•
•
ON SEQUENTIAL R-ESTIMATION OF LOCATION IN THE GENERAL
BEHRENS-FISHER MODEL
by
Pni.nab Kumar Sen
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1439
June 1983
•
•
ON SEQUENTIAL
R-ESTI~~TION
OF LOCATION IN THE GENERAL
BEHRENS-FISHER MODEL
Pranab Kumar Sen
University of North Carolina, Chapel Hill.
Key Words and Phrases: Asymptotic risk-efficiency; Behrens-Fisher
model; cost-function; point estimation; rank estimators; rank
order statistics; risk; shift model; unifom integrability.
ABSTRACT
In the context of asymptDtically minimum risk (sequential)
pDint estimation Df the shift (i. e., difference) of IDcatiDns Df
•
two distributiDns, sequential analogues Df the classical RestimatDrs are cDnsidered.
AIDng with the unifDrm integrability
and moment cDnvergence of the classical (non-sequential) IlestimatDrs, asymptDtic risk and distribution Df the al lied stDpping
times for the sequential estimatDrs are cDnsidered.
•
The general-
ized fDrm Df the Behrens-Fisher prDblem (relating tD the difference
Df locations of two symmetric distributions of possiblY different
forms) is presented and desirable sequential R-prDcedures arc
studied.
In this cDntext, the performance of the two-sample rank
estimator is studied, and, side by side, an alternative approach
(based Dn the di fference of one-sample R-estimates) is al SD cDnsidered.
Allied efficiency results are studied.
111C chDice Df
(asymptotically) Dptimal score functiDns is discussed and the nunoptimality of the usual two-sample estimates for the Behrens-Fisher
mDdel is studied.
•
•
1.
Let
{X.; i>l}
1
INTRODUCTION
be a sequence of independent and identically
-
.distributed random variables (i.i.d.r.v.) with a continuous distribution function (d. f.) F, defined on the real line
Also, let
{Y.,
J
j~I1
d.f. G, defined on
E
=
(_ro,ro).
be a sequence of i.i.d.T.v. with·a continuous
E.
In the classical
two-sa~lc
problem, one
assumes that
G(x)
so that
=
6
F(x-6),
xEE,
6 real,
(1 . I )
stands for the shift (i.e., difference of locations).
In the parametric case, the difference of the two sample means is
taken as an estimator of 6,
while in the nonparametric case, some
two-sample rank statistics may be employed to obtain some robust
estimator of 6
•
u
these are caJ.l ed R-estimatol's.
In the context
of (asymptotically) minimum 1'isk point estimation of 6
(based on
an appropriate loss function incorporating the cost of sampling),
procedures based on samples of predetermined sizes may not work out,
and mUlti-stage or sequential procedures are advocated.
For the
one-sample location problem, a detailed discussion of these sequent.ial procedures is given in Sen (1981, eh. 10, Sec. 6).
The first
object.ive of t.he current. invest.igat.ion is to present. sequential Rprocedures for the two-sample problem, sketched above.
In this
setup, one needs to study the uniform integrability and momentconvergence properties of two-sample R-estimat.ors, and these are
considered here.
In a somewhat more general setup, one assumes that
8 2 -°1,(1.2)
where
FO and GO are both symmetric about 0, 6 , 6
are the
2
1
respective location parameters, and 6 is the difference of the
location parameters. Here.
Fa and Go need not be of the same
form.
In the socalled Behrens-Fisher (BF-) model, one may take
Go(x) = FO(vx),
•
XEE,
for some (unknown)
setup, one additionally assumes that
1'0
v>O;
in the parametric
is a norillal d. f.
Ghosh
and Mukhopadhyay (1979) have considered the BF-model wherein
F
O
has not been restricted to the normal family and formulated sequen-
•
tial procedures' leading to asymptotically minimum risk point estimates based on the sequence of two sample means and variances.
In
the non-sequential setup. the robust.ness of two-sample R-estirnates
for the generalized BF-model has been studied by Hoyland (1965) and
Hamachandramurty (1966), among others.
The second objective of the
current study is to consider sequential counterparts of these estimates and to focus on their asymptotic risk and related properties.
For the classical shift-model in (1.1), along with the prelimLoary notions and basic regularity conditions, the R-estimators are
introduced in Section 2.
The uniform integrability a.od mornent-
convergence results for these estimators are studied in Section 3,
and, in Section 4, these are incorporated in the study of the asymptotic minimum risk property of the proposed sequential estimates.
Properties of these estimates under. the general BF-rnodel are then
studied in Section 5.
Section 6 deal s with an alternative form of
sequential estimates which behave very similarly to the ones in
Section 2 for the model (l.l), but have more intuitive appeal for
the general model (1.2).
•
In this context, the one-sample results
of Sen (1980) are generalized in a natural manner to the two-sample
case, when the general B-Fmodel rnay hold.
The concluding section
deals with some general comments on the proposed procedures;
2.
SEQUENTIAL R-ESTIMATES FOR THE SHIff MODEL
For the model (1.1), Jet
~ILitah
Ie estimator of
/).
of the two distribut-ions.
and a posi ti ve
J
n
-+
v2
We assume that there exist an
finite number
.2nE (Tn -tl).
v2
Tn = T(XI, .. ;,X ; Y1, ... ,Y ) be a
n
n
based on samplos of sizes n from each
2
0<\)<00,
v
2
(~l)
such that
exists for every
as
nO
n>n .
- 0'
n~.
