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ROBUST SEQUENTIAL CONFIDENCE INTERVALS
FOR THE BEHRENS-FISHER PROBLEM
by
Malay Ghosh
Department of Biostatistics
University of North Carolina, Chapel Hill
Institute of Statistics Mimeo Series No. 685
May, 1970
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ROBUST SEQUENTIAL CONFIDENCE INTERVALS
FOR THE BEHRENS-FISHER PROBLEM*
by Malay Ghosh
University of North Carolina at Chapel Hill
~.
The problem of providing a bounded length (sequential) confidence
interval for the median of a symmetric (but otherwise unknown) distribution based
on a general class of one-sample rank-order statistics was investigated in [6].
The purpose of the present note is to indicate how the techniques developed there
can be extended to the two-sample problem.
It has been shown that in particular
for the Behrens-Fisher situation (see e.g., [3] or [4]), when the proposed procedure is based on the "normal-scores" statistic, under very general conditions
on the unknown distribution function (d.f.), it is asymptotically at least as
efficient as an analogous procedure suggested in [5].
~.
Consider two independent sequences of random variables (rv's)
. {Xl' X2 ,···} and {Y , Y , ••• }, the X's independent and identically distributed
I
2
(iid) with d.f. F(x-p) and Y's iid with d.f.
G(x-p-~),
p,
~
unknown.
Our goal
is to find a corifidence interval I of width 2d for ~ such that P{~EI} ~ I-a (the
desired confidence coefficient), where, O<d<oo, O<a<l are preassigned constants.
F. and G. being unknown, no fixed sample size procedure for the above problem
seems feasible.
A sequential procedure for the above problem based on the mean difference
X-y
was proposed in [5].
This, obviously, is vulnerable to gross errors or outlying
observations, and may be quite inefficient for heavy-tailed distributions.
* Work
supported by National Institutes of Health GM-12868.
In the
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present note, an alternative procedure is proposed based on rank-order statistics,
which is not, or at least, much less vulnerable to the above criticism.
To motivate the procedure, we use the same logic as in [5].
For a fixed
sample (Xl' X2 '···, Xm) of size m=m(p) and (Yl , Y2 , ••• , Yn ) of size n=n(p) (p=m+n),
from the two populations with d.f.'s F(x-p) and G(x-n) respectively
(~=n-p;
F and
G assumed to be symmetric about zero), we construct estimators of nand p based
on one-sample rank-order statistics and take their difference as estimator of
~.
With this end in view, define the following one-sample rank-order statistics:
m
I
(1.1)
a=l
u(X )J (R /(m+l»;
a m ma
n
I
S=l
(1.2)
u(YB)J (R'a/(n+l»,
n np
m
where, u(t) = 0 or 1 according as t<O or not; Rma = I.~= l u(/Xa I-Ix.~ I), R'a
np =
Ij~l u(IYS/-IYjl), and Jm(u), In(v) (O<u, v<l) are generated by a score function
J(u)
(O~u<l)
in either of the following two ways:
(a) Jm(u) = i/(m+l), In(v) = j/(n+l),
(i-l)/m<u~i/m, (j-l)/n<v~j/n, l~i~m,
l~j~n.
(b) J (u) = EJ(U i)' J (v) = EJ(U .), (i-l)/m<u<_i/m,
m
m
n
nJ
(j-l)/n<v~j/n, l~i<_m,
where, U 1< ••'.<U (U 1< ••• <U ) are m(n) ordered random variables in a sample of
m- mm n - nn
,size m(n) from a rectangular (0,1) distribution.
Further, assume that J(u) =
~-1«1+u)/2), O~u<l, ~(x) being a d.f. defined on (_00,00) satisfying
~(x)+~(-x)
(1.3)
(a)
(1.4)
(b) -log
= 1, for all real X;
[l-~(x)]
is convex for all
x~xO
(x O real, >0).
