ASYMPTOTIC DISTRIBUTION OF UI-LMPR TESTS FOR RESTRICTED ALTERNATIVES
*
Ming-Tan M. Tsai and Pranab Kumar Sen
University of North Carolina at Chapel Hill
SUMMARY. For testing against some restricted alternatives, in the light of
the most stringency and somewhere most powerful character, the UI-LMPR tests
are asymptotically optimal. The asymptotic distribution theory of such
UI-LMPR test statistics is developed , and the same is incorporated in the
comparative study of the asymptotic powers of the restricted UI-LMPR tests
and their unrestricted versions (over the restricted parameter space).
1. INTRODUCTION
Let
~l'
•..
'~
be N independent random vectors (i.r.v.) with continuous
distribution functions (d.f.) Fl, ..• ,F , respectively, all defined on the pN
dimensional Euclidean space EP , for some p ~ 1. We assume that
F.(x)
= F(x;~N')
,
~ - - ~
where the
~Ni
~N· = «
- ~
= «c Nijt »
S.ncN·jn
»j =,
1 ... p ; JIv=n 1
J JIv 1 JIv
,
•••
q
,i=l, ... ,N,
are pXq matrices of known constants and
~
=
(1.1)
«Sj~
»
is a pXq matrix of unknown parameters. The traditional null hypothesis relates to
H :
O
~
= 0 ,i.e.,
F = .•• = F .. F ( unknown),
l
N
(1. 2)
against the global alternative that
H : S # 0 , i.e., the F. are not all the same.
l 1
(1. 3)
For this problem, granted the asymptotic optimality of the classical likelihood
ratio test (LRT), several authors have cosidered the corresponding nonparametric
versions and studied their asymptotic optimality [ viz., Puri and Sen (1985)].
In practice, often, we may have a restricted alternative hypothesis which may
be characterized by a suitable subset
r of EPq and framed as
AMS 1979 Subject Classification Nos : 62E20, 62F05, 62G10, 62G20
Key Words and Phrases: Bahadur-Cochran defficiency; Bahadur-efficiency;
likelihood ratio test; mixture of chi square distributions; rate of convergence; shortcoming; union-intersection locally most powerful rank test.
* Work partially supported by the Office of Naval Research, Contract No.
N00014-83-K0387.
2
H* :
B
E
r ,
for some proper subset
r
of
EPq .
(1. 4)
The orthant alternative, ordered alternatives and a broad class of others all
belong to the type in (1.4). Since the classical LRT,for H vs. Hl,has an
O
exponential rate of convergence ( to 0) for the error probability (under nonlocal alternatives) which does not improve for a restricted alternative,
Brown (1971) advocated the use of the classical LRT for testing H vs. H*.
O
However, some Monte Carlo studies [viz., Barlow et al. (1972)] exhibit the
•
•
power superiority of the restricted LRT to their unrestricted counterparts.
In the nonparametric setup under consideration, the current authors have
incorporated the notion of locally most powerful rank
t~sts
(LMPR) [ viz.,
Hoeffding (1951)] along with the union-intersection (UI-) principle of Roy
(1953) in the construction of some UI-LMPR tests
which are asymptotically
power-equivalent to the corresponding LRT under restricted (contiguous)
alternatives.
Like in the parametric case, the asymptotic distribution theory
~ ~:~~;
0"......
_'-, ~ '"'
. :t ~
of such restricted alternative tests becomes more involved. In fact, the simple
•• ::-
;..
(,
, ..
"
w
chi square ( central under H and non-central under the global alternative)
O
distributional approximations are generally not tenable for such restricted
cases. Our main contention is to study the asymptotic distribution theory of
the UI-LMPR test statistics and to incorporate the same in the study of the
asymptotic power properties of these tests and their classical versions.
Along with the basic regularity conditions, these UI-LMPR tests are
introduced in Section 2. The asymptotic null and non-null (under contiguous
alternatives) distributions are then derived in Section 3. The rates of
convergence of the actual distributions to their asymptotic forms are investigated in Section 4.
Some asymptotic relative efficiency and asymptotic opti-
mality results are presented in the concluding section.
e·
3
2. PRELIMINARY NOTIONS
We assume that in (1.4), r
is a positively homogeneous set. Without any
loss of generality, we may take r to be of the form
r = { S € EPq : ~ vec~ ~ ~ where A is an aoxpq matrix of rank aO( ~pq)}, (2.1)
and vecS denotes the pq-vector obtained by stacking the rows of 6 under each
other. To construct the UI-LMPR test statistics, we make the following assumptions.
* = N-~ vecC . , -*
N *
[AI] For every N , we let cN'
, and
c- N = N-1 E.-lc
- 1
- N1
1- _ N
1
*
*
N
*
-*
*
-*'
(2.2)
-CN = « cNkk'»k "k'=l ... ,pq = L-=I(c
1
_ Ni - c
- N)( cN'
- 1 - c
_N ) .
We assume that
max
max
l<j~p
l~.R.<q
IC~ij.R.