I.ater on, we shall verify these for the proposed
(2.1)
(2.2)
R~estimators.
,
TI,e
•
•
~ss
In estimating
6
by
is taken as
T
n
L (a,C) = aCT _6)2 + c(2n),
n
a>O,
n
c>O,
(2.3)
so that the risk is equal to
Pn ( a, c ) = ELn ( a, c ) = (2n) -lav2n + 2cn, V
(2.4)
n~O'
Our objective is to minimize this risk by a proper choice of
(lVhere the total sample size is 2n); this optimal
2
{v }
n
depends on
o
n (a,c)
and
a = 1
lim{
0
r-'
c+O 2n c /(v/yc)
•
v,
For known
=
n (a,c),
(a fixed), this optimal choice can be
c+O
Without any loss of generality, we
derived readily by using (2.2) .
set
say,
Conveniently, in an asymp-
as well as on (a,c) .
totic setup, where
n,
n
0
=
o
n.
Then,
c
lim{
and
1
r:--.
(2.5)
c+o P (1,c)/2VYCf = 1.
nO
c
(2.5) provides the desired solution.
{VA}
is not known, but a sequence
of consistent estimates
n
~(X , ... ,X ; Y , ... ,Y )
When
A
V
n
=
of v is available, led by (2.5), we may
l
n
consider a stopping number N , c>O, by letting
n
1
c
N
c
min {n>n :
- o
=
TN
estimating
p*(c)
6
(v +n
A
n
-h}
)
(2.6)
is a suitable constant) and adapt the sequential
(where h (>0)
estimator
2n>c
-~
c
based on
by
= E~
c
(l,c)
The risk of
Xl""'~C;
is then equal
= 2cEN
2
+ E(TN -6) .
c
c
(2.7)
We are primarily interested in showing that for a general class of
tlVo-sample rank-based (R)-estimators, we have
~~~P*(c)/P
(2.8)
O(l,c) = I,
nc
so that
•
TN
c
is asymptotically (as
dO) risk-ef.ficient.
shall study the asymptotic b.ehaviour of the stopping number
First, we consider .the fol lowing R-estimators of
6.
Also, we
N.
c
For every
n (,::1)
Y
let
Y
Rnl,· .. ,Rnn be the ranks of YI""'Yn among Xl'
Xl"" ,Xn , YI ,· .. ,Yn · Also, for every m (~1) let am(l), ... ,am(m)
be a set of scores generated by an absolutely continuous, nondecreasing and square integrable score function
~: {~(u),
O<u<l}
•
in the
following way:
or
where
k: 1, ... ,m,
HEU ),
mk
(2.9)
< ..• <U
are the ordered r.v. of a sample of size m
ml
mm
from the uniform (0,1) d.f. Without any loss of generality, we may
U
set
1
f0
~:
~(u)du
For every real
•.• , Y -d
d,
A~ :
f
Y
let
among the
n
and let
and
0
:
1
o
~2(u)du :
2n
Y
: 2n
1 <.oi:l
~ 2n a (1
. ) . (2 . 11)
2n
S (d)
is" in
n
d,
~(sup{d:
Sn(R) : Tn
and we set
S (d»O
n
+
inffd:
S (d)<O}).
~.
is the so-called R-estimator of
general regularity conditions, as
!z "
(2n) (~n (R) -~) - N(0, \J
\J
2
2
4A<p/Y
:
y : y(<p,~) :
2
: 4y
f
1
o
-2
,
2
•
(2.12 )
n
invariant, consistent and robust estimator of
where
(2. 10)
Rnl (d), ... ,Rnn(d) be the ranks of .Yl-d,
observations Xl"" ,Xn , Yl-d, ... ,Yn-d,
S (d)
n
Then
1.
It is a translation~
and under quite
n~,
(2.13)
),
and
<P(u)~(u)du;
~(u) : -f' (F-l(u))/f(F-l(u)),
(2. 14)
O<u<l;
(2. 15)
it is assumed that the d.f.F has an absolutely continuous density
function f with a finite Fisher information:
<
00
Now
<P
is a specified function, but
I(f):
y
f
(f'/f/ df
(and hence v)
•
•
,
depend on the unknown
F
through
W.
Thus, we need to verify
y (or
(2.1)-(2.2) first and also to estimate
v), so that
N
1n
c
(2.6) may be defined properly and we may be able to verify (2.8) .
Note that under
H :
O
pendent of the underlying
6 = 0,
F,
there exists two real values
ul
p{Sn, L~Sn (O)~Sn., HO) = I - an
as n increases. Let then
has a distribution inde-
S (0)
n
and hence, for every
S
n,1
and
(~l-a).
S
n,U
where
a(O<a<1) ,
such that
a
n
converges to
a
A
6
n,L
=
sup{d:
Sn (d»Sn, uJ,
(2 . 16)
=
in f{d :
Sn (d)<Sn, L}'
(2.17)
A
6
I)
n,U
=
n
rzn (~. n,U-~ n,L)
(2.18)
From the results in Sen (1981), we conclude then
•
•
vn
where
= gn
1(2T
12) + v,
ap
T
as
n-- ,
(2.19)
is the upper SOa% point of the standard normal d.f.
al2
A
For the proposed sequential R-estimates, in (2.6), for
the estimator in (2.19).
v ' we use
n
Thus, the proposed sequential R-estimator
depends on the stopping rule in (2.6) based on the
A
v
n
in (2.19)
and the point-estimation rule in (2.12).
3.
