I
I.
I
I
I
I
I
I
I
Research supported by National Institutes of Health, Grant No. GM 12868
Ie
I
I
I
I
I
I
I
.e
I
ON BOUNDED LENGTIf SEcpENTIAL CONFIDENCE INTERVALS
BASED ON ONE-SAMPLE RANK-ORDER STATISTICS
by
Pranab Kumar Sen and Malay Ghosh
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mlineo Series No. 648
November 1969
I
II
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.I
ON BOUNDED LENGlH SE~NfIAL CONFIDENCE INTERVALS BASED
ON ONE-SAMPLE RANK-ORDER STATISTICS*
by
PRANAB KUMAR SEN and MALAY GHOSH
Department of Biostatistics
University of North Carolina at Chapel Hill
1.
Introduction.
'VVVV\f\I\.f\.I
The problem of finding a bounded length confidence band for
the mean of an unknown distribution is studied by Anscombe (1952) and Chow and
Robbins
(196~.
distribution.
Farrell (1966) considers the problem for the p-quantile of a
Sproule (1969) has extended the results of Chow and Robbins to
the class of Hoeffding's (1948) U-statistics, and in the particular cases of
the signed-rank and sign statistics (which are both U-statistics), Geertsema
(1968) considers the problem based on rank estimates of the median.
In the present paper, we consider the problem of providing a (sequential)
confidence interval for the median of a syrrunetric (but otherwise unknown) distribution based on a general class of one-sample rank-order statistics.
Of
particular interest is the procedure based on the so called one-sample normal
scores statistics.
This procedure is shown to be asymptotically (i.e., as the
prescribed bound on the width of the confidence interval is made to converge to
zero) at least as efficient as the Chow-Robbins procedure for a broad class of
parent distributions.
In course of this study, several asymptotic results, having importance of
their own, are derived.
First, the elegant result of Bahadur (1966) on the
behaviour of the empirical distribution in the neighbourhood of a quantile is
extended to the entire real line (see Theorem 4.1).
Second, the weak convergence
results of Sen (1966, Theorem 1) and Jure~kova (1969) are replaced by almost
*Research supported by National Institutes of Health, Grant No. GM 12868
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
••
I
-2-
sure (a.s.) convergence results, under slightly more restrictive conditions on
the scores and the underlying distribution; see Theorem 4.3.
It
is also shown
(see Theorem 4.6) that the usual one-sample rank order statistic possesses the
martingale or the sub-martingale property according as the parent distribution
is symmetric about the origin or not.
Section 2 of the paper deals with the preliminary notions and basic assumptions.
The next section describes the proposed procedure and states the main
theorem of the paper.
Section 4 is concerned with the results stated in the
preceeding paragraph.
The proof of the main theorem is supplied in Section 5,
and the asymptotic efficiency results are studied in Section 6.
2.
~.
Let {Xl'X2"" ad inf} be a sequence of independent and
identically distributed random variables (iidrv) having an absolutely continuous
distribution function (d.f.) F (x) with location parameter 8 (unknown). It is
8
desired to determine (sequentially) a confidence interval IN = {8: ~L ,~8~~U ,N}'
where for each positive integer n,
8L,n
and eu,n are two statistics (not depending on d) based on the first n observations, such that ~U ,n -~L ,n > a a.s. and
lim P{8EIn } = I-a (the desired confidence coefficient), while N is the stopping
variable defined to be the first integer n>n
(some positive integer) such that
-0
n~
eU,n-eL,n ~ 2d.
Our procedure for determining N and (eL,N,8U,N) rests on the following
class of one-sample rank order statistics. Let c(u) = a or I according as u<O
or not.
Define
Let then
I
I.
I
I
I
-3-
(2.2)
where {In(u): O<u<l} is generated by a score-function {J(u): O<u<l} in either
of the following two ways:
(a)
J n (u) = J(n +i l ) , (i-l)/n<u<i/n,
for i=l, ... ,n;
-
(b)
In(u) = EJ(Unl.), (i-l)/n<u<i/n,
l<i<n,
where Un 1 -< ... < Unn
- -
I
I
I
I
assume that, J(u) '" \fI-l(l;U), O.s.u.s.l, where \fI(x) is a df defined on (-00,00) satisfy-
(2.3)
(a)
\fI(-x)+\fI(x) = 1 for all O<x<oo,
Ie
(2.4)
(b)
-log [l-\fI (x)] is convex for all x>x
,
-0
I
where x0(>0)
is some real number.
