}) within the class of
aligned rank tests for testing (1.2) against (1.3) when q=1 at the same level of
significance a.
Note that both the proofs of Theorem 4.1 and Corollary 4.2 are very
similar to the proof of Theorem 3.1 and Corollary 3.2. and hence. are omitted.
Also note that the last conditroon in (2.4) is needed for the aligned rank tests
but not for the intra-block rank tests ( only in the context of the asymptotic
optimality study).
5. Asymptotic power comparison of Q~l and Q~2
. The optimal property of
2
ONI
may not generally. hold when we extend the
domain to the class of tests based on rankings after alignment. Since the
rankings of Q;2 bases on the aligned observations which disregard blocks. so Q;2
should have the optimal property within a more broad class. For testing
homogeneity against global alternatives. Hodges and Lehmann (1962) were
-12successful in establishing the superiority of the r.ankings after alignment
procedure over the intrablock ranking procedure when the underlying
distributi~
is normally distributed. For a wider class of distribution. this result was
extended by Sen (1968) and studied in detail by Puri and Sen (1971). The
assertion of the superiority of the aligned rank tests to the intrablock rank
tests can be easily extended to the testing'against alternatives which put
constraints on the parameters in the linear form of lower dimensional hyperspace.
However. for a wider class of restricted alternatives. this assertion may not be
true. The following Theorem provides a partial answer when the aligned rank tests
are asymptotically power-superior to the intrablock rank tests in the restricted
alternative space.
Theorem 5.1. Let
lim
N-lllO
P{~2 ~
~*
=
A'l-~~l' ~1(~'!1)
'"
'"
"''''
x(2)IKN}' If lim
a
n-lllO
= lim
~
P{~l ~
P{~l ~ x~l)IHO}
.
*
~1(~':1) ~ ~2(~'~) whenever ~ € 00 = {~
= lim
N-lllO
+a
€ E
*
x(l)IK } and
a
n
~2(~'!~)
"'~
P{~2 ~ x~2)IHO}
=
a,
=
then
lJ(Tf. J. = 1..... a}
; ~j ~ 3 xa
Outline of the proof. Let us define
(5.1)
for j
~ j'
= 1..... p; i = 1•.... n. and
~1 = lim n
n-lllO
-1 n
.
! AI"' ~ = 11m n
i=l'" 1
~
-1 n
! ~i
i=l
(5.2)
(~1 - ~) .
(5.3)
Then via Theorems 3.1 and 4.1. we have
-1
~1
= ~1 - ~
and
-1
~
2::!.
= p
Namely.
1d ~.
:1 = p
...E...~1 = p-1 ~ and ~1 = ~ = ~. say.
(5.4)
Next. we define
2
Qk =
.-1
~]£Ao{~(J:J')~(JJ:J')~(J:J,)}l{~(J:J') ) O.
-1
~(J'J')~(J') ~ O}
where
Vk
= 1,2.
(5.5)
-13-
It suffices to show that if"
P{~ ~ x~k)IHO}
P{Q~ ~ x~l) IKn } ~ P{~ ~ x~2) I~}.
~1 =
= a,
Vk
= 1,2,
then
Without any loss of generali ty we assume
I and note that
P{n2 ~ x(2)1~_} _ p{Q2 ) x(l)IK }
~
a
-~
1 - a
n
=ctci90[IB2(J)~a(~;
(5.6)
J.!l ~.~) - IBl(J)dota(~: ~.~)]
where
(5.7)
Since P{~ ~ x~k)IHO} = a, V k = 1.2. therefore it is obvious that
x(l)
a
~ €
Furthermore. we write x~l)
=. x(2).
a
a.
!
Aor TCA
'+=-J_
=
= 1.
DO' (i) if a
fa
0
1
= xa
and ~o
""
= ""~
+ 01, where 0,) O. For
""
then we have
(J)[dl (z;
a ""
~o,I)
""
""
- dl (z;
~,I)]
a"""" ""
(5.8)
f [e-t(z~0)2_ e-t(z-~)2]dZ + f [e-t(z~0)2_ e-t(Z-~)2]dZ} ~ 0;
v21f~~0
z)~
a
-!-f
(Ii) if a = 2. by regrouping the sums and synmetric arguments, we have
~o,I)
!
fa '(J)[dl (z;
AorTCA
1
a"" ""
'+=-J_ 0
""
- dl (z;
a""
~,I)]
""""
~~2 -0~1
+
f
f
~-0-~2
+ 0(1)
~
-ell)
O.
(5.9)
By mathematical induction. thus p{Qi ~ xalKn} is non-decreasing in ~*
whenever: € DO' Therefore we have. : € DO'
-14and hence the theorem follows.
Remark. For a
*
~2
r -}
l 31vx
a
let 0*
= 2,
u {~*
€
E2 ;
= {~*
*
~1 ~
0,
€
1vx
r -' ~2
* ~ ~} u {~*€ E2 ; ~1* ~ 0, ~
E2 ; ~1* l 3
a
*
~2 ~
a}
U 00 , then under parallel arguments as in
(ii) of Theorem 5.1, we have
(5.10)
Generalizing this result to the higher dimensions is still an open problem.
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~