MOMENT CONVERGENCE OF R-ESTIMATORS
For the study of (2.1)-(2.2) with possible generalization for
the 8ehrens-Fisher model, first we consider the existence of the
A
moments of
6n (R) , where (I. I) may not hold. 111US, we assume that
are i.i.d.r.v.
XI ' ... , Xn are i.i.d.r.v. with a d. f. F and YI""'Y
n
with a d. f. G, where G and F may not satisfy (I. I). We assume
that for some
a>O
(not necessarily
?,I),
II xl adF
< ",
and
JlxladG<",
Let
•
X 1<" .<X
and Y 1<" .<Y
be the two sets of
n:
non
n:
n:n
order statistics corresponding to Xl'" "X
and Y1"",YnJ
n
respectively. On the scores a (k), 15.k~2n, we impose the same
2n
conditions as in Section 2, and, without any essential loss· of
generality, let
a
= O. Then, note that for every
Zn
exists a k (=k) such that
n
l:n
a (r)
l:Zn
( ) < 0
r=k+l Zn
+ r=Zn-k+laZj r
,
n,
•
there
(3 1)
.
(3.Z)
° < ct]
-1
] im n -lk n < limn kn -< ct z
Let us now choose d in (Z.ll), d = Y
n:n- k
and further,
~
Then
0.
for k values of j
Y, - d
]
for (l1-k) values of J'
(3.3)
As a result, for
Sn (d) =
11
d=Y
-x
+0
n:n-k
n:k
'
-If,sum of (n-k) of a (1), ... ,a (n) plus
Zn
Zn
k of a
(n+ I), ... ,a
2n
2n
•
(ZnJ)
I
n
Zn
)
< n- {l:r=k+laZn(r) + l:r=2n_k+laZn(r)
<0,
by (3.l).
(3.4)
.'
Therefore, by (Z. ]Z) and (3.4), we conclude that
3nCR)
< Y
-
n:n-k
- X
(3.5)
n:k'
Similarly, if we take
d = Y : - X : - + 0, we have by (3.2),
n k
n n k
-l k
a (r)} > 0, so that
Sn(d) ~ n {l:r=la2n (r) + l:2n-k
r=n+l 2n
ZnCR)
lIenee, to show that
n
~
no(m),
(3.6)
-> Yn:k - Xn:n- k'
E I Zn (R)
1
m
< "",
it suffices to show that for
fini te moments up to the order
n -l kn
and
E
for some
(ctl,ctzL
E I Xn :
m
kn
E!Yn : n - k +llm < "",
n
directly from Sen (1959).
1
a (>0)"
< "",
for every
E I Xn : n -
m (>0)
F and
G admi tting
for every
m
ku
o<a{ct <]'
Z
+l 1 < "",
n~, nO(m,a),
and all
I
E Yn : k
with
!m < ""
n
and these follow
•
•
A
Remarks.
is
concerned, we need not even take the two samples of equal size.
very similar argument works out when there are
,
~n(k)
As regards the existence of the moments of
A
n
first sample
1
second sample observations, 'where n /(n +n ) is bounded
and n
1
1 2
2
away from 0 and 1. This case will be faced in the B-F model. This
moment existence result extends directly Theorem 2.1 of Sen (19RO)
to the two-sample case.
Let
US
nOW return to the model (1.1) and verify (2.2).
For
this, we need to impose some additional regularity conditions which
are stated below.
For the score function
exists a
K (O<K<oo)
•
~
Also, we assume that
in (2.9), We assume that there
o(>~),
and
! (d r /du r )$(u) I
$
such that for every
K[u(u-u)] -~+o-r ,
u
E
r : 0,1,2.
F belongs to the class
(0,1),
(3.7)
FO of all absolutely
continuous d.f.'s for which the p.d.f. f and its just derivative
f'
are bounded almost everywhere (a. e.) and
~~m f(xH(l) (F(x))
are finite.
x~oo
(3.8)
Then, we may virtually repeat the proof of Theorem 2.2 of Sen (1980)
wherein we use Theorem A.4.2 of Sen (1981, p. 395) and conclude that
if in (3.7),
0 > (1+T)/(4+2T),
for some
T>O,
then for every
k < 2 (1+T),
lim E{(2n)k/2!6
n-,
where
Z
n(k)
_~\k}: VkE\zlk,
(3.9)
has a standard normal distribution and
defined by (2.14)-(2.15).
v2
=
4y -2
is
As such, if we make use of (2.13), (3.9)
and Theorem A. 4. 2 of Sen (1981), we obtain 'that under the assumed
regularity conditions
(3.10)
•
"
here
R*:
O( vn
Ten)
n
Eln
-J,;
Rn*\
k
... 0,
a.
5.,
V k:
as
n-+<o,
an d
0<k<2(1+)
_
T,
T >0 .
(3.11)
These results are strengthened versions of the weaker representation
results of .Jur,,~kov1i (1969), under more stringent regularity condi-
•
tions.
4.
PROPERTIES OF THE PROPOSED ESTIMATOR FOR THE SHIFT MODEL
By virtue of the results in Section 3, we are in a position to
verify (2.8) and consider other asymptotic properties of N,
for
c
the shift model (1.1).
Theorem 4. Z. Under (3.7)-(3.8), for every o>~ and for every
h (>0) in (2.6), as ctD, for the shift model (1.1),
1i.