Also, let
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F, G E ~O(J), the class of all absolutely continuous
d.f.'s IT(x) symmetric about zero for which both IT'(x)
(1.5)
and IT"(x) are bounded for almost all x(a.eaox) and
limx-+oo IT'(x)J(IT(x)-IT(-x»
0-.6)
is bounded.
Introduce the following notations:
(1. 7)
~ =
(1.8)
f
1
o
1
2
=J
J(u)du, A
2
J (u)du.
0
It follows from (1.1) thath(X -a 1 ) (1 - an m-component row vector with
1 -m
-m
-m
all elements 1) is
+ in
of F, symmetric about
~
Also, under p=O, it has a distribution independent
a.
E.
Defining now,
m
sup {a: hl(X -al ) > ~
-m -m
(1. 9)
, (1.10)
pel)
mZ
"(1)
we propose Pm
= inf
= ~(P(ll)+p(lz»
m
m
{a: hl(X -al ) < ~ E },
-m -m
m
Similarly let ~(Z)
n
as a point estimator of p.
be a point estimator of n based on h •
Z
of 1::..
Em},
"
Propose I::.
,,(Z) "(1)
=n
-p
p n
m
as a point- estimator
Under (1.3)-(1.6), and the assumption that
O<A<l, where A = limp-+oo m(p)/p,
(1.11)
it is well-known (see e.g., [4]) that
2
2
2/
2
2
IP
"
-
(I::. -I::.) is asymptotically (as p-+oo)
p
2
2
2
2
2
normal (0'00)' 00 = 01 A+aZ/(l-A), 01 = A /B (F), 0z = A /B (G), where,
00
(LIZ)
B(IT)
= f ~
o
J(ZIT(x)-l)dIT(x).
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4
Thus, if I
=
[~-d
6 +d], limp~ P{~£I}
P 'p
the d.f. of a normal (0,1) distribution.
= Z~(a)-l,
Equating
a
=
limp~ !pd/a ' ~ being
O
Z~(a)-l
to I-a, we find a =
Next we find strongly consistent estimators of a
2
2
and a '
l
Z
It follows from the remarks after (1,8) that there exist (known) constants
h(m) and h(m)
·lla
lZa
= E _h(m)
mIla'
such that
I-a -+l-a as
(1.13)
m
m~,
Define
(1.14)
1'(1)
PL,m = sup {a:
hl(X-m-al-m ) > h(m)},
lla '
(1.15)
1'(1)
PU,m = inf {a:
h(m)}
hl(~m-a!m) < lZa '
Then, p{p(l)<p<p(l)} I-a, and it follows from lemma 5,Z of [6] that for every
L,m
U,m
m
0>0, there exists an integer m such that for
o
m~mo with probability ~1_0(m-l-0),
(1.16)
1'2
A
A
2 2
11'2
21
-~
3
Thus, if aIm = m(PU,m-PL,m) /Ta/Z,alm-al = O(m (log m) ) with a probability
statement as above, Define, now nL(Z) and nu(Z) similarly as in (1,14) and (1.15),
,n
,n
1'2 _
A(Z) A(Z) 2 2
) /T /Z' Then, for every 0>0, there exists an integer
-n
and let a Z - n(n
n
u ,n L ,n
a
1'2
21
-~
3
-1-0
no such that for n~no' 1aZn-a Z = O(n (log n) ) with probability ~l-O(n
),
Following Robbins, Simons and Starr [5] we propose a sampling scheme which says
that if at any stage, we have taken m observations on X and n observations on Y
A
A
with p = m+n~ZPO' the next observation is taken on X or Y according as m/n~alm/aZn
A
A
or m/n>alm/a Zn '
The procedure generates an infinite sequence of observations,
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'"
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and, from the definition of 0lm
and 02n'
it is seen to depend on a, but not on
d.
Further as in [5], one can easily show
that
m/n~1/02
as
p~.
