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This research was supported by the National Institutes of
Health, Institute of General Medical Sciences, Grant No. GM-12868-04.
t
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t
I_
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.-
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ON A CLASS OF RANK ORDER TESTS FOR REGRESSION
WITH PARTIALLY INFORMED STOCHASTIC PREDICTORS
by
Malay Ghosh and Pranab Kumar Sen
University of North Carolina
Institute of Statistics Mimeo Series No. 616
I
I.
I
I
ON A CLASS OF RANK ORDER TESTS FUR REGRESSION
WITH PARTIALLY INFURMED STOCHASTIC PREDICTORS I
BY MALAY GHOSH AND PRANAB KUMAR SEN
University of North Carolina at Chapel Hill
o.
The present paper
extends the findings of Hajek to the multiple (linear) regression model with
stochastic predictors including the situations where the predictors are partially
I
,
informed.
The proposed tests are shown to be permutationally (conditionally)
distribution-free.
Their asymptotic properties and efficiencies are studied and
the asymptotic optimality is established under the conditions of Wald (1943).
1.
J
I
Introduction.
Consider a (double) sequence of stochastic matrices 2 =(2 1, ••. ,2 ),
-v -v
-VV
(1 < v < 00), whe re 2'.
-v~
= (Y v~.,
X'.)
-Vi
(1)
(p)
X . ,..., X . ), (p > 1) are indev~
V~
-
= (YV~.,
pendent and identically distributed random vectors (iidrv) having an absolutely
continuous distribution function (d.f.) F (y, x ), x E RP , the p-dimensional
I_
V
real space.
-
We assume that the p-variate marginal d.f.
FOl(~
) = F (00, x)
v
does not depend on V and possesses a continuous density function f lex).
o -
I
J
we denote the conditional density function of Y i given X . = x by f (y Ix) •
1
also assume that fv(yl~)
,.
J
Hajek (1962) has obtained asymptotically most powerful rank order tests
for simple (linear) regression with non-stochastic predictors.
I
'I
Summary.
=
f
20
([y - a -
V
V
-V~
-
V_
We
2 13 ,x]/o), where a and 0 (>0) are
nuisance parameters and 13 " " , 13 are parameters under test.
1,
p
Thus the density
function corresponding to the d.f. F (y, x) is given by
V
-
(1.1)
We want to test the null hypothesis of no regression i.e.
H
(1.2)
I
,.
O
13 = ~
against the alternatives 13
~
O.
It may be noted that under H : ~ = ~, fv(y, ~) = fey, ~) = f (~) f 20 ([y-a]/o),
o
Ol
for all V > 1 and thus Y . and XV' are stochastically independent for all
v~
- ~
I
I
1)
Work supported by National Institutes of Health, Institute of General Medical
Sciences, Grant No. GM-12868-04.
I
2
I.
I
1\
t
I
I
i=1,2, ..• ,v and all V > 1.
I_
t
I
I
t
V
-1
(k = 1, .•• ,p) and Y = V
v
The following assumptions are made:
00
(1. 3)
fio(y) = (d/dy) f
E(X~~»
(1.4)
20
(y) exists and l(f
20
) = f
-
00
= 0 (assumed without any loss of generality) and E IX~~)
4 0
1
+
< c <
00,
for some 0 > 0 and for all k=l, ••• ,p, V ~ 1;
(1.5)
L = Var(X 1) is positive definite (p.d.)
~
~V
Let
(1.6)
t
t
We shall use the notations X(k)
where
(1.7)
-1
~
F (u). = inf {x : F(x)
20
u}
~*(u) = - g' (G-l(u»/g(G-l(u», 0 < u < 1,
where G is a class of d.f. with density function g satisfying (1.3); let
(1.8)
leg) = f~ ~*2(u) duo
We introduce a class of score functions av(i), i=1,2, .•• ,v, satisfying
ravel + [uv]) -
(1. 9)
~
*(u)] 2
du = 0,
where [x] denotes the highest integer less than or equal to
X.
In particular,
we can take the following types of score functions as suggested by Hajek [12]:
* (U~i» ], i=l, ••• , V
(a)
av(i) =
E[~
(b)
a (i) =
v
~
(c)
a (i) = V filv
V
(i-l)/v
*
(i/(V + 1»
~
i=l, ••• , V
V> 1
V > l',
* (u) du, i=l, ••• , V
,
V .::: 1, where u(l) -<
vI
< U(V)
-
V
denote ordered statistics in a sample U , ••• , U of size V from a uniform
VV
Vl
distribution on (0, 1), U . 's being distributed independently of
V~
(i=l, ••• ,v; k=l, ••• ,p).
X(~)'s
V~
L~t Fk(x) be the marginal distribution of ~~) and
and Fk,k'(x, y) the joint bivariate marginal distribution of
~~)and x~~').
"
3
'.'I
We consider another class of score functions denoted by bkv(i),
lim
V ~ 00
(1.10)
I
I
I
:1
'I
't
Ie
I
,I
"
I
I
0'I
fa1
k=l, ... ,p) satisfying
*
[bkV(l + [uv]) - ~k(u)]
2
du = a,
* (u) = F*-1 (u), (k=l, ••. ,p),
k
~k
where
(r.v.)
,'~
F denoting the d.f. of a random variable
k
W(k), (k=l, ••. ,p), satisfying (1.4) and (1.5).
V
We also assume that
(W~l) , ... ,w~p» are distributed independently of YV1 ' •.. , YVV •
In Section 2, a class of permutationally (conditionally) invariant distribution-free tests have been proposed and studied.
In Section 3, the asymptotic
permutation distribution of the proposed class of test statistics (under H ) has
o
been developed.
The asymptotic non-null distributions have been obtained in
Section 4 by using the contiguity lemmas of LeCam (See e.g. [11] or [12]).
In
Section 5, the optimality properties of the proposed tests have been established
under the conditions of Wald [17] and the ARE results of the allied tests (in
Pitman's sense) have been derived.