~
I
max
l<i<N
-1
Yo
=O«N loglogN)2),
(2.3)
and there exists a positive definite (p.d.) matrix C* = «c *
,», such that ~N*
kk
converges to
* as N increases. Note that
C
(2.3)
and the above ensure that
max
l<i<N
F;s absolutely continuous with continuous density function f(x) :f(x;0),
li~-+oo
[A2]
{
.<
~=(~l""'~o)'
fi(~;or)
which satisfi.es the conditions:
is' absolutely
con~'ir:~ou~-for al~st'.al1v~
;.-
if we let
for ~ =or,
.
(c)
f iy (~ ;or)
-
: (vec r)
I
•
V'f iG (~ ;~)
.,
.r~
U~
j
, I-r x. € ~P and y € r .
ly
_ -.r
_
u~
If. (x;oy)ldx=f
ly
-
-(I)....
-
The largest characteristic root of .-I(f)
~[(a v~c 0)10g f(~;§H a v;c
8 ) 10 9
For the p-variate p.d.f.
f(~;§)]
f(~;~), ~e
-00
1 -
for j=l, ... ,p.
_j
- -
_j
1 -
-
....
=
is finite.
denote the conditional p.d.f. of
and let
g.(x.;e,lx~) = (te-)109 f(x;E)) = (a~ )log f.(x.;6 i Ix~)
J -J -J
_
If.ly (x;O)ldx
is finite.
- _
the jth coordinate, given the others by fJ.(x.;eJ.lx~)
J - -J
J
so that
rl
- r (x;O.) =lim o-l[f.(x;OY) -f.(x;O)]exists,
-
-
[A3]
;EP and r €r,
.....
For every y€r, lim
(d)
For every; (1'5.-i~N),
V'fiOC~;~) =(av;c 9-)fi(~;~) and fiY~~;O!) =(U)fi(~;O!) , then
For almost all x_ and y,
the limit
-
(b)
(a)
(2.5)
J J ...., -J
Also let f[n denote the marginal p.d.f.
Of
the J' th coordinate, and
4
f*[j](Xj;~j) =
t-ah )log f Cj ] (Xj;~j)
by g(~;~)
We denote the corresponding pq-vectors
(2.6)
j=1,2, ... ,p.
and
f*(~;~)
respectively,
and assume that there exists a p.d. matrix H such that
g(~;Q) = Hf*(x;O)
-
for all x € EP
--
--
(2.7)
~
CA4] Suppose for each j"(=l, ... ,p)and
(=l, ... ,q), fCj]Q,(xj;Q)
is
differentiable with respect to x. on (a.,b.), where
J
J
J
_co<
a.< -suP{x.;FC'](x.;O)=O}, co>b. =inf{x.;FC'](x.;O) =l}, and
(i)
fC'](X';O) >0 on its support (a.,b.) .
J JJ J
(i i )
Let ftj]i(xj;Q) = (a~.)f[j]i(Xj;Q),
J-
J
J
J -
J
J
and
J
if[~]Q,(Xj;Q)1 a~. o(N r ),
sup
l<i<N
J
J-
assume, for some r€(O,~)
as N -+
(2.8)
co.
Let Rij be the rank of Xij a~6ng·Xlj,X2j;~.. ,X Nj , for i=l, ... ,N
and j=l, ... ,p; we dent)te th~";p\d'f~fa~k collection matrix by 8 . Let R*
N
-N
be the p x Nmatrix obtained frdm 8N by permuting the columns in such a
way that the top row is in natural order, and for any given
8~,
let
S(8~) be the set of Nf possible rank":collection matrices which can be
reduced to 8~ by column permutations.
rank permutation principle applies to
Then the Chatterjee and Sen (1964)
~N
and under HO' we have
(2.9)
We denote by P
N
the permutational probability measure in (2.9). For every j
(=l, ••• ,p), let Xj(l) < ••• < Xj(N) be the order statistics corresponding to
*
Xlj' ..• '~j ; defining the f[j]Q,
as in (2.6), we let
*
f[j]i(Xj(r);~)
bNjQ,(r) = E{
We define the pxq matrix
6
TNji
N
*
!~
=
I HO }
« T~ji»
= Ei=l c Niji bNji(Rij)'
!N =
o
vec!N·
, r=l, ... ,N; j=l, ... ,p;Q,=l, ... ,q. (2.10)
of linear rank statistics by
j=l, ... ,p; i=l, ... ,q;
(2.11)
(2.12)
e·
5
Also, let
I
*
*
*
= E{ ~ (~i;~)[~ (~i;~)]
*
««9 kk
'»k,:k'=l, ... ,pq
=
* = Diag( E{ f[,](X.,;O)[f[,](X'j';O)]
*
*
D(I)
- -
~j
- j
-
- j
l-
'I
I
HO} ,
(2.13)
H } ' j=l, ... ,p) ,
O
(2.14)
and, for every i (=l, •.• ,N), let
(2.15)
Let then
v = (N-1)
-N
-1
N
'
Li=l ~Ni~Ni
«
=
( 2.16)
v Nkk ' »k,k'=l, ... ,pq ,
*
* ' f kk , » .
L =
cNkk,vNkk'»
and E =
c kk
-N
Finally, consider a sequence of (restricted) alternatives {K }:
N
«
«
-~
8 =N
y,
~:
(2.17)
01.. t: r .
