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ASYMPTOTICALLY MOST POWERFUL RANK ORDER TESTS
FOR GROUPED DATA*
by
Pranab Kumar Sen
University of North Carolina
and University of Calcutta
Institute of Statistics Mimeo Series No. 487
August 1966
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*Work supported in part by the Army Research Office, Durham,
Grant DA-31-124-G432.
DEPARTMENT OF BIOSTATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
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ASYMPTOTICALLY MOST POWERFUL RANK ORDER TESTS FOR GROUPED DATA*
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by
Pranab Kumar Sen
University of North Carolina, Chapel Hill,
and University of Calcutta
Summary.
The object of the present investigation is to extend the findings
",......-...-~
of Hajek [5J on asymptotically most powerful rank order tests (AMPROT) to
grouped data where the underlying distributions are essentially continuous but
the observable random variables correspond to a finite or countable set of contiguous class intervals.
In this context, the two sample problem for grouped
data is considered and various efficiency results are also studied.
Let us consider a sequence of random vectors X = (X , ••• , X
)
"''1
'II
VN
v
consisting of N independent random variables, where X has a continuous cumV
Vi
ulative distribution function (cdf) F (x), for i=l, ... ,N ' 1:::; V < 00. As in
v
Vi
H'jek [5J, we consider the model
1.
Introduction.
~
(1. 1)
where a..,
=F(O'
-1
[x-a.-~c
Vi
J),
i=l, ... ,N, 1<'1<00
V
~
and 0'(>0) are real parameters, (c , ... , c
) are known quantities,
VI
VN..,
concerning which we make the following assumptions:
N
(1. 2)
(1. 3)
V
L. i=l cVi
=
0,
N
L. V
i=l
c2
Vi
2
= CV'
sup C2
V V
<
00;
Max
c 2 !C2 = 0(1).
l<i<N Vi V
--V
Hajek [5J has considered the class of cdf's for which the square root of the
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* Work supported in part by the Army Research Office, Durham, Grant DA-3l-l24-G432.
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probability density function possesses a quadratically integrable derivative
1.
e.,
00
f
(1. 4)
[f'(x)!f(x)J 2 f(x) dx = A2 (F) <
00,
-00
where f(x) = dF(x)!dx and f'(x) = df(x)!dx.
Throughout this paper, we shall
also stick to the assumptions (1.1) through (1.4).
In Hajek's case,. X is
"'v
observable, while in our case, we have a finite or countable set of class
intervals
(1. 5)
a. < x:s a + , j=O,.l, ... ,oo (without any loss
j 1
J
of generality),
I.:
J
[where
_00
= a
O
<
a
1
<
a
2
< ... <
00
is any (finite or countable) set of ordered
points on the real line (-oo,oo),J and the observable stochastic vector is X* =
"'V
(X* , ... ,.X*
V1
V~
), where
(1. 6)
(1. 7)
Z ..
1.J
={
1,
0,
if X
Vi
€
I.,
J
otherwise,
for all i=l, ... ,N, j=O, •.. ,00.
V
Thus, having observed X*, we want to test the null hypothesis
"'v
(1. 8)
H:
o
~O
i.e.,
against the set of alternatives that
no regression,
~>O
(but infinitely close to 0).
It may be noted that if actual practice, even if the parent cdf's are
continuous, the process of data collection mostly introduces such a set of class
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intervals on which the data are recorded.
data, where the usual nonparametric methods (for continuous variables) are not
strictly applicable, and the present author is not aware of any optimum nonparametric procedures for such data.
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The object of the present investigation
is to consider some permutationally distribution free tests for regression for
grouped data and by a generalization of Hajek's [5] ideas, to show that these
are AMPROT for the same problem.
grouped data is also studied.
studied here.
In this context, the two sample problem for
Two different types of efficiency factors are
First, how these AMPROT compared with the tests by Chernoff and
Savage [2], Hajek [5] and Gastwirth [3]?
Secondly, how the various bounds for
the efficiency factors available in the continuous case compared with the parallel
bounds for grouped data?
Finally, in the two sample case, it is also indicated
how Capon's [1] results can also be generalized to grouped data.