In the remainder of this section, we prove a lemma which will be used
repeatedly in the subsequent sections.
LEMMA 1.1.
Let {X .; i=1,2, ••. ,V,V > I} be a double sequence of random variables
- - V1.
-
such that for each V, XVI"'"
for which Elx . ,2+0 < c <
V1.
X are independently distributed with d.f. Fv(X)
VV
for some 0 > 0 and all i=l, .•. ,v.
00
Xv - E(X ) ~ a almost everywhere (a.e.)
v
The proof of the above lemma is based on the following result :
LEMMA 1.2.
Let XVI' XV2 ' XV3 , ••• ; Xvv be samples from distributions with d.f.
If there exists n(> 2) such that
E Ix .I n < c <
V1.
p{!xv
11
I
~l;
(i=l, •.. ,v,v
X
V
-1
= V
PROOF.
V
L:
i=l
-
00,
~vl
for all i=l, •.• ,V, V > 1, then given any
> E} < K/(E n V
-1
X . and].l = V
V1. - - V
n/2 )
E
> 0,
for all V > Va and some positive K, where
V
L:
. 1
1.=
E(X .).
V1.
The proof of the lemma follows exactly in the same lines as that of Brillin-
ger [5] (who however has considered the case of iidrv Xl' X2 ' .•• with Elxl,n <
00)
,
t,
6
Ie
I
*
v! permutations of the columns of ~v(o) would be uniform under H ' i.e.
(2.4)
*.
P{~v(o) = :v(O) IL:(~v(O»'
irrespective of F (x).
01 ~
I
Let (j>
Ho }
o
= (v!)
*
-1
for all :V(O)E L:(~v(O»
denote the permutational (conditional) probability
\1
v
measure generated by the conditional law in (2.4).
Then, after some algebraic
computations, we get the following results:
"
(2.5)
I
(2.6)
E(Sk)
) = 0, k=l, •.. ,p;
UJ)
Vkk ' , v = Cov (SkV' Sk'V'
(v/(v-l» {V -1
I
,
I
cP v
{v- l
v
(k)
L: (b (R
)- b )
kV Vi
k
i=l
(k')
(bk,v(R Vi )- bk,v)}
V
2
L: (a)i) - a) }
i=l
,
(k, k'=l, ... ,p).
We propose the following test statistic:
(2.7)
'Ie
Mv = ~V' ~v* ~v '
where V*
is a generalized inverse of V
= «V
~V
~V
kk
,,».
v
We shall show in the next
section that under the assumptions (1.3), (1.4), (1.9) and (1.10), V converges
~V
I
l
in probability to the matrix
(2. 8)
with O'kk =
I
O'kk' =
I
I
leg)
, where
a
« O'kk ,»
J1o 1}Jk*2 (u)
du , k=l, ••• ,p,
Jla Jla ",*
~k (u) ,10*'
~k'(u )
and
dH kk , (
u,u' ), 1 2 k ~ k' 2 p ,
(k)
(k')
Hk,k'(u, u') being the bivariate joint d.f. of Fk(XVl ) and Fk,(XVl
),12k~k'2P'
The following test procedure is proposed based on the critical function sl(Zv):
I
I
I
e
I
~O =
~O
if M > M
V,E
V
1
(2.9)
sl (~V) =
°lVE
a
ifM
= M
v,E
ifM
< M
v,e::
V
V
where M
and 0lVE are so chosen that E6'v(Sl(~V» = E.
v,e::
This implies that
= E, E(' IH ) standing for expectation under H i.e. sl(Z ) is a
o
similar size E test.
0
~V
However, this procedure requires evaluation of v! possible
I
I.
,I
I
,I
I
I
I
7
realizations of M . The task becomes prohibitively laborious for large V and
v
this leads us to the study of the asymptotic distribution of the permutation
test statistic M
V
3.
V
theorem generalizes Theorem 4.2
score functions ¢* (u) or
of Puri and Sen (1966) in the sense that unlike
*
~k
(u)'s and the boundedness conditions thereon.
uniform boundedness of the (2 + 6)th order moments of
for some 6 >
OJ
W(~)
's (i=l, ... ,v, k=l, .•. ,p)
V1
I~:
the conditions
the latter implies
In
integrability of ¢* and the
proving the result all we need to assume is the square
(u)
I~
[u(1_u)]-1/2 + 6'
for some appropriate 6' > a as assumed in [16].
v~V
THEOREM 3.1.
PROOF.
Now, a
I(g) ~O
fa1
in probability
aV]2
E [a (i) -
. 1
V
-+
V
-1
V
1=
I
V
-2
E
- a
i=l
fa1
a (l+[uv])du =
v
V
= v- 1
[a (l+[uv]) v
V
¢* (u)]du
Using now (1.9) and the Schwarz inequality, we get, a
(since
V
-+ 0
as
fa1 ¢* (u)du=O).
V -+
Further,
00.
-1 V
2
V
E a (i)
I
I
I
V
i=l
fa1
[a (l+[uv]) - ¢* (u)] 2du + 2
v
fa1 ¢* (u)
[a (l+[uv]) - ¢* (u)] du +
v
fa1 ¢*2 (u)du
Using now, (1.9) and the fact that
I
I
I
This
Puri and Sen we do not have to assume the existence of the derivatives of the
Ie
I'- e
We shall now prove a theorem which
V
will be used in deriving the asymptotic permutation distribution of M .
I
I
Asymptotic permutation distribution of M •
-1
with I(g) <
00,
we get, V
V
Thus,
E
i=l
(3.1)
-1
V
V
E
i=l
-
[a (i) - a ]
VV
2
-+
I(g).
Proceeding exactly in the same way and using (1.10), we can show that,
(3.2)
V
-1
~
i=l
[b
(R(k»
kv vi
-
b ]2 = v- 1
kv
~
i=l
Using (2.6), (3.1) and (3.2) we get,
[b
kV
(i) -
b ]2
kv
-+
OkOk (k=l, ••• ,p).