NinO
c
c
1,
E(N InD)k
c
c
+
1,
Y k • [0,1);
(4.1)
(4.2)
Theorem 4.2
If in (3. 7) ,
o > (1+T)/2 (2+T), .where
•
T > ] + 2h
,
and h (>0) is defined as in (2.6), then for the shift model
(1.1), the asymptotic risk efficiency in (2.8) hoZde.
Proofs.
Note that by virtue of (2.16)-(2.18) and Theorem A.6.1 of
Sen (1981), we conclude that for every. T>D,
adequately large, so that for every
[}
P {I \)~ -\) >£
n
where
c
£
«00)
<
-
C
£
there exists an
£>0,
n -l-T ,
(4.3)
may depend on
£.
Also, by virtue of Theorem 4.3.1
of Sen (1981, p. 93), we may conclude that for very
there exists a
0:
0<0<1
nO'
and an
nO «00) ,
£>0
and
Ti>O,
such that for every
n>n ,
-0
•
•
Once these results are obtained, the rest of the proofs of Theorems
,
4. I and 4.2 follows virtually the lines of the proofs of Theorems
n~nO'
for every
where
£'(>0)
depends on
£,
and
£'+0
as
£+0.
,
3. I and 3.2 of Sen (1980), and hence, we omit the details.
It may be remarked that by definition in (2.6), whenever
h
1S
~,
chosen as greater than
-lo..-h
c -Jzv N < 2N C - c -N C ,
(4.6)
c
1
A
c-"V
N -I
(4.7)
> 2 (N -1)
c
c
where by (2.5),
(as
°c
c+0), so that
(N -n )/In'0
c
•
(2nO/~(VN
<
where for
c
c
-
c
1-V)/V + c-J'(N _1)-h/fnO
c
(4.8)
c
h>~, c-~(N _1)-h//nO - (2/V)/n O (N _l)-h ~ 0,
as c+O,
c
c
c
c
by (4. I), and using a version of (4.5) for the lower as well as the
upper confidence limits in (2.16)-(2.18), we may conclude that
o ~I VN -vN -1 I
(2n)
c
c
c
A
A
(2n~)~(VN
&0,
-v)/v
as
c+O.
Hence, whenever, for
c+O,
V N(0,S2),
(4.9)
c
for some
S:
O<S<oo,
(nO)-~(N _nO)
c
c
c
we conclude from (4.8) that
-+
V
N(0,S2).
(4.10)
Fortunately, (4.9) follows from (4.1) and the recent results of
v
,
Huskova (1982), and hence, the asymptotic normality of the stopping
time
N'
c
follows from (4.10).
5.
PERFORl-lANCE OF
~(R)
FOR THE B-1' MODEL
With respect to the generalized Behrens-Fisher model in (1.2),
•
we shall study the performance of the sequential estimator
In (2.12) "hen
N
,c
.is defined by (2.6) "ith
{v}
n
LlNc(RJ
in (2.19)
It
follows from the results of Ramachandramurty (1966) that. for the
~
non-sequential case,
tin (R)
tI(; 6 -6 ),
2 1
estimator of
in (2.12) is a valid and consistent
and further, as
,
•
n->oo,
(5.1)
where
v*; 2A*/y*,
If
A*2;
_00< x<y<oo
If
+
_00< x<y<oo
FO(x) [l-F (y)]cj>(1) (1I (X)cj>(l) (lI (y))dG (x)dG (y)
o
O
O
O
0
GO(x) [l-GO(y)]cj>(l) (1I (X)cj>(1) (1I (y))dF (x)dF (y),
0
O
0
. O
(5.2)
(5.3)
and
HO(x)
;
J,(FO(x)+GO(x)) .
cj>
that the score function
:=
Note that in this case, we assume
is skew-symmetric (i. e .•
cj> (u) + cj> (l-u)
'rI O<u<l) -- this was not needed for the shift model in (l.l).
0,'
~
The existence of the moments of
tln(R)
has already been established
for the B-F model in Section 3.
To obtain results analogous to
•
(3.9)-(3.11), we define
"n
1
S* ; 2n" "J:
n
1;1
00
{J
_00
[I(Y1"'::X)-G (x)]cj>(1) (1I (x))dF (x)
o
0
O
00
- J [I (\.::x) - FO(x)]cj> (1) (110 (xlldGO(x)},
(5.4)
_00
and note that when (1.2) holds with
ES* ; 0
n
where
A*2
and
2nE(S*2); A*2
n
is defined by (5.2).
6 > (1+T)/2(2+T), T>O,
dGO(x)
"2k 2
n /
Now,
2dH (x),
O
S (d)
n
(5.5)
Further, under (3.7) with
using the fact that
dFO(x).:: 2dH (X) ,
O
we have
E{[s*l k ltl;0} <
n
tI;O,
00
,
'rIk::'2(1+T)
(5.6)
in (2.11) is a particular case of a general linear
•
•
,.
rank statistic, and under the model (1.2), we may virtually repeat
the proof of Theorem A.4.2 of Sen (1981, pp. 395-396), and letting
(for
k>O)
J
o«~)
fOT every
Idl~- ~ (log
; {d:
nk
and
£>0,
a finite positive constant
k
n)},
we obtain that when
there exist a positive integer
nO
n~nO'
where
s < (1-20)/20
and
Also, by the Cauchy-Schwarz inequality,
n~ls (d)1 < !2il{ll:2n [a
n
-
n 1= l
2n
and
such that
KO«~)'
s
P{dsJuP Is (d)-S (O)+J.-,dy*!>£/!i1) < KOn- ,
E nk
n
n
for every
6=0,
(5.7)
y*
is defined by (5.3).