2.
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This results in so called grouped
Asymptotically most powerful parametric test.
This test is considered in
brief, as it will be essentially required in the sequel.
(2.1)
F. = F([aj-a.]/O'),
(2.2)
6.. = [f(F-l(F
J
J
.»
J
p. = Fj+l-Fj
J
f(F
-1
for
(Fj+l»]/P j ,
Let us define
j=O, 1, .•. ,00;
j=O,.... ,00,
00 6.2 p ..
A2 (F, {I.}) = r. j=O
J
J
(2.3)
Now, 6.. can be written as
J
(2.4)
and hence,
cll( u) = -fieF
-1
(u»/f(F
-1
(u», 0
<
u
<
1,
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00
A (F~ (I.}) = L: '=0
(2.5)
J
J
I
F
f
1
.-I
2
F J'+1
Hu)du
/
j
F
f
F J'+1
du
j
I
I
F'+ l
00
<L:·
- J= O f
J
F.
J
uniformly in {I }
j
Again for
~
i.e.~
for all possible
sufficiently close to
zero~
-00
= a O < a < a < ... <
2
l
I,
I
00.
we have
F (a.+l)-F (a.) = P.{l + (~/cr)c
6. + o(~)},
Vi J
Vi J
J
Vi J
(2.6)
(by virtue of (1.2) and (1.3».
j(=O~ ... ~oo),
uniformly in
Thus, for
~
sufficiently
close to zero, the likelihood function is
N
nv u:.:
L(X*I~) =
V
i=l
(2.7)
z.. P. [1+(~/cr)c
J- 0 1.J J
Vi
6. + o(~)]}.
(2.8)
L(X*j~)/L(X*I~O) = 1
V
V
+
(~/cr)
00
L: c
L:
i=l Vi j=O
6. ZiJ'
J
+
o(~).
Let us denote by T
V
(2.9)
I\,
=
L:. 1
V
1.=
00
T
c
Vi
L:. 0 6. Z...
J=
J
1.J
Hence, from Neyman-Pearsonts fundamental lemma (cf. Lehmann [6,. p. 65]), it
readily follows that for testing H :
o
most powerful test function is
~O
against
~
_I
J
Consequently,.
NV
I
I
> 0,
t~
asymptotically
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1jrl(~) =
(2. 10)
Tv ,E
where
and
rE
1,
if
T
>T
r E,
if
T
= T
V,E
0,
if
T
< TV,E
V
V
V
V,E
are so chosen that E{1jrl(X*)/H
NV
0
r=
0 <
E,
E
< 1,
E
being the
desired level of significance of the test.
(It may be noted that in actual practice both
unknown, and as a result, 6.'s are also so.
J
~
and a in (1.1) are mostly
However, as in (3.1) - (3.4) of
Hajek [5], we may estimate ~ and a and work with the estimated 6.'s.]
J
00
Now L: j=O Zij6 y i=l, .•. ,
~
are independent random variables, and hence
Tv
is a linear function of N independent random variables, where (1.3) guarantees
V
the condition for the central limit theorem to be satisfied by the coefficients.
Consequently, we get on using the classical central limit theorem and avoiding
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the details of derivation, the following.
THEOREM 2.1
For
~
a,([T
V
infinitely close to zero
- (~/a)C2A2(F, (I.})]/C A(F, (I.})~ N(O,l),
J
V
J
V
where C2 and A2(F,{I.}) are defined in (1.2) and (2.3), and
V -
J
-
£(Z)~N(O,l) indi-
cates that Z converges in law to a normal distribution with zero mean and unit
variance.
Thus, from (2.10) and theorem 2.1 we get that the asymptotic power of the
test (2.10) is given by
(2.11)
1 -
¢>(T
€
-
(~/a)C
V
A(F,{I.}),
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where ¢(x) is the standardized normal cdf and ¢(, )=l-e.
e
3.
Asymptotically most powerful rank order test.
Let us define
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00
(3. 1)
N j ' for j=O, ... , 00, so that N = 2: j=O N j;
v
v
(3.2)
2:t~O
N
<Il(u)dul
J
FN '+1 =
V' J
-
INV
VI,
v
for j=O, 1, ... ,00.