,
8
b
k'v
(R(k'»
Vi
-
b
kv
~
}
k'V .
(3.4)
We introduce the following notations.
(u, v)]
(u)
= v-I
(3.6)
H_
(3.7)
CkV(i/(V+I»
-leV
[# of F
k
= bkv(i);
(X(~»
VJ.
< u] ;
-
(i=l, ... ,v;
Hence,
= (I) + (II) + (III) + (IV) + (V) , (say).
1(1)
I ~ {J~[CkV(VHkV(U)/(V+I»
-
~:
(VHkV(U)/(v+I»]2dHkV(U)}1/2 .
I
I.
I
I
I
\1
I
I,
9
-1 V
*
2
1
*
2
Now, V E [bkv(i) - ~k(i/(V+l»] = J O [bkv(l+[uV]) - ~~((l+[uV])/(V+l»] du
i=l
Since
J1
O
*
[~k
((l+[uv])/(v+l»
from (3.9), (I)
1(111)
I~
'I,..,'
Ie
a as V +
+
J1O
*
~k(u)]
00.
*
2du
+
a
Similarly
{~k(VHkv(u)/(V+l»
as V +
(II)
Using now (1.10), we get
00.
a as V +
+
00.
*
2
1/2
- ~k(U)} dHkv(U)]
.
But,
as V
(3.11)
+
00.
Also,
(3.12)
** (A)
~k
where
is a function defined on 0 < A ~ 1 by
** (A) = ~k
* (i/(V+l»
~k
(3.13)
so that
** (i/(V+l»
~k
inequality (a+b)
,.
I
[
-
*2 (VHk'V(v)/(v+l»dHk,V(v)] 1/2
J1O ~k'
"
II
is a monotonically non-decreasing and square integrable function
of u, using lemma a (p. 164) of [12], we get,
(3.10)
I
I
I
I
I
I
*
~k(u)
(3.14)
E
=
2
for (i-l)/V < A 2 i/V , (i=l, ... ,V)
*
~k(i/(V+l»
for all i=l, .•. ,v.
~ 2(a 2 + b 2 ), we get from (3.12),
J; [~~(VHkV(u)/(V+l»
-
~~(u)]2dHkV(u)
Then using the elementary
I
I.
I
I
I
**
/(V+l»
2 E [ljJk (R(k)
vI
- 1jJ* (u)] 2du =
Now,
k
*
Since ljJk(u)
is a monotonically non-decreasing and square integrable function of u,
using lemma
**
(k)
E [ljJk (RV1 /(V+1»
(3.15)
< 2 /:2 (v- l / 2 max
l<i<v
Ie
I
as V
~
0 as
V ~
00.
Again
~
as V
,-
-1 V
I ljJ:*(i/(V+l»
E
Also, V
00.
(ljJ:*(i/(V+l»-
i=l
**
1jJ k(i/(V+l»
~
~*I)(V-1 ~
_
=
v-
1 V
*
ljJk(i/(V+l»
~
~
fa1
~:*)2)1/2,
*
ljJk (u)du = 0
i=l
-1
V
~
1jJ
i=l
**2
~
(i/(V+1»
cr
0
<
kk
00,
and
max 11jJk**(i/(v+l»I = v 1 / 2 max I f i/v ljJ*(u)dul < max { f i/V 1jJ*2(u)du}1/2~0
l<i<v
l<i<v (i-l)/v k
- l~i~V (i-l)/V k
as V ~
(3.16)
V
**
2
- ljJk (U vl )]
i=l
V- l / 2
I
I
I
2
**
fl [ljJk
(u) - 1jJ* (u)] du
0
k
a, p. 164 of [12],
using lemma 2.1 of [12], we get,
-**
where ljJk =
I
I
I
I
_ ljJ**(U )]2
k
VI
* 2du
- ljJk(u)]
I
I
I
I
10
~
since cr~k <
00
Now, from (3.15),
Thus, from (3.14),
00.
fa1
00.
[ljJk* (VH
kv
(U)/(v+1»
* 2 dHkv(u)
- ljJk(u)]
~
0
as V
~
00.
Using (3.11) and (3.16) with the fact that convergence in mean implies convergence
in probability we get now from (3.10),(111)
Similarly (IV)
(V) = v-I
~
i=l
E
I 1jJ:
~
1jJ:
0 in probability as V
~
00.
(Fk(X~~») ljJ:,(Fk'(X~~'»),
(Fk (X~~) » 1jJ:, (F k'
(X~~ , ) » I
1
+~
2
~
0 in probability as
Again,
Since,
V~
00.
I
I.
I
I
I
I
I
11
< ()() ,
of large numbers, we get,
(V) ~ v-I
I
in probability as V
~
()()
The above theorem is used in proving the following theorem:
THEOREM 3.2.
normal
Under the permutational model
(j) , S is asymptotically p-variate
V
-V
(~, I(g)·~O) in probability.l
PROOF.
Let e = (e , ••. ,e )' .,0 be a vector of finite real constants.
l
p
E(e' S I~ ) = 0, E[(e'S )2 1UJ ] = (e'V e)I(g).
-v
V
- -v
V
_ -Vprobability.
Then
We have to show that for every
We have,
(3.18)
I
I
I
••
I
E E[~*(F (x(~») ~* (F (X(k'»)]
i=l
k k -~1
k' k' -~i
vector ~ as above, ~'~V is under Q)v asymptotically normal (O'(:'~O:)I(g» in
Ie
I
I
I
I
v
Hence (3.4) is proved.
(3.17)
I'
using Markov's weak law
(k)
p
tn
Under IT v' {E e k (b kV (:~Vi )
k=l
is equiprobable over the
V! permutations of the columns of
*
~(O)
while {av(i), i=l, ••• ,v} is equiprobable
over the v! permutations of the integers 1,2, ..• ,v independently of the permutations of {
f ek(bkV-L
(R~~»)
k=l
i=l, ••• ,v}.