V d,
(i)-a ]2)~
2n
1
= O(n~),
with probability 1,
(5.8)
so that using (5.7), (5.8) and proceeding as in the proof of
•
•
Theorem 2.2 of Sen (1980), we obtain that under the stated regularity conditions,
(5.9)
where
(R*)/vn -+ 0 a.s., as
n
n-,
and for
0 > (In)/2(2+,),
,>0,
(5.10)
At this stage, 'we make use of the following result due to Ghosh
(1972):
and an
For every
nO «~) ,
0 = (1+,)/2(2+,),
such that for every
,>0,
there exist a
KO«~)
n~nO'
l-
n"2lFn (x)l- rex) I
p{sup
x
(5. 11)
is the empirical d. f. based on. Xl"" ,X ; asimi1ar
n
Tesult holds fOT the second sample Y1" .. 'Y
too. If H ;
n
nO
J,[FnO+GnOl stands for the combined sample empiTica1 d.f., then
wldcT 6=0, by (5.11), on letting
J,
1 0
An = {x: n IHno(x)-Ho(x)I/{Ho(x) [l-HO(x)]}~- ~ K 10g n}, we may
0
where
•
0
> K log n) < 2n - 1-.
F(x) [l_F(x)]}J,-o - 0
write
FnO
•
(5.12)
where by (5.6), (5.8) and (5.11), the second term on the right hand
side of (5.12) converges to 0, as
n~,
whenever
k < 2(1+T).
For
the fi rst term on the right hand side of (5.12), we use the ChernoffSavage integral representation for
S (0)
n
and by the usual partial
integration claim that under the assumed regularity conditions, over
the set
A,
n
(5.13)
so that (5.12) converges to 0, as
n~.
As a result, by (5.9),
(5.10) and (5.12), we conclude that
0*
nY*(L"n(R)-ll) = 2nS~ + Rn
A
where for
8 > (1+T)/2(2+T),
Eln-~R~*lk
4
0,
(5.14 )
•
>
T>O,
V k < 2(1+T).
(5.15)
•
Finally, for
S*,
n
involving two sets of i.i.d.r.v.'s, it follows
by the well-known moment convergence results that
(5.16 )
where
Z has the standard normal d.f.
As such, by (5.14), (5.15)
and (5.16), we conclude that
V k < 2(l+T). (5.17)
Note that for
S
and
n,U
lim~2n S
n,U = Ta / 2 ,
n~
Sn, L'
defined in (2.16)-(2.17),
lim,l2n S
n~
n,L = ~Ta/2'
(5.18)
so that by (2.16)"(2.18) and (5.7), we conclude that under (1.2),
8
stochastically converges to 4 (y*) -I T / , and hence, defining
An
a 2
\J
as in (2.19), we have under (1.2),
A
n
•
•
~
v
+
n p
2/y* = v*,
say,
as
(5. 19)
n+oo.
As such, keeping (2.6) in mind, we may define
as
For the model (2.1),
theory.
Note that
(1.2),
lI=O),
{S*}
n
n*
c
(5.20)
dO.
plays a vital role in the asymptotic
in (5.4) form a reverse martingale (when in
so that the "uniform continuity in probability" in
(4.4) can easily be prov.ed for the
S* under (1.2), and using
m
I
(5.12)-(5.15) [insuring the a.s. convergence of n~IS*-S (0)1 to 0
n
as
n+oo] along with the above, we conclude that (4.4) also holds
for the model (1.2).
By virtue of (5.7), (5.9) and (5.10), we have
then (4.5) for the model (1.2).
needs to be replaced by
•
n
v*
The proof of (4.3) wherein
v
also follows along the same line as in
the proof of (4.3), and hence, parallel to (4.1)-(4.2), we obtain
that under the regularity conditions of Theorem 4.1, when (1.2)
holds, as
c+O,
N'/n* + 1,
c c P
~
E(N /n*)k
c c
A
(2n~) (liN (R)-lI)
c
+
1,
(5.21)
Y k • [0,1]
V N(O, (2A*/Y*) 2),
(5.22)
and
~
.
p*(c) = E(~ (R) -lI)
c
2
(5.23)
The last expression wi 11 be useful in companng the asymptotic
risk of the sequential R-estimator
lINc(R)
in Section 2, under
(1.2), with some alternative ones to be considered in the next
section.
F GO' y*=y
O
agrees with (2.5), where v=2/y.
6.
•
Note that for
and
A*2=1,
sO that· (5.23)
ALTERNATIVE SEQUENTIAL R-ESTHlATORS FOR THE B-F MODEL
The estimator lIn(R) in (2.12) (or its sequential version) is
based on equal sample sizes for the two samples. For the general
B-1' model in (1.2), this choice of equal sample sizes is rather
counter-intuitive, and may lead to increased risk.
Par this reason,
•
we consider here another generalization of the procedure in Sen
(1980) based on signed-rank statistics.
As in Sect ion 5, we assume that the score function
symmE'tric and define
ep+(u) = epUl+u)/2), O<u<l.
RY~ (b))
IYi-bl
among
+
n
a (k)
ep
nl
be the rank of
nl
Let then
is skew-
replaced by ep+. and assume that (2.10)
. X*
+2
Jl +
2
2
(or.
A
= o[ep (u)] du = Aep = 1. Also, let R. (b)
be defined by (2.9) with
holds, i.e..
ep
Ix. -b I
among
1
IXl-bl, ...• lxn-bl
i = 1, ... ,n ,
!Y I-bl, ... , IYn-bl), for
b
(or
real, and
let
= n
Y
T (b) = n
n
.