If now N > 0, we define
Vj
(3.3 )
FN , j+l
= J V
FN '
v,J
where
F
-1
FN , J'+1
du,
V
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FN '
v'J
is the inverse of F(x) and
~(u)
is defined by (2.4).
If N
V, j
=
0,
we conventionally let
A
6
(3.4)
= <Il(F
Vj
.) .
NV' J
Our proposed test-statistic is then
(3.5)
S
=
V
N
V
~'1
1=
't'
and we shall see later on that S
V
against
3.1.
~
C
Vi
00 A
2:. 0 6
J=
Vj
Z. "
1J
provides an AMPROT for the hypothesis
~=O
> O.
Null distribution of S.
V
Since, we are dealing with grouped data, even
under Ho in (1.8), the distribution of Sv will depend on the unknown 6 (j=0, ... ,00)'e
j
However, under a very simple permutation model, Sv will provide a distribution
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free test.
Now, under H in (1.8), X* (i=l, •.. ,N ) are independent and
o
Vi
V
X* has
identically distributed random variables (i.i.d.r.v.), and hence ,.,v
a joint
distribution which remains invariant under any permutation of its N arguments.
N
V
Thus, in the N -dimensional real space R V, we have a set of N , permutationally
V
V
equiprobable points and the permutational (conditional) probability measure
defined on this set is denoted by (p.
V
Hence, under ~, all the N I equally
V
likely realizations have the common probability l/N
V
'.
Now, N
in (3.1) and
Vj
6
in (3.3) are unaffected by the permutations of the coordinates of X* i.e.,
vj
/\Iv
v
A
they are permutation-invariant.
Hence, by some simple reasonings it follows
that
E.." (Z'j} = N IN
(3.6)
lTV
~
Vj
V
for all i=l, ..• ,N, j=O, ... ,00;
V
' 1, ... , Nv' JrJ
.~ .• 0
E!'v (Z if Zij' } = 0 f or a 11 ~=
= , ... ,00;
(3. 7)
E...", (Z .. ·Z .•. ,}=N (N
-5 . . ,)/N (N -1)
rV
~J
~ J
vj Vj' JJ
V V
(3.8)
for all i'fi'=l, ... ,Nv and j,j'=O, ... ,oo; where 5
jj
• is the usual Kronecker delta.
By analogy with (2.3), we define
A2(F
(3.9)
~'
(1}) = E 00 6,2 N IN
j
j=O Vj Vj V'
and it is easy to see that (3.9) also satisfies (2.5), uniformly in (I.} and for
J
all (F }.
N
From (3.5) through (3.9), we have
V
E".. (S } = 0
(3. 10)
~V
and
V
where C2 is defined by (1. 2).
V
E"'t) (S2} = [N I(N -1)]C 2A2(F
~V
V
V
V
V
Now, under (p ,. S
V
V
NV
,(I.}),
J
can only assume N , possible
V
I
-8-
00
values (actually, there are N 1!~ N ! distinct equally likely permutations
V j=O Vj
of X*), and hence, for small values of N, the upper tail of the permutation
V
~V
distribution of S, can be evaluated to formulate the test function:
V
\jr2(~) =
(3.11)
1,
if
5 ,
if
O~
if
€
.>
S
s
V
v~€.
S = S
V
V~€
S
V
< sV~ €'
where Sand 5 are so chosen that E(\jr2(X*)jdP } = €, the level of significance.
V,€
€
...v
V
This implies that E(\jr2(X*)J H } = €, 1. e., \jr2(X*) is a similar size € test.
NV
0
..,V
Let us now define
(3. 12)
Under
W =
Vi
A
@,
/: :V, j
V
~.
00
A
0 /:::, Zi. for i= 1, ... , N .
J=
Vj J
V
's are all invariant while Z.. 's are stochastic.
that N of W 's are equal to
Vj
Vi
eracy condition on F(x) as
~J
A
6.
Vj
,
for j=O, ... , 00.