-
I
implies that there exists a to(>O), such that
I
I
I
(2.5)
I
I
••
I
are the n ordered random variables from the rectangular (0,1) d.f.
Also, we
ing the conditions
Note that by definition, J(O) = 0 and J(u) is t in u: O<u<l. Also, (2.4)
1
(2.6)
(2.7)
M(t) =
fo exp[tJ(u)]du
< 00 for all t < t ;
-
0
J(u) 2. K[ -log(l-u)], 0 < u < 1;
J' (u)
.s. KI(l-u),
0 .::. u < 1, 0 < K < 00.
Note that (2.4) [and hence, (2.6) and (2.7)] hold for the normal, the logistic,
double exponential and many other df's.
The statistic Tn when \fI is the standard
normal df is termed the normal scores statistic, and when \fI is uniform over
(-1,1), it is termed the signed-rank statistic .
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
••
I
-4-
We denote by ~ the class of all absolutely continuous df's {F(x)} symmetric about 0 for which both the density function f(x) and its first derivaAlso, let ~(J) be the class
tive f'(x) are bounded for almost all x (a.a.x).
of all Fe: ~o for which
lim f(x)J'[F(x)-F(-x)] is bounded.
x-+oo
(2.8)
That is
(2.9)
:;o(J) = {F:
Fe:~, (2.8) holds} .
Throughout the paper it will be assumed that Fe(x) = F(x-e), where Fe: ~(J).
[For better insight on (2.8), the reader is referred to Lemma 7.2 of Puri (1964)].
Introduce the following notations:
(2.10)
(2.11)
-
-1\ n
-1\ n
En = n Li=lJn(i/(n+l)), A~ = n Li=lJ~(i/(n+l));
II =
f
1
o
J(u)du and A2 =
f
1
J 2(u)du.
0
Note that if e=o, Tn (X ) has a distribution independent of F.
two (known) quantities T(l) and T(2) (=E _T(l)), such that
n,a
n,a
n n,a
~n
(2.12)
P {T(2) < T (X ) < T(l)} = I-a ~l-a as n-+oo.
e=o n,a - n ~n - n,a
n
For large n, it is known that
(2.13)
where
(2.14)
=
1
(2TI)-~
X
f
-00
exp(-~t2)dt.
Hence, there exists
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
••
I
- 5-
It is also known (cf [3, Theorem 2]) that under the assumptions made earlier
(2.15)
n-II.nIIJ (R j(n+l)) - J(R j(n+l)) I = o(n-~),
1= n no.
no.
which implies that the use of either of the scores will lead to the same
asymptotic results.
3.
(where 1 = (1, ... ,1)) is
-n
Define as in [13],
in a: -oo<a<oo.
~
(3.1)
eL,n = sup{a'.
T (X -al ) > T(l)}
n -n -n
n,a '
(3.2)
S
= inf{a:
U,n
T (X -al ) < T(2)}.
n -n -n
n,a
It
follows immediately that Su,n-SL,n > 0 a.s. and Pe{SL,n .::.e~su,n} = I-an'
The main theorem of the paper is as follows:
TIffiOREM 3.1. Under the assumptions of section 2,
(3.3)
N(=N(d)) is a non-increasing function of d, N(d) is finite
a.s. and EN (d) <00 for all d>O; limNed) = 00 a.s. and lirnEN(d)=oo;
d-+O
d+O
= 1 ~. ;
(3.4)
lim N(d)jv(d)
d+O
(3.5)
lim p{e£IN(d)}
d+O
(3.6)
lim E[N(d)]jv(d)
d+O
(3.7)
B(F)
=
f
00
o
= I-a;
= 1,
c& J[2F(x)-I]dF(x) .
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
-6-
The proof of the theorem is postponed to section 5.
In proving this theorem,
several results are needed which are proved in section 4.
4.
~
-oo<a<oo}.
{Tn(~n-a!n):
Since F(x) is absolutely continuous, the random variables Y.=F(X.),
1
i=1,2, ... are distributed uniformly over (0,1).