Then,
-1 n
+
x*
n
nl
E. lSgn(X.-h)a (R . (b)),
1=
1
1:::
and
n
1
Y
T (b)
n
are both "
= l:!(supfb:
TX(b»O} + inffb:
en , 2
= l:!(supfb:
Y
T (b»O} + inffb:
(6.2)
b,
in
and we set
(6.3)
n
n
f
and
by
ljJf and ljJ g' respectively, and Y
rE'spectively. Then,
(or 8
)
n,2
n"l
consistent and robust estimator of 8
1
e
A
Let now the first sample be of size
2
(6.1)
nl
8n, I
n
real
-1 n
+
Y*
E. lSgn(Y.-b)a (R . (b)).
Corresponding to the densities
size
b
Y
-T (b)<O)).
(6.4)
n
ljJ
g,
•
in (2.15) is denoted'
in (2.14) by
Yf and Yg'
is a translation-invariant,
(or
8 ), and as
2
n->oo,
n
and the second of
l
(n =.n + n ), so that we have the estimator of ~
2
l
(6.6)
and under (2.1)-(2.2), for each estimator, we have for
adequately large,
n ,n
l 2
•
•
p
nln Z
(c) = E(~
nln Z
_tl)Z + c(n +n )
1 Z
2 -I
Z -1
+ (n Y )
+ c(n +n ) +
= (n IYf)
Z g
1 Z
as
Note that for known
dO,
I
n
(6.7)
(l (-)
the optimal
are
gi Yen by
Z Z
2 Z
-I
nlY f - n ZYg - c
C6. R)
.,.'
Tilus, if we lE't
-\.. -1
n le - c Y f '
Ii
(e) - 2c (Y
P
n I cnZc
-1
f
+Y
-I
g
n
t I,,'n for
-
Zc
dO,
(6.9)
).
and t.his represents t.he asymptotic minimum risk fur the ,"stimat.ion
h3!'cd on the' difference of "individual J{-<:,sl.jmatcs of locatiun.
•
Y
Since
f
t.heir
w~jng
and
Y
g
are unknmYn,
e~t.imates,
consider the following stopping rule.
8 =0 (8 =0), T~(O) C<CO)) has a specified
Z
1
distribution, symmetric about 0, and hence, for every a (O<a<I),
Note that under
there exists a positive constant
a
>
n''.' '
\"here
as
n ,a -+ , a/2
Defining the
TY(O).
n
such t.hat.
n
n-+OO ,
and
n Jet
TX(O)
n
j8 =Ol,
(6.10)
1
may al so be replaced by
as in (6.1)-(6.Z), we let
T (b)
n
Ay
8
n,L
= sup{b:
TYfb»T
n .
n ,cx
l ,(6.11)
AX
8
X
= inf{b: T (b)<.T.
X
V =
x e. . x
(i?n,U_eXn,L ) /2Tn,o. -Iil (8. . .n,
u- n, L) /Z, a. /2'
(6.13)
/n
,.,y
",y
Y
Y
(en,U
-an,L
)/ZT
-Iil
e
(e
Un,cx
n,
n, L)/Z, a. /2'
(6.14 )
n,U
n
Let
Tn,a'
> P{ITX(O)!>T
AX
8
= sup{b:
n,L
•
proceed as in Sen (1980) and
I;"
=
n
n.a
l,
Ay
8
n,U
Y
= inf{b: T (b)<-T
n
11,0.
l;(6.1Z)
nO (.::Z) be an initial sample size and h (>0) be some constant.
to be defined more forma 11y later on. Define then for every c>O
(6.15)
n>c
-_.
-lz
Y
(Vn +n
ch
)
•
(6.16)
TIlen, the proposed sequential procedure consists in drawing one
observation at a time from each of the two populations, until for
the first time, one of the stopping numbers
N ,N
is reached,
lc 2c
and then drawing observations sequentially only from the population
for which the stopping number has not been reached and terminating
when both the stopping numbers are attained.
tial R-estimator of
The proposed sequen-
is then
b
(6.17)
en,l ,8n, 2
where the
are defined by (6.3)-(6.4)_
TI,e risk of
•
is
+
= p*
*
c,l.+ P c,2'
cEN
lc
+. cEN
+
2c
say .
(6.18)
Note that with the definitions of the
e
and the
N ,
c
Theorems 3.1 and 3.2 of Sen (1980) apply directly to the individual
sample estimates and risks, so that we conclude that under the regularity conditions assumed in earlier sections, as
dO,
For
j = 1,2,
N. In.
Jc
P~,l
Jc
~
I
1,
in probability as well as the first mean, (6.19)
1
2C,\~,
P~,2
!z
and the asymptotic normality of
Thus, it follows that as
-1
(6.20)
2c Yg ,
dO,
(n] +n
c
1<
2C
A
1 2(bN
N
lc 2c
-bl
hOlds.
•
•
,
p* / P
ecl
c
"I c n 2 c
.. 1,
(6.21 )
which establishes the asymptotic risk-efficiency of the proposed
sequential procedure for the general B-F model.