(2.18)
Then, following the lines of Puri and Sen (1985), we have
E(
I
!N
~N
-
~
We adefine
~N
=
,
PN ) = 0
L
E(
-1
~
as in
(~N)~N
!N!N
~7~14) ~
~N'
( 2.19)
~pq(~').
; l:) ~ ,.,A ,=,. veey
,_if...
-
(2.20)
PN
~
{}
and!.N
~(EN)
~ ~
,
) =
with !*,rep1aceg
-1
and C'~~, ~.1~:~
(EN) ~ijB
~Y"~M')' and
let
-1'
(~N)ot,·
(2.21)
Also let J be any subset of AO ={l, ... ,a,O} , ia~.g;.J'obe its complement.
a
For each of 2
0 set J, we parti.tion ~N and'~N ~s
U = [~N(J) 1 ,
-N
~N
~N(JI)J
=
[
~N(JJ)
~N(JJ ')
~N(J'J)
~N(J'J')
):.
(2.22)
For each J (</l~J~AO), we define
~N(J:J') = ~N(J) - ~N(JJ' )~-lN(JIJI) ~N(J')
6
-N(JJ:J')
= ~N(JJ)
-1
- ~N(JJ')~ N(J'J')~N(J'J)
Thus, the UI-LMPR test statistics for testing
(defined in (2.1)) and for testing Ho:~=2
02 = T I 0: -1 _ D-1 (l.: )AI 6-, AD -1 (l.: )) T +
N -N -N
-
-N - -N __
-N -N
(2.23)
(2.24)
H : 3 = 0 against H*
(3e r
O
versus H,: ~12 are of the forms
(2.25)
b
and
2
'-1
RN
=
~N ~N ~N'
(2.26)
respectively, where l(B) stands for the indicator function of the set B.
3. UI-LMPR T!ST STATISTICS : ASYMPTOTIC DISTRIBUTION THEORY
The task of studying the exact distribution theory (even under H ) becomes
O
prohibitively laborious as N increases. For this reason, we take recourse to
the asymptotic distribution theory. Let
Q*2
= W,(~-l_~-l(E)~,~-l~~-l(~))~
+
+
L {~J:JI~j;JI~J:JI} l{~J:JI > ~}l{~j~JI~JI.s.~}
(3. 1 )
¢2:J~AO
where Wis the random vector having a pq-variate normal d.fwith mean vector
Ey and covariance matrix Z (i.e., W-<Ppq(';Iy,l:)), Z = AO-l(Z)W,
--
-
-
-- -
-
--
- -
j.
-
= lim
N~
E(A
~N
IH)
a
and the partitioned vectors and matrices are defined as in (2.22)-(2.24).
THEOREM 3.1. Under the Assumptions [Al]:- [A41 ,
l'
N~
2
p{ QN ~ x
I
HO}
=
*2
p{Q
~
a
x
2
O
I y =....~} = k(J)=O
r
~(J)P{Xpq-ao+k(J)~X}'(3,2)
2
where xpq-ao+
k(J) represents a Chi-square random variable with pq-ao+k(J)
degrees of freedom, and ek(J)=
the sum over all set
Proof.
I*
k(J)
PH {ZJ'J'>O' 6J-~J,ZJ'~O} with
J(¢~JsAo) s~ch
I* denoting
k(J)
that the cardinality of J is k(J).
a
~.
-
-
-
-
Since the first term and the second term of r.h.s. of (3.1) are
independent, so
= E
HO {
eitW~«(l_Q-l(~)~,(l~Q-l(~)~
(i t
E e
} HO{
1-
Q*(J)a*(J),
J
"5 cAO
(3.4)
where
-1
Z
Q* ( J) -- ZI
-J:JI~JJ:JI_J:JI
Note that the matrix E(E- 1-0-'(E)A't::.- 1AO- 1(E))
- -- -
-- - - -
(3.5)
is idempotent, and thus
rank[~(~-'-Q-'(~)~'~-'(~))J = tr[~(~-l_Q-l(~)~,~-l~Q-l(~))J = pq-a o (3.6)
Hence we have
(3.7)
-7-
If we write
R(J) =
{~EEaO; ~J:JI >~t ~J~\JI~I 2.~}
(3.8)
a
O
then the collection of all 2 set R(J) is a disjoint and exhaustive
a
partitioning of E 0
Thus we get
Q*(J)a*(J),
l
J
it
)
= J...
Je ¢SJ~o
a
Q*(J)a*(J)
E O
=
L
¢sJ~o
J... J e 1"tQ*(J) dtP
R(J)
( l; 0 ,t.)
aO - - -
( 3 .9)
Kudo (1963) showed that, for each J (¢~~AO)' Q*(J) and R(J) are independent
under HO'
Thus (3.9) can be rewritten as
Q
,
(3.10)
so
EH {e
a
i tQ*2}
aO
= L
k(J)=O
(pq-aOfk( J)) .
2
e k(J)(1-2it)
(3.11)
Therefore t using the Fourier inversion formula on (3.11)t we obtain that
a
lim
N~
p{ Q2 <
N- x
I H0
}
=
O
I:
k(J)
e
k(J)
. p{
i'~q-aO+k(J) -< x
}
(3.12)
for every real x, and this completes the proof of (3.2).