Since V
-V
~
EO in probability, to
-
prove the theorem it is sufficient to show that the sequence {
~ ekbk(R(~»}
k=l
V1
satisfies the Noether condition (see e.g. [10]) in probability and the sequence
{av(i)} satisfies the condition
1
By saying
~
is asymptotically p-variate normal
we mean for every vector a = (al, ••. ,a )'
(a'x -
a'~)/(a'Ea)1/2has ~symPtoticall~
a
~
(~,~),
where E is positive definite,
° of finite real constants
no~al
(0, 1) distribution •
I
12
II
I
I
I
I
I
I
max
l_<i l _< ••. _<ik,:::..V
v
(3.19)
(3.20)
v-+
~
kjV::;: 0
lim
v-+
oo
2
2
Let aV(l) ~ aV (2) <
V
2
V
i::;:l, ••• , v.
n (u) ::;: a (i) for
V
But,
-+ leg) <
as V -+
00
J 1
Hence,
00.
nv(u)du -+ 0
l-kv/'J
as V -+
00
since then kv/V -+ O.
max
l<i <••• <i
1
v -+ 0 as V -+
Also a
00.
Hence,
max
k
1 <i < ••• <i <v
-1
k-
<v
v
_
v
<
2
[a (i) - a ]
v
v
-+OasV-+OO.
Putting in particular k
v : ;:
1, it follows that the sequence {av(i)} satisfies
the Noether condition i.e.
(3.22)
Also,
(3.23)
,I
o
av ]2
V
i::;:l
I
,I
v
v
a
2
be the ordered a (i) 's and let us define a step function n (u) as
l:
I
I
[a (i) -
which implies the Noether condition,
and then apply Theorem 4.2. of [12], p. 514.
v
I
t
oo
v
0,::;:1
lim
i::;:l
-
I
I
kv
2
l: [a (i ) - a ]
v
-1
<
{ pl: e [ h «k»
max
k V R.
V1
l < i < v k::;:l k
v
-1
max
l<i<v
_ -b
kv
] }2
I
I.
I
I
I
I
I
I
I
13
-<
(p
P
2
max
max
v-I [b (i)
e )
k 1 < i < V 1 -< k .s.P
kV
k=l
L:
_ b ]2 + 0 as V+oo
kv
(the proof being similar as in the case of a (i) 's.) •
v
Further,
(3.24 )
(using lemma 3.1 and the fact that
is p.d.).
Hence,
P
(k)
2
max
{ L: ek[bkV(RV. ) - bk-v]}
1
1 < i < V k=l
(3.25 )
V
{
L:
i=l
- b
kV
]}
An immediate consequence of the above theorem is
1/ V
V
Asymptotic non-null distribution of M .
lemma:
PROOF.
(4.1)
We first prove the following elementary
V
LEMMA 4.1.
V
in probability.
2
2
Under ~ , M is asymptotically a central X with p degrees of freedom.
THEOREM 3.3.
4.
+ 0
~ e [b (R(k»
k=l k kv Vi
Hence the theorem.
I_
I
I
I
I
I
I
I
e
I
I
~O
(i)
V-1/2 1
(ii)
V
ma.x
Ix(k.)
~ 1 ~ V
V1
I+
0 a.e. as V +
-1
for all k=l, .•. ,p.
as V +
We first prove (ii).
00
,
for all k=l, ••• ,p.
We can write,
V
-1
L:
i=l
(k) 2
00
(X(k»2 _ x(k)2
vi
v
= Elx(~) 14+0
(k) 2
Since (XVI) , ••• , (XVV )
V1
< c <
00
,
using lemma 1.1,
(4.2)
Again,
(4.3)
(k) 1 2+0
ElxvI
x(k) -+E (x(k»
V
vI
So,
=
o
a.e.
as
V -+
00.
(ii) follows now from (4.1) - (4.3).
To prove (i), we first recall two equivalent forms of Noether condition given
by Hoeffding (1951).
We state these in a slightly different form convenient for
I
14
II
I
I
I
I
I
I'
re
I
I
I
I
I
I
I
.I
our purpose:
-+
0
as \)
a.e.
lim
a.e.
\)-+00
-+
00
for some r>2.
The right hand expression in (4.4.) can be rewritten as
(4.5)
= 0 a.e. for some r>2.
We shall prove first (4.5).
Taking r = 2 + 8' where (0 < 8' < 8/2) and using an
elementary inequality, we get
(4.6)
,,
Taking now,
( E' X (~)
V1
1
8
2 8 , 2+8"
+ )
8 - 28'
2 + 8' ,
we have,
< E 'X(~) ,(2+8')(2+8") = E'X(~) ,4+o<c<00. Hence, by lemma 1.1,
V1
V1
a.e.
(4.7)
-(k)
Also Xv
-+
a.e.
0
-+
Using now (ii), (4.6) and (4.7), we get (4.5).
from (4.4), (4.5), (4.7) and the fact
Let P
as V
X(k)
V
-+
0
a.e. as V
v and Qv be the probability distributions of
~V
-+
00 •
00 .
(i) now follows
Q. E. D.
under the null hypo-
thesis and alternatives
(4.8)
, (k=l, ... ,p).
Let PV1, and Q"l'
be the respective d.f. of Z
. under Hand
HV (i=l, ••. , V;
v
~V1
0
Let PVi(') and qVi(') be the corresponding densities.
= {qVi (X)/POVi (x)
(4.9)
i f PVi (x) > 0
otherwise
Consider now the following two statistics:
V
(4.10)
=
2
L:
i=l
V
Lv =
We also introduce the notations :
II
i=l
r , .
V1
Define
(i=l, ••• ,v).
V ~ 1).
I
I
e
I,
I
I
I
15
,
(4.11)
y' = (Y1'"'' Yp ).
~
We need show that the distributions Q are contiguous to the
1ennnas of LeCam.
v
distributions P
We adopt a procedure analogous to that of Hajek (1962).
v
following lemma will be needed:
E(W
LEMMA 4.2.
PROOF.