Note that this
asymptotic risk-efficiency is with respect to the opitmal nonsequential procedure based on the same R-esiimators with the sample
sizes
,n ' if Y and Y were known.
lc 2c
f
g
It may be remarked that if in (1.2), we consider the usual
n
location-scale model, viz.,
GO(x); FO(dx), for some d>O, then
2
and
go(x) = dfo(dx), so that I(gO); d I(f )
y ; dy f · Thus.
o ~,-l
-1 g
(6.9) is asymptotically equivalent to
2c ~f (l+d ). In such a
case, if we now choose
~(u)
; kWf(u), O<u<l, k#O,
defined by (2.15), then by (2.10),
•
where
Wf;W
is
-1
; [I(f )] , and hence, by
O
(2.14), Yf ; [I(fo)]~' Thus, for this specific choice of ~, the
asymptotic risk is given by 2c ~[I (f ) r\l+d- l ), which represents
O
the lower bound (information limit) to the risk of any Jloint cst ima~,
tor of
k
2
for the location-scale (B-F) model and is attainable
(under additional regularity conditions) by the maximum likelihood
8 ,8 , Thus, the proposed sequential procedure,
1 2
based on the optimal score ~=Wf' becomes asymptotically optimal
estimates of
within a bound class of estimation rules.
The choice of
~=Wf
in
this context is in agreement with that of the, conventional testing
or estimation problems in the non-sequential case [viz., Hajek and
Sidak (1967)].
~(u) ;
<P(u) ;
(u-~),
Il:2
-1 '
<I>
For a given score function
(u),
sign statistic:
[viz., Wilcoxon:
~(u); sgn(~-~),
Normal scores:
the inverse of the standard normal d. f.1. will'''
is unknown (and hence,
l
2c\;I(I+d- ) ;
•
~
Wf
f
is also so), we may write (6.9) as
{2C~[I(fo)r~(I+d-l)}{I(f0)/Y~}\
(6.22)
where the first factor represents the information limit. while by
the Rao-Cramer bound, y 2 ; {I;: nE(8 1-8)2}-1 < I(f). Thus,
f
-2
n
n,
0
Yf l(fO) > 1
and this explains the effect of non-optimal choice
of the score function ~, on the asymptotic risk in (6.9), '(6.18)(0.20). Recall that
(6.23)
•
is the Pitman (asymptotic-) efficiency of the conventional rank
statistic based on the score function
mal sC\lre function
= kljJf'
~
holds when
ljJf'
P(~,ljJf)
and
~
with respect to the opti-
2. 1, where the equality sign
As such, if we define the asymptotic
(k~O).
risk efficiency by the ratio of the asymptotic risk for the optimal
~,
score (ljJf) and the score
then, by (6.22) and (6.23), we have
this equal to
1
{e(~,ljJfn':i
';Pitman efficiency.
=
(6.24)
With this identity, the results on the Pitman efficiency, studied
in detail elsewhere [viz., Puri and Sen (1971)], provide us with
the parallel ones for the location-scale (B-F) model under consideration.
Ghosh and Mukhopadhyay (1979) have considered the B-F
model and the asymptotically risk efficient sequential point estima~
tion of
based on the sample means and variances.
that asymptotically the risk of their estimator is
2
where
Ox
ti.vely.
and
It follows
k
2c 2(OX+Oy)'
2
•
0y
are the variances for the d.f. F and G, respec2
-2 2
Note that under the location-scale model, 0y = d oX' so
that the above reduces to
~
2c 0X(l+d
-1
).
Hence, the asymptotic
risk-efficiency of the R-estimator with respect to the mean estimator is given by
k2
k2
1
1
2c (OX+Oy)}/{2c (Y +Y; )}
f
=
where
t
0XYf
=
';Pitman efficiency (R~,t),
stands for the Student t-test and
based on the score function
~.
(6.25)
R~
for the rank test
In particular, if we use the normal
score statistics and the derived estimates, then it is well known
that
O~Y~ ~
1
where the equality sign holds only when
is a normal distribution.
F itself
This explains the asymptotic superiority
of the proposed sequential procedure over the conventional parametric
procedure.
•
•
For the genera) R-F model in (1.2), we obtain, by (6.9) and
(6.23), that as
c~o,
;
(6.26)
'.
where by.the Rao-Cramer inequality, Oir(f) ~ 1
2
2 2
2
Actually, 0XI(f)p(<j>,ljif) ~ YfO X'
yl(g)P(<j>,ljig)
°
if we
~
2 2
OyY ~ 1, where the
g
equality sign holds (in either place) when the d.f. R (or G) is
use the normal scores, then
normal.
2 2
1.
>
y~x ~
1,
Thus, comparing the first expression in (6.23) and (6.26),
we again conclude that for the general B-F model in (1.2), the
proposed normal Scores prodedure is asymptotically more risk-
•
efficient than the conventional procedure when at least one of
and
F
G is di fferent from a normal d. f.
7.
SO~IE
GENERAL REMARKS
We study nOl' the relative performance of the sequential procedure in Section 2 and the alternative one in Section 6, when·for
both of them, the common score function
the shift model (1.1).
1hus (5.24) reduces to
For ·this model,
\
1
4c v;.
used.
and
First, consider
Y'
= Vf = Yg .
Also, by (6.18)-(6.20), the asymp-
totic risk of the sequential estimator in (6.17) reduces to
\
2c (V
-1
-1
\.-1
+Y ) ~ 4cy . Thus, the two sequential procedures are
f
g
f
asymptotically risk.equivalent·for the shift model (1.1). However,
the two-sample approach does not require the symmetry of
F,
under
(1.1), while for the validity of the one sample estimates in
Section 6, the symmetry of
F is a part of the assumptions for the
a 1ternati ve procedure in Section 6.
lIence, for the shi ft model, the
two-sample approach in Section 2 rests on relatively less stringent
regularity conditions.