Next, we consider the non-null case and restrict ourselves to a sequence of
restricted ( contiguous) alternatives [ as in (2.18)]. Note that under {K
in (2.18), for any J (
~
SJ
~AO)'
N
}
Q*(J) and R(J) are no longer stochastically
independent, and this introduces complications in the form of the asymptotic
distributions under consideration. We have the following.
THEOREM 3.2. Under {~} and the Assumptions [Al] through [A4J.
o
(3.13)
where
r
I
*
J
dh = 1
e-t~ e* (n)
ifh=O.
aO -
e-t~
(3.14 )
~)
Y
d(k(J) ,j;n
j+k[J)=h
.:/-
if his odd, k( J) =1 , ... ,a
I e-t~{e: (~)(;/2)h/2[(h/2)!rl+
I
a
I
k(J)+j=h
if
with
~
r'[E-~~-l(~)~,~-l~~-l(~)~J~, ~
=
=
o.
d(k(J);j;n,~)lJ
h
- -
is
even, k(J)=l , ... ,a '
O
~~-l(L:)Z\,
k (J~
"tJ
(3.15)
=0 , 1 , .. • ,a 0 •
(3.16)
and
d(k(J) ,j;n,ll) =
~
- R"m::JQ
(3.17)
H2M=j
where
1-1
C(l,t;n,ll)
= f((Hl)/2)2 2
(tlliT)-l
I
)
e*a·
(n I)
k(J)=l
0-1- J
e
-;~J:JI~
-1
JJ:JI~J:JI
~ ! - 1 - .Mil
C(k(J),t:~,~) = r t+~(J)J22 TT 2 curl
1-
r
,
-tn
I
-.
6- t
L)Q. k(J)-2
k(J)-i-1 e
de ,d6 2'" d6 (J)-1 ~ ,
Jk
~J:J'_JJ:J' _J
IT
,"os
Ji
J
J
-,
i =1
I
0·1-9)
'It
such that the cardinality of J is k(J), and for each J (
~J
n
-1
-
for all k(J) ... 2, ••• ,a O ' 1: (J) denotes the sum over all subsets J (
k
vector
.:.
([e;(JI)(~JI)e -J:J'_JJ:J1_J:J'
k(J)
(
~3J:J,~t.q
(3.18)
-t
t
(~J:JI~JJ:J')
I
is defined as
~
c:. J
C A )
O
0 C
J ~A )
O
, the
( . G
i Sin
-J1
i\C~SGJ1SinGJ2
_Tfl
:
Icas GJ1
•.•.•••. casG Jk(J)-1 sinGJk(J)-1
l cas GJ1
•••••••• casG Jk( J)-1
1
i'
IT
.".
-2" ~ ~Tj ~
-IT
=
V j 1 , ... ,k (J) - 2
~ qk(J)-1 ~
IT
(3.20 )
(3.21)
wher-a
EKN{eit~'(~-l_Q-l(~)~'~-l~Q-l(~)~}
=
I
e-t~
-pq-a +2m
o
2-m(m!)-1~m(1_2it)
(3.22)
2
m=O
and
I
it
KN{ e
E
=
Q* ( J )a* ( J )}
¢eJcA O
Q~ ( J )a* ( J )
=f ..a ·Je--
--
E
L
o.:.J.:.A
L
it
¢cJcAo
o
*
. , itQ*{ J)
,',
' .
:.
ek(J ' )(TlJ,)fz '-'
e
d~k(J)(ZJ'J' ,Tl J . "'\-:lluJJ'J') •
-J:J I >2
-. - ' f ~_-:~ •
-
(3.23)
O
Let aJ(t) be the integral of the r.h.s of (3.23). Then
-1
I
=
e-t~J:JI~JJ:JI~J:J' ~ (~!)-l
~=o
f t
~JJ:JI ~J:J'>2
(2)
-k(J)
2
e-Hl-2it)Q*(J)
IT
~
-t
I
(~J:JI~JJ:J'~J:JI) d~J:J'
(3.24)
Therefore we have the following:
(a)
If k(J)=O, then QJ(t)=l.
"J ( t) =
Q
co
\'
(1
I )
L~'
! -1
-1 r ( .H1 ) 22
~=O
(b)
2
IT
-t
If k(J)=l, then
21- 1
:"I
-J : J I-JJ : J I -J : J '(
-tn'
e
I
,\ -
t
1.
,9, (
~J:JI~JJ:JI}
1-2it
(3.25)
And (c)
if k(J)=2, ... ,a
O
'
then make a transformation to the polar
coordinates (PJ,e J1 ,. .. ,6 Jk (J)-1)'
1
. ) - (-.:.....-) •
Let
(3.26)
2
10
where
-~ ~
9 Ji
~
I
and
-IT
~
9Jk (J)-1
~
Jacobian of this polar transfonnation is
IT
V i=l ,2, ... ,k(J)-2 . Then the
k(J)-l k(J)-Z
k(J)-;-l
oJ
,rr
cos
9
.