1
Define the empirical process
(4.1)
-h
Note that Vn (t)=O for t$(O,l).
k
Let gk(n) = n 2(log n) ,k>l.
Define
(4.2)
(4.3)
K (*)
n
= sup (K (t)).
O<t<l n
It is known that Kn (*) weakly converges to zero in the Prohorov sense, viz.
~jek and Sicfuk (1967). However, for our results in section 5, we require the
strong convergence of Kn (*) . This is accomplished here by extending a result
of Bahadur (1966, p. 578) to the entire real line.
TIIEOREM 4.1.
For every finite s(>O), there exists a positive constant c~l)
and a sample size ns ' such that for n>ns '
P{K (*) > c
(4.4)
Proof.
n
-
(1) -!t;
s
n
(log n) k } < 4n-s .
-
Let~.J ,n = j[n~]-l(o<;<[n~]),[x]
being the largest integer less than or
.-.J_
equal to x.
Since for the uniform df., the density function is equal to 1 (02.t2.-1),
by the same technique as in Bahadur (1966; (13), p. 579), it follows that for n
large,
I
I.
I
I
I
I
I
I
I
-7-
(4.5)
P{Kn (s,J,n »c*l n
-!t;
k
k
(logn)} -< 4 expo (-vn), j=1,2, ... ,[n 2 ] ,
where ci>O and
(4.6)
vn = -!t; log n + [ci2n~ (log n) 2k ]j[2{nk2 (10g n) k + cin!t; (log nk}]
=
-~
k
-!t; -1
log n + ci 2 (log n) (l+ci n )
.
Since ~1, by proper choice of ci = }:~1), say, vn can be made greater than
(s + ~) log n for any given s (>0) . Thus, using the Bonferroni inequality, we
obtain that for sufficiently large n,
(4.7)
1 (1) -~
k
~
P{Kn (s·J,n ) > ~3
s n (log n) for at least one j=1,2, ... ,[n ]}
<
-
4[n~ ] exp(-vn ) < 4n -s •
Ie
(4.4) will follow from (4.7) and (4.8) which we prove below.
I
(4.8)
I
Note that if t and t+a both belong to the same interval (s·
I
I
I
I
I
I·
I
K (*) < 3[max
n
-
1
l.9~Jn~]
K (S·
n J,n
)].
, s'+l ), as
J ,n
So, we need consider
J ,n
S·J+ 1 ,n-SoJ,n = n-~ < gk(n),IVn (t+a)-Vn (t)1 -< 2Kn (s·J,n).
only the case where t and t+a belong to two different intervals.
that t£[sJ' n' sJ'+l n] and (t+a)£[sr n' sr+1 n]'
"
whenever r
(4.9)
~
"
Then, lal2Zk(n)~sr n- sJ'+l ~k(n),
,
j +1 (otherwise, interchange j and r).
Hence,
IVn(t+a)-Vn(t)I ~ IVn(t+a)-Vn(sr,n)\
+ \Vn (sr,n)-Vn (Sj+1,n)I +
IVn(Sj+1,n)-v~t)1
~ Kn(sr n) + 2Kn (sJ'+1 n) ~ 3
,
,
Suppose now
~ ~ Kn(SJ' n)'
l.9~Ln
]
,
,
I
I
.-
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.I
-8-
. ~
for all J,r-l,2,
... ,[n].
Hence (4.8) follows from (4.9).
The second part of
the theorem follows from (4.4) by letting s>l.
Remark.
If F(x) is such that sup f(x) = f <00, we have IF(x+a)-F(x)l~folal,
x
0
-oo<x<oo, and hence from (4.4), we have for every s>O, n>n
,
-s
(4.10)
(1)
> Cs
n
-~
log n}
~
4n
-s
,
where the empirical df Fn(x) is defined by
(4.11)
Fn(x) = n
-1
n
iIlC(x-X i ), -OO<X<oo.
We also need a strong convergence result on sup IVn(t)I which we state below:
O<t<l
LEMMA. 4.2. p{ Sup Iv (t)1 = O((log n)~) > l-O(n- s ) for every s>O.
O<t<l n
The proof proceeds on the lines of the theorem 1 after using the same grid
points as in the theorem, and hence is omitted.