•
•
For the general B-F model in (1.2),symmetry of
needed for hoth the procedures.
F ' ·G
1S
O O
By (5.23) and (6.18)-(6.20), the
asymptotic Pisk efficiency of the two-sample procedure in Section 2
relative to the alternative procedure in Section 6, both based on
the common
q,.
•
is equal to
-1 - l }
*2 }-l
; {y*(y f +Y g J/2 {J-,(l+A J
.
(7.1J
Note that by the elementary inequalities among the arithmetic mean
(A.~L),
geometric mean (G.M.J and Harmonic mean (H.M.J of non-
negative quantities, we have
(7.2J
where in each place the equality sign holds when
Yf;Y '
g
Thus,
(7.1J is bounded from below by
(7.3J
For the specific case of the Wilcoxon scores (i.e.,
112(u-J-,), O<u<l),
q,(l\::m,
/12"
m < fO'
12
< fo' gO>' Yf;
so that
fO>
JJ {Fo(x) [l-FO(y)]dGO(x)dGO(y)
Y*;
and Yg;
q,(uJ;
m J fO(x)gO(x)dx
m < go'
gO>'
•
;
Further, A*2 ;
+ GO(x) [l-GO(y)]dFO(x)dFo(y)} 5.-
x:'Y
3
JJ {dGO(x)dGO(y)
+ dFO(x)dFO(y)} = 3.
Hence, (7.3J is botmded
x5.-Y
from below by
(7.1)
"here
O~Cl(fo,goJ5.-1.
To obtain an upper bound for (7.1), we make ·use of (5.17J, for
k;2,
and the Rao-Cramer inequality (along with the fact that
is unbiased for
6J, and obtain that
'E.
n (IlJ
•
•
(7.5)
where the last step follows from (6.23), and both the
p(~,W)
are
As such, if we define
< 1.
(7.6)
e*~l,
and note that by (7.5),
we have from (7.5) and (7.6),
that
(7.1) is bounded from above by
'-I -1
{y*\(y +y ) /A* =
f
g
•
.'
-1-1
\(y f +y g )/(A*/y*)
< {(e*(~'Wf'W )/min{e(~,w ),e(~,w.)})}
g
g
g
Now, for the location-scale model:
y
f
= d
- I
e(~,1JJf) = e(~,1JJg)'
y g'
,
[e* (~,1JJf,1JJg) /e(<!>,1JJ )]
f
Note that
A*=l.
\(l+A*2)?. ·A*
!<
(7.7)
2
GO(x) = Fa (dx)
d>O,
for some
and hence, (7.7) reduces to
\
(7.8)
and the equality sign holds only when
Thus, (7.1) is bounded from above by the first line of (7.7),
where the equality sign holds when
A*=l,
while the last equality
sign i.n (7.7) holds for the location-scale model.
an attainable upper bound only for
~=Wf (=W ),
g
(7.8) is
~l,
e(~'Wf) = 1,
while
A*=l.
e*($,1JJ
Hence', (7.8) is
In parti.cular, whvlI
e 1JJ g )
~ 1,
and hence,
where the' equality sign holds [refer to (7.5)] when
l(f ) = l(gol
(i.e., d=l) and the first equality sign in (7.5)
O
holds. Thus, for the location-scale model, for optimal ~, the
t"o-sample approach in Section 2, for
d;ll,
does not have the
asymptotic risk optimality, which is possessed by the alternative
•
approach in Section 6.
(7.5)
Note that for the location-scale model, by
•
(7.9)
so that by (7.61 and (7.9), for the location-scale model,
•
(7.10)
which is equal to I for
it converges to
a
as
d=l
d+O
and otherwise is less than one, and
d~.
or
Thus, (7.8) is bounded from
above by
(7. ll)
and it clearly shows that for every
there exist two values of
d 1: [dO,dOl,
(7.11) is
cP,
for which
d, O<do<l<dO<oo,
<1,
c(cP'¢f) > 0,
•
such that for
indicating the asymptotic risk-in-
efficiency of the two-sample approach for extreme scale variation.
For the general B-F model,
e*(cP'¢f'¢)
g
is the asymptotic
relative efficiency of the rank order estimate
A
6 (R)
n
in (2.12)
so that (7.7) is comparable to (6.24) or (6.25).
ACKNOWLEDGEMENT
This work was partiallY supported by the National Heart, Lung
and Blood Institute, Contract NlH-NHLBI-71-2243-L.
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Statist. A 8, 637-652.
•
•
•
Hii.iek, J. and Sidak. Z. (1967).
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Thea!,!! of Rank Te.9ts.
New York:
Hoyland, A. (1965). Robustness of the Hodges-Lehmann estimate for
shift. Ann. Math. Statist. 36, 174-197.
v
",
ll11skova, M. (1982). On bOlffided length sequential confidence
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B. V. Gnedenks et aL.) Amsterdam: North Holland, Pl'.
In
v
.Iureckova, J. (1969). Asymptotic linearity of a rank statistic in
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Puri, ~1. L. and Sen, P. K.
(1971). Nonpar>ametI'io Methods in
New York: John Wiley.
MuZtivar'iate AnaZysis.
Ramachandramurty, P.V. (1966). On some nonparametric estimates
for shi ft in the Behrens- Fisher situation. Ann. Math.
Statist. 37, 593-610.
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