Thus, it follows
(3.13)
N
Thus, writing 8 equivalently as E .V c W , it follows from the well-known
V
~= l
Vi Vi
permutational central limit theorem by Wald-Wolfowitz-Noether-Hoeffding-H~jek
(3.14)
&m"'V (8!C
A(FN ,(I.))
V V
V
J
~
N(O,l), in probability.
Consequently, from (3.11) and (3.14), we have
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We now impose the nondegen-
(cf. [4]) that under (1.2), (l.3), (3.9) and (3.13)
_.
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S
(3.15)
V,E
where T
E
~ T
C A(F , (I }),
NV
j
5
€ V
€
~
0, in probability,
[It may be noted that ~ being a conditional
V
is defined by (2.11).
probabili ty measure, (given (N , j=O, ... , 00),) (3.14) and (3.15) hold in probVj
abili ty 1. e., for almost all (N j' j=O, ... ,00). ]
v
3.2.
ASymptotic optimality of
*2(~~
The main contention of this paper is to
establish the asymptotic equivalence of *1(~~) and *2(~)' in (2.10) and (3.11),
respectively.
LEMMA 3.1.
For this, let us first consider the following lemmas.
~
Under (1.1) through (1.4) and for
infinitely close to zero (being
_1
of the order N 2)
V'
A2(F , (I.}) converges in probability to A2(F,.(I.}), uniformi1y
NV
J
J
in (I.}.
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PROOF.
We shall prove the lemma only for
~O
as the rest of the proof will
follow by the contiguity argument of H~jek [5].
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real and positive numbers
lim
(3.16)
V=oo
(~}
~
Let us select a sequence of
in such a manner that
= 0 but lim
Nk
v=oo
V
~
= 00.
For any given N, the set of class intervals (I.} is divided into two subsets
V
J
G(l) and G(2) where
V
V '
(I .: P
(3.17)
J
j
<
-1
Tj
V
}.
Let F (x) = N
[Number of X < x] be the empirical cdf of X. Then by making
N
Vi ... v
v
V
Sup
Sup
use of the fact that . jF
-F.! ~ x jF (x)-F(x)! and the well-known result
N
N
J
Vj
J
V
on Ko1mogorov-Smirnoff statistics, we readily get that
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Sup
(3.18)
1
{N2/FN
.
J
.-F·I} is bounded in probability.
V' J
V
J
Further, it is easily seen that
(3. 19)
F.
SUPI!
:S.
J
J
1
F.
N,]
1
. -F. 12
$ 2()
u du 12/ FN
V' J
J
:s
1
A2( F )SUPj
. F
. -F. 12
N
J
V' J J
V
Hence, from (2.2), (3.3), (3.18) and (3.19), we have for all 1. EGO),
J
V
_I
(3.20)
where
(3.21)
Consequently, from (3.16), (3.20), (3.21) and some simple algebraic manipulations, we have
"'2
(3.22)
/
L: (1)6 .N . N
G
VJ VJ
V
_
2
- L: (1)6.P.
G
J J
V
+
0
P
(1).
V
Again, using (2.4) and (3.3), we have
F j +l
(3.23 )
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6C;P.
J J
= !
F.
J
F j +l
$2(u)du -
!
F.
J
[$(u) -6.J2du;
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FN ,. j+l
3,.2
(3.24)
N
Vj Vj
INV
V
f
[$(u) -~ .J2du,
F
VJ
N , j
V
A
where 6.. and 6.
J
FN
'+1]'
v,]
~(u)
on [F.,F.+ ] and [F
.,
NV' J
J J 1
Now, using (3.18) and some routine analysis, it is easily
are also the conditional mean of
Vj
respectively.
seen that
F j +1
(3.25)
f
L: (2)
G
~2(u)du
F.
J
+ 0(1),
P
and we shall show that the second integral on the right hand side of (3.23)
converges to zero (as
V~)
and the second one on the right hand side of (3.24)
converges to zero, in probability.