Ji
1 =1
We also let R*(J) be a set of restrictions on oJ and 8 'S of the form
J
~3:J'
:J >
~,
2
Then, for every k(J) : 2
QJ(t) =
I
~
oJ
and
_.1
(3.27)
0
2 ao '
k(J)
I
-
1
(Z!)- 1e 2nJ'J'~JJ'J,nJ'J'
~.
~ . ~.
<.=0
(2~)
f···
f
R*(J)
-t
)f
~J:J'~JJ:J'~J
') 2
(
-k(J) Z+k(J)-l -t 1-21t oJ
oJ
e
('
klJ)-2 k(J)-;-l
IT
cos
9J . d~JdoJ
; =1
1
where d~J denotes
"
:X>
Q)t)
= 1.~0
(3.28)
d8 Jl d8 J2 · .. d8 Jk (J)':'1'
After some simplication, we have
. . . •
•
.
-(t+k(J» /2
c(k(J) ,i;~,~) (1-2it)
(3.29)
Using (3.21) through (3.29), we conclude that
r
EX
,e
Nl
'Q*2}
1t
a
','.1
ex>
= le-~~ ~ 2-m(m!)-1;m(1_2it)
L
m=O
~o
a0
e*a' (n)+
~
~
= (1 - 2it)
aO
+
I
Thus, we obtain that
!
J
(H k( J ) ):
c(k(J),1.;n,6)(1-2it)
2
I'
~
-
,
~
_(k(J)+j)-r
d(k(J) ,j;n,6)(1-2it)
2
- -
-t(pq-a
d (1-2it)
h
2
_
t
ex>
h=O
(PQ- O+2m) -;
-t(pq-a)
ex>
0 e- t ~ e * (n) I (m!) -1 2-m;m +
aO - m=O
k(J)=l j=O
= I
I
k(Jl=l 2.=0
ex>
I
ex>
_
+h)
0
J
(3.30)
11
(3.31)
This completes the proof of Theorem 3.2.
2
Note that in the unrestricted alternative case, we have a central X distri2
but ion under the null hypothesis and a non-central X under the (local) alternative.
However, the picture is different here. It is a mixture of central chi square
d.f.'s under the null hypothesis, but the non-null case may not be reducible to
the same mixture of non-central chi square d.f. 'so
4. RATES OF CONVERGENCE FOR DISTRIBUTIONS OF UI-LMPR TEST STATISTICS
Let GN be the distribution function
TN
defined in (2.12).
We
suppose M is a measure on the Borel a-field on EPq, and denote B(w,~)
the open ball with center
'N
and radius::, i.e., :::
>
0,
( 4. 1 )
'.vhere i[' II denotes the Euclidean norm.
ForanyJ. (~~.J~AO) and any
real-valued Borel measurable function gJ
o~
R(J) defined in (3.9),
we define, '1E:>0, the oscillation function S
J(B(w~s))
9
--
as
S J( ';e:) = S J(B(w,e:))
g9
(4.2)
-
and also define SgJ as the supremum of the average modulus of oscillation
of gJ with respect to a finite measure ,\1 over all translate of gJ by
SgJ(e:,M) = sUP{fR(J)
SgJ(~,S)C\ld~),~E R(J)}
(4.3)
Y
where
J
J
g/~) = 9 C~+~),
(4.4)
For simplicity, we assume that
N
[BIJ
i~l
N
cNiji = 0 Pond
iI
l
CN~j~
=
1
"/j
)
=l, ... ,p; ~ =l, ... ,q.
(4.5)
12
LE~~
4.1.
Let h be a bounded Borel measurable function defined on
EPq, then under the assumptions [Al]-[A4] and [Bl], there exist constants
dl ,d 2 and d3 (not depending on N, cNij 2.,h)
such that
~ 9:
5. dl sup ;h w)!!.~
f..
L :C."*3+6,,5
'0:
11
W EPq
- i=l j=l(.=l 111J ....
(w;O,:" )1
\'J d(G (w)-4>
If Pq h()
N
E
I pq - - -I"
I
(
92. I; c",
* '" iI 3+-:5 ,,,8.?
(. ,n L- " ))
1,
=1 111JA.
pq
~-n
't'
Proof.
,
(4.6)
, ... ,
It follows along the lines of the main theorem of HuskoV8 (1980)
,
and hence is omitted.
THEOREM 4.2.
Let X.,
1< i< N,be N independent p-dimensional stochastic
-1
F(x'~~I')'
vectors having continuous c.d.f.
-
-,11
where -,11
:", defined in (1.1).
Consider the UI-LMPR test statistic O~ given in (2.25), then under the
assumption [A1]-[A4] and [81], there exists a constant d such that
4
sup
X€
E
IPH
{O~':' x'r->;~{Q~2
~;x
0
0
}\
~d4':I=1 ~:JJ =1
1
J
I=
1
ICNijii3+cN5
(4.7)
Proof. For every J (0 5J SAO)' defining Q*(J) as in (3.5), we let
Q(J)
= W'( ~-l
-
~
- D-l(r)A,~-lAD-l(~))W
- ---
Then, we have for every x
~
--
<
(PH {ON(J)aN(J)
O
0
.: ~=.J=.AO
I IfO(J)~x
+ Q*(J).