V
Ip V)
~
- l4
l(f
20
The
) (y' L y)l
W
V
1 2
. (.l.= 1 "
h Vi = v- / y'x
~ ~Vl.
Y~i'
•• , ")
v
an d
20
(4.12)
max
I (k) I 1/2 )( PL IYk)I
1 < i < V XVi / V
k=l
(using lemma 4.1(i)).
-+
0 a.e. as V
2
Also, E(hVl.. Ip V ) = y'
L Y which is non-zero and finite.
~
~ ~
We get from (4.12) after some algebraic simplifications,
00
where g (h .) =
Vl.
a.e.
J
_
Let,
(4.13)
Proceeding as in Hajek [12], we get the following lemma:
LEMMA 4.3.
Proceeding
00
now exactly as in ([12], lemma a, _p. 211), we get the result.
,e
I
SpO);
fl/2 (x), we get,
Ie
I
I
I
I
,I
I
I
= (S10"'"
The asymptotic distribution of M will be obtained by using the contiguity
V
Further writing (Y Vi - a)/a =
'I
I
I
§O
Var (W
V
- y' T i p )
~
~V
V
-+
0
as V
-
-+
00
,where
~V =
--
Q. E. D.
-+
00
I
16
I.
I,
Using lemmas 4.2 and 4.3 t it now follows that
(WV - y'
T + l4 l(f ) y'
~ y)2
~
~V
~
20 ~
E
I
PV ] ~ 0
(4.14)
I
I
I
I
I
I
LEMMA 4.4.
PROOF.
Under
Tv is asymptotically multinormal (2t l(f
Pvt
:'!v
real constants t
is asymptotically normal (Ot l(f
20
)
)
~
~) ~.
0 of finite
~' ~ ~).
We can write t
(4.15)
,
(4.16)
Al
¢(F 20 (YVi »
E
max
so) 1 ~ i ~
I
V
~ e
k=l k
Ip v ] = Ot
X(k) / v l / 2 1
vi
<
~
0 as
V
~
00
from lemma 4.1 (i) and
00
V
E
-+- e' E e
a.e.
as V
~
from lemma 4.1 (ii) •
00
i=l
a.e.
(4.17)
as
V
~ 00.
i.e.
the sequence { ~ e X(k)} satisfies the Noether condition
k vi
k=l
,
(k)
p
a.e.
. '
Now t under Pvt YVi's are independent of k:l ek XVi's and so are ¢(F 20 (YVi »'s
,
which are Borel-measurable functions of YVi's.
,
satisfied and
(F 20 (Y Vl
conditional on the
»"" ,
,
(~20(YVV»
Since (4.16) and (4.17) are
are all identically distributed,
X~~) 's fixed e' Iv is asymptotically normal
Since
a.e' t we get t
e'
T
is unconditionally asymptotically normal (Ot (e'
~
~V
~
:)I(f
20
»a.e.
Using the above lemma t we get now from (4.l4)t
,.
I
20
We have to show that for every vector e = (elt ... te )'
p
Ie
I
I
I
I
I
I
I
which implies
(4.18)
W is asymptotically normal (v
t l(f20 ) y'
We next prove the following lemma.
~ Yt l(f 20 ) y' E y).
Q. E. D.
I
17
I.
I
I
I
I
I
I
I
I
I
I
I
I
max
I
I
1 < i < V PVC qVi(~Vi)/PVi(~Vi) - 1 >
-
-
PV(lqVi(~Vi)/PVi(~Vi)
PROOF.
<
€
=
€
-1
-1
I
)
= 0
> €)
E
00
E
00
f
_
(i=l, •.• ,v)
00
So,
< c- -1 I 1/2 ( f )
max
Ih I
~
20 1 < i < V E Vi
<
-1
€
I
1/2
max
I I
(f 20 ) E (1 < i < V hVi ) .
We know, 1 <max
i < V Ih Vi I
+
0
a.e.
and
ma.x
Ih.1 < ,,-1/2{ ~
max
I (k) I} (1 <max
I I
u
l<'1. < V ~\~
k < P Yk )·
I 2. 1. 2. V V1. - v
k=l
(4.20)
••I
- 1
€
But it can be shown after some elementary algebra that
Ie
I
I
lim
V + 00
LEMMA 4.5.
VoL
But, ~V
-1/2
(where Ak
Ak
_
max
I (k)
I
f -1/2
max
1 < i < V XVi (w) dPOk(w) = ~V
(1 2. i 2.
Ak'
(Qk'
={1 i f W
0 if w
(w)
I
E:
_
~ k'
E:
~
E:
Q
k
V
IXVi
(k)
I
(w)
(w) )I~ (w)dP
Ok
POk) being the measure space and
- ~)
2. [ f v-1 {1 <m~x< vl~~)(w~}2dPOk(w)]1/2[ f I~(W)dPOk(W)]1/2
=
1
{v- E[l
<m~x< vlx~~)]2}1/2 {POk(~)}1/2
2. {v-1 (v 2/ (2v - 1)) 0'~}1/2 p~'2(~) 2.O'k p~'2(~)
POk(~)
+
O.
Hence, integrals of r.v.
1 2
v- /
l
+
0 uniformly in vas
<m~< vlx~~) (w) I
are uniformly
I
18
I.
I
I
I
I
I
I
I
Ie
I
I
continuous for each k=l, ... ,p and using (4.20) and the L -convergence theorem
r
(4.21)
E(l
([15J, p.163), with r=l, we get,
max
i < v IhVi I)
+
<
as
0
V +
00
•
The theorem now follows from (4.19).
We now state LeCam's second lemma (see[12J, p.205) which is an immediate
consequence of (4.12) and lemma 4.5.
LEMMA 4.6.
The statistic
log L satisfies
v
(i)
1
log Lv - Wv + -4 l(f 20 ) (y'
L: y)
~~~
(ii)
log Lv is aSymptotically normal (-
(iii)
the distributions Q are contiguous to the distributions P .
v
v
+
0 inV
P --",=-p.;:.ro;..;b;.;a;,:;;;b:.;:i;.;;:l:;:;i..;;,-ty,,-.
t
l(f 20 )(r'
~ r),I(f 20 )(r' L: y))a.e.