For an unessential simplification of this
proof we shall assume (as in lemma 2.2 of [4]) that ~(u) is
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from the existence of (1.4) that if P <
j
lim L:
(3.26)
V=oo
G(2)
V
~2(F .)P. = lim
J
J
V=oo
L:
(2)
G
V
~
for all I
j
i
G~2),
€
~2(Fj+1)PJ' = f
G(2)
in u.
then
~2(u)du.
V
(3.27)
Therefore
Fj + l
(3.28)
L:
(2) f
G F.
V
J
[$(u)- 6..]2 du ~r; (2)[~2(F'+1)P, - ~2(F.)P.]~ 0
J
G
J
J
J J
V
It follows
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as V~ (by (3.26».
Simi1a~lY,
from (3.16), (3.18) we have for all I
j
€
G~2)'
F
'+1 - F
. < n + 0 (N-2) < 2 n for adequately large N. Consequently,
N,J
N, J
V
P Y
V FN J'+l
V
v
V
V'
P
as in (3.26) and (3.27), we get that L: J
[Hu)-6 J2 du ~ O.
A
F
N , j
Hence,
Vj
V
(3.29)
L: (2)
G
62 yJ.
N
Vj
INV = L: G(1)6::;P.
J J
V
+
0
P
(1).
V
Hence, the lemma follows from (3.22) and (3.29).
Under H in (1.8), E(S -T }2 ~ 0 _as V~, where Sand Tare
LEMMA 3.2.
V-V
VY
0--
defined in (3.7) and (2.9), respectively.
PROOF.
N
N
= N---3L..
-1
V
Y
00
.IN -
c 2 .E [ L: (,8, . -6.)~
i=l Vl Y j=O VJ J
VJ
L:
V
00
(L: (,8,
j=O
IN
-6)N
vj
j
vj
}2]
V
where E
'
and E stand for the expectation over the permutation distribution
V
and the distribution of the order statistic associated with X*, respectively.
fly
"'v
Thus, it follows from (3.30), that we are only to show that
E (L:
V
.00
J= 0
(6
Vj
-
6.)2 N IN}~ 0 as V~.
J
Vj V
We rewrite the expression in (3.31) as
(3.32 )
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(3.30)
(3.31)
_.
E (L:. 00
V J= 0
62
N
Vj V j
IN} - 2
V
00
6.E (6 N
j= 0 J V V j V j
L:
IN}
V
00
+
L:
j= 0
6::;P .•
J J
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Now, ~j:O ~j NVj/N is nothing but A2(F ,{I j }), defined by (3.9), and hence,
V
Ny
it is bounded by A2(F), defined by (1.4), uniformly in all (N ,j=O,l, •.. ,oo).
Vj
Further, by lemma 3.1, it converges to ~j:O 6jP j , in probability. Since, for
a bounded valued random variable convergence, in probability, to a constant
implies the convergence of the expectation to the same constant, we readily get
(3.33)
E
V
U:.J= 0
00
.....2
b;.
Y
N
j Vj
I}
N ~ ~.
V
00
J=
2
0 6 P. as V~·
j J
Precisely, on the same line, it follows that
00
.....
E {~. 0 6.6 . N
(3.34)
V
J=
J YJ
./NV}~ ~.J= 0
00
VJ
2
6.P. as V~·
J J
(3.32), (3.33) and (3.34) imply (3.31), which in turn asserts the truth of the
lemma by (3.30).
Hence, the lemma.
LEMMA 3.3.
Under H in (1.8), (T, S ) converges in law to a bivariate normal
o
V
V
distribution which degenerates on the line T = S •
V
PROOF.
V
By virtue of lemma 3.2, any linear function as
V
+ bT
V
converges in mean
square to (a+b) T, and hence, from theorem 2.1, has asymptotically a normal dis'll
tribution with zero mean (under H) and variance (a+b) 2C 2A2 (F, (I.}).
o
V
J
The rest
of the proof follows from theorem 2.1, lemma 3.1 and lemma 3.2, and hence is
omitted.
THEOREM 3.4.
Under the sequence of alternatives in (1.1) (i.e., with
_1
to zero at the rate N 2),
V
&'([S
V
- (~/(J)C2A2(F, (Ij})]/C A(F, (I.}»~ N(O,l).