Q*2 ~ x }
o
ep=.J=.A
-
(4.8)
0,
PH {
=I L
-
X
~
}
X
PH
-
}
-
{I
I
o o=.J=.A O
O(J)a*(J) <
PH {O(J)a*(J)
a
~,
X
X
}JI
I
d(lP H [rN:5.. w} - il(w;O':")JI
Ol- - -11
R(J)
=
I
15
4l=.Jc A 'EPq
O
1{O(J) <
-
X}a*(J)dip
{!N'~W~
l Ha
-J -
?(w;o,:,,)! I
~ -
-11
)
}I
13
N
P
q
I I I
i =1 j=l 1..=1
* . ,11',3+6 N
3
IeN'
i
1J
,
I
J..,
+
(4.9)
where
J
9 (~)
=HQ(J) ~x
-1
} l{ ~J:J' > ~'~JIJI ~J' .5. ~}.
(4.10)
By noting that ( under H as well as {K } )
N
O
1 {~J:
J' >~, ~J 1, J' ~, ~ g}
= 1 {~J: J' > g}
1 {~J; J'
~J' ~ g} )
(4.11)
we have
<
.. 1 •
!"--
~
~,;
and
.•
(4.12)
/..
(; ; )
I
J
= ~~J~AO l~~~(J)
=
L
sup
~~JcAO ~€R(J)
f
sup
EPq
~€R(J)
ItgJ(~+~)
~O L ((aAJ(w)f+
gJ(~+~)\ .II~-~!I
< E.
~E R(J)ld¢(~;~'~~1
y)
(4.13)
~ x , ~J:J' >~, ~J~JI~JI ~ o}
(4.14)
-'-"
-
-
where
AJ(~) = h€
and (aAJ(~»)~
of
AJ(~)
EPq; Q(J)
is the set of all points whose distances from the boundary
less than
LetC o be the class of all convex Borel subsets of
E.
Pq
E , then for every J (~=.J=.AO)' AJ(~)€CO'
Bhattacharya and Rao (1976) and some
Hence,
manipulatio~,
using Corollary 3.2 of
we get
1 5
I
~eJ=.AO
Taking
E:
S J (E: ,~) ~ I~N 1222" fl.Q9; 1) lr (T) r 1 E:
g
=d3
•
( 4.15)
-I
I t I IcN;.~13+oNC
i =1 j=l Z=l
J
and combining (4.9), (4.12) and (4.15),
we obtain that (4.7) holds. Q.E.D.
Next we investigate the rates of convergence of distribution function
of Q~ to its limiting mixture central x 2 distribution under a sequence of
Under {K N} t the distribution
function depends on unknown parameter iNi = ((Yji cNij.Q.)) = ((eNij~)J, so
restricted contiguous alternative {K N}.
we make some extra conditions on the distribution function and the unknown
parameters eNi j ~ 'tJ i =1 , ... ,N, j=l,
[B2] Fo!, every j=l t... tP and 2.=l,
N .
~~~=1 eN,·n
~J 7v
=
N
a and
E,~= 1
,p
and
•
2.=l, ... ,q.
,q
2
eN'·n
~J7v
=1
(4.16)
.
[B3]
For the jth marginal density function fC'J(X' ,;8.), we assume that
J 1 J -J
exist constants dS and d6 such that for max max max !SNiJ'i l ~ d >
S
l~j2P l~~~q l<i<N
there
(4.17)
Z&~A
4.3.
Let h be a bounded Borel measurable function defined on EPq
then under the assumPtions!:tA1]~tA4] ~nd CB2i~CB3j, there exist constants
<
-
+
Proof.
9
d7 sup ih(w)i t..~ ~I.. L
r.;q
i=l J'=l (,=1
w€E
d -5 I d
9
It
hl
~
~
9 I....
8 i~l j~l L~l
follows from
' '2. !1 3+8 N5(1 +:.('il:3+6)
N
i lJ
' J
i...
IC
I 3+r5 N5(1
CNijZ'
I
+'Yj~1
13+6) '" ( ~ A )" ) 1
,'rrt';~N_''::N J
(4.18 )
Lemma 4.1 and (7) in remarks of Huskova (1980).
•
r 2 \ l~dl0 t..~ ~t.. 9L I....
I
3+~) .
2
} -PK1Q"'~X
supi PK {QN~x
cN"i 'l 3+6 No(1 +Y'"I
x€E
N
N\
J
i=l j=l £=l'J
J<..
t
15
The proof follows directl.y from Theorem 4.2 and Lemma 4.3, and hence is omitted.
5. ASYMPTOTIC POWER COMPARISON OF
Q~
~
and
While testing against a restricted alternative, in the parametric case, one
may compare the classical LRT with its restricted alternative version , so as to
gather information on the gain in the sensitivity of the test. In the nonparametric
Q~ and ~ with the
case, we intend to compare the asymptotic power functions of
same objective. The power superiority of the restricted tests over the unrestricted ones in some specific nonparametric problems has been studied by Chatterjee
and De (1974) and Chinchil1i and Sen (1981), among others. De (1976) has shown
Q~ and the traditional R~ both
that for a randomized block design problem, his
have the same approximate Bahadur-slope for any alternative in the restricted
parameter space. We may observe that this feature is generally true for the LRT.