We are now in a position to derive the asymptotic non-null distribution of
~V
by using LeCam's third lemma (see [12], p. 208).
We first define the following
statistics:
-1/2 V
(k)
*
'
= V
L: [b (R . )- b J ¢ (F 20 (Y"i))' k=l, ••• ,p;
k
k
i=l
V V1
V
v
(4.22)
(4.23)
*
-1/2 V
Tkv = V
L:
i=l
*
~k(UVi)
*
¢
(F
'
20 (Y vi )), k=l, ••• ,p.
Then,
(4.24) E[(Sk*" - T*,)2
kv
v
'I
I
I
I
I
I·
I
Lo~ve
of
Ip ,,]
=
v
E[(V-1/2'~
i=l
{b
kv
I P V}
(R(k))_ b
vi
kv
= V-1
V
< [2 v-I L:
i=l
But l(g) <
00,
E[b
bkv
+
(R(k))_ ~*(U )J2 + 2
kv vi
k vi
0
as V +
00
(4.25)
kv
J l(g)
and proceeding as in Theorems 1.5. a (p. 157)
and 1.6. a (p. 163) of [12J
i=l, ... ,v.
b2
+0
as V +
00,
for all
Hence, from (4.24),
I PVJ
E[(S*
T*)2
kv - kv
* TkV
*
Hence, SkV-
+
+
0
as
V +
00,
for all k=l, ••• ,p.
0 in P -probabi1ity (and hence in Qv-probabi1ity because of
v
contiguity) for all k=l, ••• ,p.
I
II .
19
Using
a
.(4.26)
I
I
I
I
I
I
I
Bonferrouni inequality, we get,
S* - T*
~v
~v
-+
*
0 in P -probability (and hence Q -probability),
v
~
v
* )' and
S = (Sl*V , ..• , Spv
where
~v
T
* =
~v
(T
*
l V , ... ,
S
- s* = v- l / 2 ~ [b (R(k»_ b ] [av(R(iO»kv
kv
i=l kv Vi
kv
-\)
T
*
pv ) ' .
av ]'
Further,
k=l, .•. ,p.
Hence,
Ie
I
I
I
I
I
I
I
,I
2
. (0)
*
2
. (0)
*
2
- bkV ) {E[av(Rvl )- ~ (UVl )] +(E[aV(l\;l )-~ (UVl )]
E[av(~~»- ~*(UV2)]2)1/2} •
But,
and E[a (R (0)
. )V
V1.
~
*(U
.)] 2
V1.
-+
0 for all i=l, ... ,v.
Hence, from (4.27),
P ] -+ 0 as
(4.28)
v
V -+
00
for all k=l, ••• ,p.
Hence,
(4.29)
S
~v
* -+
- S
~v
0 in P
~
v (and hence Qv )- probability.
(4.26) and (4.29) give,
(4.30)
~v
-
*
~v -+
0 in Pv (and hence Qv)- probability.
1
L: Y
From (4.14) and lemma 4.6(i), log Lv - y'
T + -2 l(f ) y'
~
~v
20 ...., ....., ....,
.
-+
0 in
I
20
II
I
I
I
'I
I
I
Ie
I
I
I
I
I
I
I
II
P -probability (also used in proving lemma 4.6(ii)).
Hence, for any vector
'J
e
~
0 of finite real constants, (log L'J , e'S
) has asymptotically the same
~ ~'J
distribution as (y'
~'J -
21
l(f 20 ) y'
~
~'~'J)'
y,
1 l(f
asymptotic distribution as (y'
T
- -2
~
~'J
20
) y'
~
Using (4.30) the latter has the same
~
~
* ).
y,
e'
T
~
~
~'J
Let
-1
(4.31)
F (u), k,k '=1, .•. ,po
k
We prove the following lemma:
LEMMA 4.7.
(Y'~'J
Under P'J'
-
t l(f 20 )Y'E y
~'!~)
,
is asymptotically bivariate
1 1 *
normal (- 2 l(f 20 )y' Ey, 0, 1(f 20 )y' ~ y), (~' ~O ~)I(g),(~' ~ooy)fo ¢(u)¢ (u)du).
PROOF.
We have to show that for every (a,b)
constants, a
Y'~'J + b~'~~
~
(0,0) where a,b are finite real
2 1 *
b (e' ~O :)1(g) + 2ab(:' ~OO y) f O ¢(u) ¢ (u)du).
a y'T
~
~'J
a2(y'~
is asymptotically normal (0,
y)1(f 20 ) +
Now,
*
+ be'T
~ ~'J
'J
L:
*
Z'Ji ( say) ,
i=l
where
* + Z2'Ji
* (i=l, .•• ,'J),
Z* . = Zl'Ji
'J1.
i=l, ••• , 'J.
* = 'J -1/2 L:P aYk~k(U .),
k=l
'J1.
Zl"1.'
v
We may also note that E[ay'T
~
~'J
* IH 0 ]= 0
+ be'T
~ ~'J
*
2
2
and Var(~y'~'J + b:'~'JIHo) = a (y'~ y)1(f 20 )+ b (:'~O:) 1(g)
+ 2ab
(e'~OO
now is that
y) f 1O ¢(u) ¢ * (u)du.
*2
'J
E[ Z . X * (o)lp ] + 0 as 'J + 00, where for any 0 > 0,
i=l
'J1. Z'Ji
'J
L:
=
But,
As in [12] (pp. 217-218) all we need to show
1
{ .0
*
iflz.l>o
'Jl.
otherwise.
+ XZ~'Ji(O/2).
Hence,
*2
L: E[Z . Xz
'J
i=l
'J1.
*
(~)Ip ]
'Ji v
'J
I
21
<
II
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
II
v
*2
So, it is sufficient to show that E[ E Z'V,X Z* (O/2)!P V]
i=l J 1. jVi
+
0 as V +
00
for j=1,2
(these conditions being equivalent to the Lindeberg condition of the classical
Central Limit Theorem for
V
1,
E Zl ' and
i=l
V1.