V
V
J
~
tending
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The proof is an immediate consequence of lemma 4.2 of Hajek [5] and our
lemma 3.3, and hence is not reproduced again.
Consequent of theorem 3.4, the test o/2(X*) in (3.11) has asymptotically
...v
the power function
1 - cp(-r
(3.35)
which agrees with (2.11).
E
- (13/cr)C A(F,(I.}»~
J
V
Thus, S
V
provides the AMPROT for H : 13=0 against 13>0.
0
This result also applies in particular to the two sample
where 0
V
Further~
is real.
with Capon's [1] technique and use his
~(u)
where
the results derived here can also be extended to the
problems of symmetry and scalar alternatives.
instead of the
prob1em~
in (1.4).
~(u)
For that one will have to work
(defined by (iv) on p. 89 of [1])
The rest of the procedure will be very similar to
the one considered here, and hence is omitted.
Fina11y~
the impact of these
findings on AMPROT for truncated/censored two sample problem will be considered
in the next section.
4.
Asymptotic efficiency.
Suppose now for the AMPROT we work with the assumed
density function f(x) instead of the true density function g(x) (=G'(x)J where
00
(4.1)
A2(G) =
J [g'(x)/g(x)]2
dG(x) <
00.
-00
We define G. and
J
A1so~
we let
~
J
as in (2.1) with F replaced by G(x), and F
. as in (3.2).
NV' J
_.
I
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
-.
I
I
I.
I
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I
I
I
I
I
I_
I
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(4.2)
~*(u)
=
gt (G
-1
(u»/ g(G
-1
(u», 0<u<1,
Gj +1
6* =
(4.3)
j
f
G.
J
~*(u)du/~,
J
6**
j
=
Gj +1
f Hu)du/P'!!,
G.
J
J
00
00
(4.4)
00
C(F, G, {I.}) =
(4.5)
J
(4.6)
THEOREM
~
j~O
6*j 6":* P'!!;
J
J
p( (I.}) = C(F, G, (I.}) /[A(G, (I.})· B(F, {1 })].
j
J
J
J
4. 1.
Under (1.1) - (1.4) and (4.1), the asymptotic power of the test
(3.11) is equal to
PROOF.
Defining
and following the same approach as in sections 2 and 3, we get that
(4.9)
(i)
(4.10)
(ii)
(4.11)
(iii)
(4.12)
(iv)
E{[S -T**J2jH }~ 0 as 'Y~,
V V
0
~([ '1*-(13/0" )C 2A2(G, (1 )) J/C A(G, {I.}» ~ N(O,l),
V
V
j
'Y
J
~(['1**/C B(F, (1.})JIH )~ N(O,l),
V 'Y
J
0
~('1*,'1**IH ) tends to a bivariate normal distribution with a
'Y
'Y
0
correlation coefficient p({1 }), given by (4.6).
j
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The rest of the proof follows directly from lemma 4.2 of H~jek [5] and (4.9).
Hence, the theorem.
As in Hajek [5], we can interpret [p({I.})]2 as the efficiency factor of
J
(3.11) with respect to the asymptotically most powerful parametric test; the
interpretation for the two sample problem again being the ratio of the sample
sizes needed to attain the same power.
REMARK 1.
The loss in efficiency due to grouping of data, as obtained from
our theorem 3. 4 and theorem 1. 1 of Haj ek [S],. is equal to
(4. 13)
00
= L: '-0
J-
F J'+1
J
F.
1
[Hu) -6.]2du! J ~2(u)du,
J
0
J
~(u)
are defined by (2.2) and (2.4), respectively.
(4.13) can be
made arbitrary small, provided the Lebesque measures of all the class intervals {I.} are also arbitrarily small.
J
Again, from our theorem 4.1 and theorem
6.1 of Hajek [5], it follows that the loss in efficiency in the case where the
assumed density differs from the true density, is given by
(4.14)
where p({I.}) is defined by (4.6) and p by (6.3) of [5].
J
(4.14) may be greater
than, equal to or less than (4.13), depending on p and p({I.}).
J
REMARK 2.