Towards this, we assume that
~i'
•
¢
,N_,~
i=l, ...
.
__
,"..,
,_
are
•. _
,~.
i:~;d.r.v.
• I ,__
with the d.L
"'
~,
o
0
(.; 6, E ), where E is assumed to be given. Consider the usual LRT and define
p q , .
"', ' • (
.:' " .L -, ",' • 'J ·:i
-1'
0
0 ..
KN(~;~'~) = N {log[L(~;~,~ )/L(~;~,~ )]}; ~= (~t~""~N)'
where
o
L(X;6,E )
(5.1)
stands for the likelihood function. By the classical Law of Large
Numbers, we obtain that
= 1(6;0)
a.s. when
o
(.;6,E) holds,
¢
pq
(5.2)
--
where 1(6;0) stands for the Kullback-Liebler Information; it is defined as
I(~;~)
= E6 {
For testing H :
O
1(6;0)
o
=0
~
0
log[f(~;~,~ )/f(~l;~'~ )]}
vs. K: 6
= 6'(E O)-lS
F0
, for every 6
F0
.
(5.3)
, it is easy to check that
for every
S
+0
.
(5.4)
Thus, the unrestricted LRT achieves the Kullback-Leibler Information in (5.4).
For testing H :
O
~= ~
against K*: A6
> 0, where A is defined by (2.1), it is
-
-
easy to show that the restricted LRT also achieves the same Information. For
this restricted alternative problem, the LRT statistic is given by
*2
QN =
N~«~
0
1
)-
0
1
0'
0 -1
0
0
0 -1 0
-~'(~)- ~)~ + 0~~SAO{~J:J'~JJ:J':J:J,}l{~J:J'>~'~J'J'~J'':'~},
(5.5)
where the partitioned vectors and matrices are defined as in (2.22)-(2.24) and
16
and
(5.6)
Thus
a
*2
-1 *2
O
0
2
=
p
{
>
Nd
Q
>
d
PH { N QN
= [k(J)=O ek(J) P{X pq _a +k(J) ~ ~t }
N
H
O
0
0
0
where the ek(J) stand for the sums 0 f the multinormal orthant probabilities
~J:J'
corresponding to the
• for which the k(J) have the common value
k (=0 •...•
a O)' Therefore, by some routine steps, we obtain that for evry t( >0),
li~.~ { -N- 1 10g P {N-1Q*~
H
O
N
= t/2
> t}}
N
a.s.
<P
o
pq
(.;S,[).
--
(5.7)
Moreover, under the same setup,
-1
-1 "
4
•
N- 1Q N2 ~>31(l:O
L
",~J.0o
! a'
,1\)6 +
_A't,°
.0-1
0
l
r 0
\
f
0-
1
0
~J:Jl~JJ:J'~J:J'J1V!J:J' >2r1l~J'J'~J' ~
I
2]
(5.8)
where )J0
=
A3.
--
. easy to s h ow t h at (5 . 8) equa 1s to 13' ",0
-1 13 , so that the exact Bahadur
It ~s
~
slopes for both the unrestricted and restricted LRT's are the same. Thus, we
need a finer asymptotic
compqri~,Q.n
to.. discriminate the two LRT's. In this context,
the concept of approximate 6ahadur-Cochran deficieQcy, developed by Chandra and
Ghosh (1978), may be adapted-to force this distinction.
Let k
o
( k ) and a
(a ) be the critical vaLue and size of the unrestricted
lN
2N
N
*2
*2
(restricted) LRT sequence ({~ }. {QN }) when the power at B is B*. Then, using
N
the results in Section 2 of Chandra and Ghosh (1982). we obtain that
(S.lO)
and further that
k
o
N
- k
N
= o(N
-1
), so that
w.eha~e
1. I
::t
ZN
+[(pq-aO+k(J))/2-l]U~k;.j)-1 +o(N- 1)))
o (eO -1
[QgJ
Nk N -t
ao
r. 2
1 0 1+ o(N-)
1 ]
O r
1-2(NkN)= a
e
1+
2
0
r..J2g::h
1N ao
ea
f[ 2 J
(1
so that a ZN
]
[
t
and hence
log a
ZN
=
log a
lN
l
[
(5.11)
+ o(N- 1)
1
J'
O
+ log e o + 0(1) .
(5.1Z)
aO
Consequently, by using Theorem Z.3.l of Chandra and Ghosh (1978) we conclude that
the approximate Bahadur-Cochran deficiency of the unrestricted LRT with respect
to the restricted one is -2(S'(ZO)-lS)-11cg eO. When ~o is not known, the LRT
- aostatistics, Q~2, for HO: ~ = g against K*: ~~ ~ g, has the same fGrm as in (5.5)
with ZO being replaced by
?