In order to prove that
rn [eU,n- 8L ,n] -+ATa / 2/B (F)
in deriving (3.3)), we also require the theorem 4.3.
convergence result was proved by Sen (1966).
a. s. as
n+oo
(to be used
The corresponding weak
Assume, without any loss of
generality that 8=0; the translation invariance of the estimates in (3.1) and
(3.2) permits it.
(4.12)
THEOREM 4.3.
Then, define the process
1
W
n (a) = n~[Tn (X-n)-Tn (X-n -al-n )-aB(F)], -oo<a<oo.
Under the assumptions of section 2, for every s(>O), there exists
positive constants (k(l), k(2)), and a sample size n such that for all n>n ,
s
s
s
-s
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.I
-9-
(4.13)
Proof. Define
(4.14)
H(x) = F(x)-F(-x), Ha(x) = F(x+a)-F(-x+a) = P
(4.15)
Hn (x)
=
Fn (x) -Fn (-x), Hn,a (x)
=
{IX-al~x}
Fn (x+a) -Fn (-x+a).
Note that as FE ~o' for all Ia! <gk(n), and n large,
Ha(X)-H(x) = O(n
(4.16)
-1
(log n)
2k
), for all -oo<x<oo.
Also, we may note that
IHn,a (x) - Hn (x) - Ha (x) + H(x) I
(4.17)
-< IFn,a (x) - Fn (x) - Fa (x) + F(x) I
+ IFn,a (-x) - Fn(-x) - Fa(-x) + F(-x)l.
We now prove the following 1enuna.
(4.18)
with probability greater than 1-4n- s for any 5(>0).
The proof of the lemma follows directly from theorem 4.1 and (4.14)-(4.17).
Returning now to the proof of the theorem we may write after using (2.15),
(4.19)
T (~ )
nn
00
=
00
f0 In(-~lH
(x))dF (x) = f J( ~lH (x))dFn(x)
'n
n
nOn n
1
+ o(n-~),
I
-10-
II
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
II
and a similar expression for
(4.20)
Tn(~-a!n)
. Hence,
1
n~ [Tn (X-n) -Tn (X-n - a1-n)]
where
(4.21)
00
(4.22)
= n~J n
{J( ~1 Hn(x)) - J( n~1 Hn,a (x))}
nO
I 2(a)
We shall only consider the case
the same line.
elF (x+a).
n
as the case of negative a follows on
O~a<gk(n)
Then, we can write I n1 (a) as
(4.23)
where
Since P{X(n)~O}
= 2- n ,
we have on integration by parts and on using theorem 4.1,
X
(4.24)
I (1) (a)
n1
= J
(n)
o
n
X
= J
o
1
n~[F (x+a)-F (x)]J'(H(x))dH(x)
(n)
n
1
n~[F(x+a)-F(x)]J' (H(x))dH(x)
+ O(n-~(log n) k)
X(n)
J
o
J' [H(x) ]dH(x) ,
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
I·
I
-11-
with probability ~ l-4n- s -2- n . Now, using the fact that H(x) = F(x)-F(-x)~F(x)
and for 0>0, P{[l-F(X(n))] < n-(2+0)} = 1_{1_n-(2+0)}n ~ n-(l+o), we obtain that
the second term on the right hand side of (4.24) is of O(n-~(log n)k+l), with
probability ~ l-n
-s - (n-l)
~ l-5n -2
,
(4.25)
-s
-n
-2 ,by letting
l+o~s.
Hence, with probability
II~i)(a) - n~aB(F)1
X
(n)
2.
II
+
II
o
00
1
n~{F(x+a)-F(x)-af(x)} J' (H(x))dH(x)
I
n~af(x)J' [H(x)]dH(x) I + O(n-~(log n)k+l).
X(n)
Now, using the fact that Fe:~ (J) and (2.7) holds, we obtain that the first term
o
on the right hand side of (4.24) is O(n-~(log n)2k+l), while the second term is
bounded by [lim sup Ih(x)J' [H(x)] I] (log n)k[l-F(X(n))]' which by the fact that
(2.8) holds and P{l-F(X(nY~ cn-~} = (l-cn-~)n ~ O(n- s ) for any s>O, is also
O(n-~(log n)k), with probability ~ l-O(n- s ). Hence, we obtain that as n-+<x>
(4.26)
for any s>O.