*
V
E Z2Vi)'
The proof is exactly the
i=l
same as in Lemma 4.4 and hence is omitted.
Using now LeCam's third lemma (see [12], p. 208), lemma 4.6 and the remarks
,
,*
just above it, we get, under QV , e~ T
is asymptotically normal
~V
y)
((:'~OO
*
~V
i.e.
(~OO
f 1O ¢(u) ¢ * (u)du, (:' ~O :) leg)) for every : ~
~V)
(and hence
Yf 1O ¢(u) ¢ *(u)
9 of
finite real constants
is under Q asymptotically normal
v
du, ~O leg)).
Also, from lemma (4.5), ~v
+
~o leg) in
probability.
Recalling definition of M in (2.8) and using Slutsky's theorem,
v
we get that under Q , M is distributed asymptotically as a non-central chi-square
v
v
with p degrees of freedom and non-centrality parameter
(y
5.
,
,-1
~OO ~O
~OO
y)
*
1
2
(fO ¢(u) ¢ (u)du) / l(g).
ASymptotic optimality of tests and ARE •
Let
(5.1)
A
(Yv
A
A,
= (Ylv, ••• ,Y )
pv
being the maximum likelihood estimator (m.l.e.) of
y)
be
the likelihood ratio test criterion for the testing problem considered here.
Let,
(5.2)
If we make the assumptions on the likelihood
function L(Z~v !Y)
as in Wald [17],
.
~
(see also [9]), which are not restated here for the sake of brevity, we get the
following result:
THEOREM 5.1.
PROOF.
Same as in Theorem 2.4. of [9] with V,
Nv ' ~v* ' ~v' A2 (F, {lj}) and Sv respectively.
~v' ~,
l(f 20 ) and M* replacing
v
I
22
The theorem implies that Mv* + 2 loge A
V
II
I
I
I
I
I
I
of contiguity).
0 in Qv-probability (because
So if the assumptions (1.3), (1.4), (1.5), (1.8) along with
the regularity assumptions of Wald are satisfied, a test procedure similar to
the one we have considered based on M* possesses the optimal properties of the
v
likelihood ratio test as described in Theorem 2.3. of [9].
We might also remark that using LeCam's third lemma, it can be shown that
under the alternativesJMV* is asymptotically a non-central chi-square with
p degrees of freedom and non-centrality parameter,
(r'
~
y) I(f 20 ). Hence,
the ARE (in the sense of Hannan [13]) of our proposed test based on M relative
v
to the one based on M* will be given by
V
(5.3)
e
=
where
~OO ~~l ~OO
[ (y'
P2
=
J6
r)/(y'
¢(u) ¢*(u)du/[(
~
r)]
J6
p;
¢2(u)du)1/2(
J~ ¢*2(u)du)1/2].
We may also observe that in the particular case, ¢* = ¢,
'1~
~k
=
~k'
for all k=l, ... ,p,
e = 1 i.e. the test procedure based on the statistic M as we have proposed is
v
asymptotically optimal in Wald's sense.
I_
We may note that the general efficiency expression depends on y = a
-1
T.
However, using a well-known theorem of Courant (see Bodewig [1956]), we get,
I
I
I
I
I
I
I
I-
max
(5.4)
e = maximal eigenvalue of 2:
Y
~
min
-1
,
-1
~OO ~O ~OO
'
-1
e = minimal eigenvalue of 2: -1 ~OO
~O
Y
~OO
.
It is interesting to observe that instead of considering the rank statistics
Skv(k=l, •.. ,p) as defined in (2.2) if we consider the mixed statistics
S~v
l 2
= v- /
~ x~~) [av(~~»- "av ]'
k=l, ••• ,p,(which would be more logical to
i=l
consider when the
x~~)
(i=l, ••• ,v,
k=l, ••• ,p) are observable,) the appropriate
statistic will be the corresponding quadratic form in
S~v 's and the efficiency
expression will be
(5.5)
e
=
2
P2
Thus in the particular case when ¢ * = ¢, the resulting test is optimal according
to criteria described earlier.
cannot be used when the
.
However, as already mentioned, the mixed statistics
X(~) are not observable. We may also remark that unlike M ,
v~
the statistic based on X(k) 's is not unconditionally distribution-free when p=l.
Vi
I
+
V
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
••
I
23
We next investigate some particular cases where the efficiency expression
can be simplified considerably leading to some interesting results.
Case I
(5.6)
p
=
1.
Here, e
=
2 2
PI P2 ' where,
1
*
1 2
1/2 1 *2
1/2
PI = fa ~l(u) ~l (u)du/[(f o ~l (u)du)
(fa ~l (u)du)
]
We may note that this efficiency expression is the same as we would have obtained
by an extension of Hajek's ideas to simple linear regression with stochastic
predictors.
The test statistic which we would consider in that case will be
simply SlV'
This will be unconditionally distribution-free since in this situa-
tion under the hypothesis of no regression (or independence according to our
model), the permutation distribution of SlV becomes identical with its unconditional distribution.
optimal in Wald's sense.
For two-sided alternatives, the test procedure is
For one-sided alternatives, this will be asymptotically
~
uniformly most powerful (liMP) as in Hajek
[1962] in case
¢*
=
¢ and ~l*
= ~l.
It might also be remarked that the test considered here is an alternative approach
to the study of independence of two random variables as has been considered by
Bhuchongkul [2].
Bhuchongkul followed the Chernoff-Savage [7] approach in proving
the asymptotic normality of her test statistic and as such had to put rather
stringent conditions regarding the existence of the derivatives of the score
function and on its bounds.
But unlike our case the asymptotic normality of
her statistic follows even when regression is non-linear.
However, her observa-
tion that the normal scores test is the locally most powerful rank test under
the alternative that the variables under consideration have a bivariate normal
distribution is equally valid in our case.