I
I
I
I
I
I
_I
J
where 6. and
--I
If we take I 0
: x <0
x, while
1 ,0
•.. ,1 to be all of sufficiently
0
1
small Lebesque measures, the results will relate to the AMPROT for truncated
I
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I
I
I
-.
I
I
I.
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I
I_
-17-
case (x < x
-
0
similar way.
being truncated).
Again, in the two sample problem~ Gastwirth [3] has considered
N*«
V
the censored case where only
sample are
More than one truncation can be dealt in a
observable~
and Nk/N
V
V
N ) of the ordered variables of the combined
V
approaches p(O < p < 1) as
N* is given but the corresponding truncation point is
V
V~.
random~
In his
case~
while in our case,
the truncation points are given and N (j=O~ .•• ~oo) are random. In spite of this
Vj
basic difference~ the power properties can be studied by the same formulae (viz.,
theorems 3.1 and 3.2 of Gastwirth [3], and our theorems in section 3 and 4).
However, he has only considered the optimality of his censored test and the power
can be traced with the aid of our theorems 3.4 and 4.1.
REMARK 3.
In the particular case of f(x) being assumed to be normal, the cor-
responding S will be termed the grouped normal store statistics.
V
ing test in (3.11) will be AMPROT for normal alternatives.
The correspond-
On the other hand,
the grouped Wilcoxon's test also belongs to the class of tests considered here
I
I
I
I
I
I
I
I·
I
(namely when we work with logistic distribution).
This may be written as
(4.15)
where Zij'S are defined by (1.7) and
(4.16)
The asymptotic normality (permutationally as well as unconditionally) can be
established in the same manner.
The test [similar to (3.11)] based on W can
V
be shown to have the asymptotic power equal to
(4.17)
1 - ~(T
E
- (~/cr)C w)~
V
I
-18-
••I
where
_
(4.18)
~(u)
and P. being defined by (2.4) and (2.1), respectively.
J
1
~]2
4
,
Consequently, from
theorem 3.4 and (4.17), the efficiency factor of the grouped Wilcoxon's test
with respect to the grouped normal score test is equal to
(4.19)
For normal cdf, (4.19) is easily seen to be less than one, uniformly in (I.}.
J
But there are certain advantages of the test based on W.
V
First, this test
is valid for all continuous cdf's, while the asymptotic optimality of the normal
score on other tests, deduced in this paper, is implicitly based on the assumptions that (i) A2(F) defined by (1.4) is non-zero finite (which localizes the
scope only to cdf's having continuous first and second derivatives, satisfying
(1.4», and (ii) 6., defined by (2.2) is not a constant (otherwise, T= 0).
J
V
For many distributions, for which the range is not extended to infinity on both
sides, either of these two conditions may not hold.
For example (i) for simple
exponential distributio~ f'(x)!f(x) = ~ and hence 6. = 0 for all j, or (ii)
J
for uniform distribution f'(x)
=0
a.e., and hence, A2(F)
= O.
In such a case,
S in (3.5) becomes equal to zero and hence, the tests are not constructable.
V
But the so called grouped Wilcoxon's test can be used. This is AMPROT for
logistic densities.
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-.
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I.
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I_
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I·
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REFERENCES
[ lJ
CAPON, J. (1961),
rank tests,
[2J
CHERNOFF, H. and SAVAGE, I. R. (1958), Asymptotic normality and efficiency
of certain nonparametric test statistics, Ann. Math. Statist., ~
972-994.
[3J
GASTWIRTH, J. (1965), Asymptotically most powerful rank order tests for
the two-sample problem with censored data, Ann. Math. Statist., ~
1243 -1247.
[4J
" J. (1961), Some extensions of the Wald-Wolfowitz-Noether theorem,
HAJEK,
Ann. Math. Statist., ;g, 506-523.
[ 5J
HAJEK, J. (1962), Asymptotically most powerful rank order tests,
Math. Statist., ll., 1124-1147.
[6J
LEHMANN, E. L.
Asymptotic efficiency of certain locally most powerful
Ann. Math. Statist., ;g, 88 -100.
I
(1959),
Testing Statistical Hypotheses,
Ann.
Wiley, New York.