= (N-1)-lZ~=1(~i-~N)(~i-~N)"
In this case, following
Perlman (1969), we obtain that under H ' for -ev~;y _~} 0,
O
p{ Q02 <
N -
x } =
2
2
Ek(J)=O ~k{'JY pL~qJa r:'Ht(jrJXN_pq-':' x }.
aO
0
(5.13)
o
By very similar steps, it· follows:'that in this-caS'e, 156ththe unrestricted and
restricted LRT's have the same Bahadur slope, and the approximate Bahadur-Cochran
deficiency of the former with respect to tne latter js still positive.
We may note that in the above discussion, we have allowed both a
to converge to 0 , and for a fixed power
S*
lN
and a
ZN
,we have drawn the relative perfor-
mance picture. This conclusion may not necessarily apply to the conventional case
where the level of significance is held fixed, and for contiguous alternatives
[ such as in (2.18)], the relative power pictures are studied. In situations where
the Pitman-efficiency measure is adoptable, the two sequence
of competing statistics
have the same type of distributions [viz., non-central chi square or normal]
differing in some noncentrality parameters, and the relative picture of these
noncentralities convey the efficiency picture. However, in our case,
Q; and
~
have different asymptotic distributions and the Pitman-measure is not adoptable.
Another plausible approach for this local comparison is to go through a second
order local scheme. i.e., to compare the slopes of the asymptotic local power
18
functions at the null point. For this, note that under
{~}
in (2.18)
,
00
h -h
-1
2
> x} ,
(5.14 )
13 2(a;y) = exp(-{~'E.~}/2) L:h=o(:'E~) 2 (h!) p{ ~q+2h
a
R
2
where x' stands for the upper 100a% point of the central X d.f. with pq DF. Also,
a
Q~ , under {~} in (2.18), the asymptotic
by Theorem 3.2, for the UI-LMPR test
.
power is given by
x } ,
a
>
where x
(5.15)
*2
is the critical level for Q . Since both (5.14) and (5.15) are smooth
a
functions, by the Taylor expansion, we have for small (A'L:A)
S
RZ
(a;y) = a +
Q({( A'L:
-
A)~}2 ) ;
(5.16)
- --
!<: 2
!<:
+ b*(A 'EA) 2 + Q({( ~'n) 2} ) ,
SQ*2(a;y) = a
(5.17)
where
x ~
a'
*
[ \
e*
L
k( J)
p{ X
pq- ao+k(J)+1 ~
-1
with A* = Sl:
and
*'
A-t
d]
~J : J I ~JJ: J'IJ 2J
.1..
2
~JJ:J,!}9
a } )f
(5.18)
,for
f y0 - 1
t k(J)=O
P{x~q-ao+k(J) ~
.
-
X
n* = sl-1 ~
aon*
J
(0)
)
1
2
x
k(J I
sl
> O. If 6; = I, then (5.18)
( ao - 1 ) [ { 2
k(J)
P Xpq- ao +2+k(J)
1
xa }]; > 0, where n* = ao
1
~
reduces to
}
xa
ao
i~ln!
(5.19)
In general, b* is not very simple in form. However, we may simplify this further as
.
b*
= -;::::;:::::;:;::;:- f[ <J n* , 6; I A*' L:A* t 6; -1 Z~ 0 ao
+
L
k(J):1
1
Z del> ( Z;
ao -
*
(L J
n* I 6; - 1 Z del>
k(J) R(J) -
-
-
ao
Q,
6;)
)
P {x 2
pq-ao
~
X
}
a
(_Z; 0_, ~) ) P {x 2
~
~
pq- a o+k(J)+l
X
a
}]
+
19
80
*
+ /,
(L
I
*'6- 1 Z d<P (Z 0 6) )[ p{X 2
k(J)=1 k(J) R(J) ~J'-J'J'-J' 80 - ; - ' pq-ao+k(J)
By noting
I
p{X~q-ao+k(J)+1 ~
J},
Xa }
~ x }
where R(J) defined in (3.8).
Ct.
(5.20)
that
n*' 6- 1 Z d<P
6-1Z~O
-
-
~*'~-1~ d<Pao(~;
ao
(~;
g,
g,
~)
)
6) ) p{x 2
~
- p q -8 0
X }
a
+
ao
L
k(J)=1
*
( L J
k(J) R(J)
p{X~q-ao+k(J)+1 ~ xa }
(5.21 )
> (
we conclude that the slope of the asymptotic power function at the null point
for the restricted UI-LMPR test is larger than that of the unrestricted one.
This result gives us the asymptotic local power superiority of the restricted
test Q2
2
N
space
to that of the unrestricted one R
n0 = {13
N
E
EPq : 11
.
> O}
We may finally remark
, in the restricted alternative
~hat
for testing the homogeneity against alternatives
which put constraints on the parameters in the linear form of lower dimensional
hyperspaces i.e., H : 13
O
=0
vs. HI: ~
E
r* = { 13
2
local (contiguous) alternatives, both Q and
N
RN2
Pq
E E
: Avec 13
=0
}, under
have non-central chi square d.f. 's
with pq-a and pq OF respectively, and with the "same non-centrality parameter.
Since, the power function of a non-central chi square variable , for a fixed
non-centrality parameter, is a non-increasing functton of its OF, we conclude
that for all B E
r*,
Q; is asymptotically locally more powerful than
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•I
I