In fact, in case of linear regression
our result is slightly more general in the sense that whatever be the parent
distribution, if
*=
~l
~l
asymptotically optimal.
and
¢* = .¢,
the resulting test as considered by us is
We give expressions for e in some particular cases:
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
I·
I
24
TABLE 5.1.
Showing some efficiency expressions.
cjJ(u) = 0~
1
<I>
-1
(U),
~l (u) = 01 <I>-l(u), <I>(x) = (2n)-1/2 JX exp.(-t 2 /2)dt
_00
cjJ *
I
I
1
u-2
~l*
-1 <I>-l(u)
°00
2 u -1
2 u - 1
°2
<I>-l(u)
-1 -1
000 <I> (u)
1
2
u
<I>-l(u)
°2
!
I
I
e
(3/n)2
3/n
3/ n
1
Case II. p=2
(i)
*
~k(u)
1
u - 2
(k=1,2),
rf,*()
1
~
u = 2U-,
~k(u)
= ok <I>-1 (u), (k=1,2)
rf,()
)
~ U
= 00-1 ....-l(
~
u,
P12 = corr (x(l)
vI'
X(12».
V
2
Here P = 3/n and we get after some simplifications ,
2
~o
1/12
=
(1/2n)sin
(1/2n)sin
L:
-1
-1
(P 1 2/2)
=
using these, we get after some lengthy algebra,
- -r 8)/
e = (3/n) (8'
where, 8 '
=
(~' ~O ~),
(ylol' YZoZ)'
~O = ~lZ ~l~
and
r
n/3
=
~O
2sin
2sin
-1
-1
(P 12/2)
(P l zl2)
n/3
-1
~O
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
25
Then,
eM
(3/n)
(3/n)
em =
min e
6
(3/n)
2
2
2
(1+P 12 )/(1+(6/n)sin
(1-P
12
)/(1-(6/n)sin
(1-P 12 )/(1-(6/n)sin
-1
-1
-1
(P 12 /2», 0 2 P12 < 1
(P 12 /2», -1 < P12
2 P12
(P 12 /2», 0
= (3/n) 2 (1+P 12 )/(1-(6/n)sin -1 (P 12 /2», -1
< P
12
20
< 1
< 0
We can draw the following conclusions as in Bickel [3].
.912 < e < .917
- M-
(1)
.
As a function of P, eM is symmetric about P = 0, concave
for 0 2P < 1 and -1 < P 2 0, attains its minimum of 0/n)2 at P = 0, approaches
it as Ip I -+ 1 and attains its maximum of . 917 at P = .78 •
(2)
e
m
considered as a function of P is symmetric about 0 and decreases from
a maximum of 0/ n)
(ii)
*
2
at P = 0 to an infimum of (3/n)sin(n/3) = • 827 as P
1
= u - 2'
(k=1,2),
~k(u)
¢* (u) =
CY
-1 ep-1 (u) ,
~(u)
¢(u) =
OO
2
2
P2 = 1 and so e = PI
Here
CY
=
CY
k
¢
-1
(u),
-+
1.
(k=1,2)
-1 ep-1 (u).
o
Hence for eM' em we get exactly the same expressions
as in Bickel (1965) and hence, all his conclusions hold in this case.
(iii)
CY
k
¢* (u) = 2u-1,
Here,
2
-1
-1
ep-1 (u).
2
PI = 1.
Hence,
¢(u) =
P2 = 3/n and
¢
CY
o
(u),
e = 3/n = .955
(iv)
le<5.7)
*
*
-1 (u),
~k(u) = CYk ep
¢ *(u) =
I
I
I
I
max e
6
=
e
CY
-1
OO
ep
-1 (u),
"-I
~k(u)
O"k
¢
(u) , (k=1,2) ,
-1 -1
¢ (u) = 0"0 ep (u).
Then,
= 1.
It can be shown in the general situation that if
O"k ep
= u e
-1
*
~k(u) =
2
2
(u), (k=l, .•• ,p), then PI = 1 so that e = P2
1
2'
= P22 (£' r
~k(u)
= O"k ep
-1
6)/ (6' £0 6),
*-1
O"k
ep (u),
Also, when,
(u), (k=l, .•• ,p), it can be found as in the case p=2,
I
II
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
II
26
r = ~O ~
where, ~O = «P kk ,»),
V
. -1 1
(2
«(2 S1n
P ,»).
kk
-1
~O'
We shall next show that
l - V-I is positive definite.
r~O
To prove this, it is sufficient to show that
~O ~ ~ ~' V T
T'
But
VT
T'
-
P
L:
=
for all
T "
p
T'
~O
T
T ,
T =
T. T , [2 sin
j,j'=l J j
L:
j
j
2n-l
L: c n Pjj ,
n=2
the same for all j,j'=l, ... ,p.
- T'
~
r
~O
T
=
L:
n=2
,
where c
(P
jj
,/2) - Pjj , ]
c
n
are positive constants and is
Thus
p
00
VT
~ ~
T'
_
-1
00
j,j'=l
(5.8)
0 (see e.g. Fraser [8], p. 55).
2n-l
L:
T.T., P ,
jj
n j,j '=1 J J
Using now a result of I. Schur (see [1], p. 96)
and B = «b .. ,»
JJ
we get that if A = «a
are positive definite matrices, C = «c .,»
jJ
is positive definite.
Hence, since «P
is positive definite for all r=l,Z, ••.
jj
,»
jj
= «a .. ,b .. ,»
JJ
JJ
r
is positive definite, «Pjj'»
Thus from (5.7),
T' V T - T' ~O ~ > O.
Since
i.e.
6.
-1
~O
-
7-1
is positive definite,
6' ~O ~ > 6'
Acknowledgement.
r 6.
,»
2
Hence, from (5.7), e < PZ.
Thanks are due to Professor G. E. Simmons for bringing
into the attention of the authors the paper by D.R. Brillinger [5].
I
27
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
I·
I
------.1..
_
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[2]
Bhuchongkul~
[3]
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