Also, in a similar fashion as [5], we define the following three stopping
rules:
Stop with the first
N~2PO
such that if r observations on X and s observa-
tions on Y have been taken with
r+s~2PO'
then
"'2
-1
(1.17)
"'2
01 /r+o /s<b
r
2s -
,i.e.,
(1.18)
"'(1) "'(1)
(PU,r-PL,r)
A
A
A
2 + (n'"
U, s
_n"'(2»2<d 2
A
-
L, s
A
A
r>bO (01 +0 ) and s>b0 (01 +0 ), i.e.,
- lr
r 2s
- 2s
r 2s
(1.19)
It follows that if N are the stopping variables corresponding to R (R=1,2,3),
k
k
'"
'"
then, NlSN2SN3' The confidence interval prqposed is IN(d) = [~N(d)-d,
~N(d)+d].
~.
THEOREM 2.1.
The basic theorem of the paper is as follows:
Under (1.3)-(1.6), and following the sampling scheme and any of
the stopping rules in the earlier section, we have,
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N(=N(d»
(2.1)
is a non-increasing function of d, N(d) is
finite a.s. and EN(d)<oo for all d>O; lim
d40
N(d)
= 00 a.s. and lim d+O EN(d) = 00;
(2.2)
(2.3)
Proof:
Since (1.11) holds, from the remarks following (1.16), one gets that for
~
every u>O,
there exists an integer Po such that for
.
O(p -~ (log
p) 3 ) with probability
ho 1 ds for
A2
21
I02n(p)-02.
-1-0 ).
~1-0(p
P~Po'
A2
I0lm(p)-Ol
21
=
A similar probability statement
Because of the above, using the same technique as in
lemma 5.5 of [6], under rule R , we get EN(d) <00 for all d>O. Since Ni(d)~N3(d)
3
(i=1,2,3), the same is true for R and R also. (2.1) now follows from the
l
2
'definition of the stopping rules and the monotone convergence theorem.
The proof of (2.2) follows in the same manner as the corresponding result
in [5].
(2.4) follows along the lines of [6].
The details are omitted for
brevity.
/-
Since, vp
A
(~p-~)
2
is asymptotically normal (0, 00)' and (2.2) holds, to prove
. (2.3), it is sufficient to show that the sequence {~ } is uniformly continuous
p
with respect to p
-~
(see e.g., [1]).
Use the inequality
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I - (~ p ,-~p )
.
sup
P{,·,
I<~ I p
p -p up
A
A
,
>~}, ~ some positive constant
(2.5)
+
1'-PI'
p -p I<~up vn(p)
{
SUP
(n n (p ,)-nn (»
p
A
A
I
>
S~} •
2/p
The proof is now completed by using (1.11) and the same technique as in lemma
5.3 of [6].
The theorem shows that our procedure is "asymptotically consistent" and
"asymptotically efficient" in the sense of [2].
We compare the asymptotic relative efficiency (ARE) of the proposed procedure
to the one proposed in [5].
For a definition of ARE, see [6].
If Rand P stand
respectively for the procedures in the present paper and in [5], let e
of R with respect to P.
R,P
= ARE
Then (see (2.4) and (9) of [5]) under the assumption
that the variances 0,2 and 0,2 of the two populations with d.f. F(x-p) and
1
2
~G(x-n) respectively, are non-zero and finite, we have,
(2.6)
which is independent of d.
G(x)
= F(cx),
c>O, i.e.
In particular, for the Behrens-Fisher situation
2= 0ilc,
The two d.f. F and G differ only in scale, O
Hence, from (2.6), e
R,P
= 0'2
B2(F)/A 2
1
ARE of a general rank-order test with respect to student's t-test.
which is the
The conclu-
sions made in [6] can be repeated in this case, and in particular for J(u)
=
~-1«1+u)/2), eR,p ~ 1, uniformly in F with a finite secona moment, equality
being attained if and only if F is normal (0,oi 2) d.f.