We now rewrite I~i)(a) as
(4.27)
n-~~.nl{J(
L1=
nn+l Hn (IX·I))-J(H(lx.\))}{c(X.)-c(X.+a)}.
1
1
1
1
Note that c(X.)-c(X.+a)
= 0 unless -a<X.<O.
1
1
1-
Hence, using theorem 4.1, (4.27),
lemma 4.2, and after noting that J' (u) is bounded in the neighbourhood of zero,
we have,
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I.
I
I
-12-
(4.28)
-k
L
-k
k+k
= O( (log n) k )'O(n"(log
n)"2) = O(n 2(log n) 2),
with probability ~ l-O(n- s ), as n~.
Thus, it remains to prove that 0 Sup (n'l11 (a) I = O(n-~(log n)k+l), with
<a<gk~!I 2n
s
probability> l-O(n- ). Define x* by l-H(x*) = 4c(1)n- 3/ 4 (log n)k, where c(l)
-
n
is defined by (4.4).
s
s
Then
x*n
=
(4.29)
n
J
00
+
o
J {J(
x~
~l H (x)) - J(n+n l Hn,a (x))} dF n (x+a).
n
n
We first consider the following lemma the proof of which follows directly from
the theorem 1 of Hoeffding (1963) after noting that Hn(x) and Hn,a(x) are
bounded valued random variables and (4.16) holds.
LEMMA 4.5.
For every s(>O), there exist positive constants c~l)(>O), c~2)(>0)
such that for sufficiently large n,
7
(4.30)
P{IHn(x~)-H(x~)1 > c~1)n-8(lOg n)k} < c~2)n-S,
(4.31)
A direct use of lemma 4.4 and lemma 4.5 leads to the fact that for all
x < x*, with probability> l-O(n- s ),
-
n
(4.32)
and. hence for any 8:
-
1 [l-H (x)],
l-Hn (x) -> -3
n,a
0<8<1 and O<x<x*,
--n
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
I·
I
-13-
(4.33)
1-Hn,a, e(x): l-{eHn (x)+(l-e)Hn,a (x)}
1
: e[l-Hn (x)] + (l-e)[l-Hn,a (x)] -> -3
[l-Hn,a (x)],
with probability ~ 1-0(n- s ). Hence, by using Lemmas 4.4,4.5, (2.7), (4.32)
and (4.33), we have with probability ~ 1-0(n- s )
(4.34)
(1)
sup
II 2 (a) I = sup
O<a<gk(n) n
O<a<gk(n)
kl f
n
x*
2
0
n
n
ll+T [H (x) -H
(x)]
n
n
n,a
J' [n~1 Hn a e (x)] dFn a (x), (for some en: o<e n<1)
, , n
'
x*
< O(n-~(log n)k)
-
f
0
n
J'( ~1 H
e (x))dF
(x)
n
n,a, n
n,a
< O(n-~(log n)k) 3K
-
< O(n-~(log n)k) 3K
-
= 0 (n -~log n) k
-~
k
3Kn
-1
x*n
f
0
00
f0
[1 - ~1 H (x)]-ldH (x)
n
n,a·
n,a
[1 - ~1 H (x)]-ldH (x)
n
n,a
n,a
~
L (1
. 1
1=
i-1
- -+1 )
n
n
\-1
~
O(n log n)
6K L i
i=l
~
O(n -~ (log n) k) (l+log n) = O(n -~ (log n) k+1 ).
Finally, we consider I~~) (a), for which we have
(4.35)
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
I·
I
-14-
Now, for every a20,
(4.36)
1
00
< n~ r J( n+ H (x))dH (x),
- x~ n 1 n
n
n
as dFn (x)
~
dHn (x) and J(u) t in u:
(4.36) is bounded above by
O<u<l.
1
nzrIn (x*)
n
1
(4.37)
J -K
H (x*)
n~
n
n
1
log(l-tijdu<
-
By (2.6), the right hand side of
{-K log(l-Hn(x~))}
= O(n-~(log n)k+l), with probability ~ l-O(n- s ),
by lemma 4.5 and definition of
H(x~).
Similarly, the second tenn on the right
hand side of (4.32) is also O(n-~(log n)k+l) with probability ~ l-O(n- s ). The
proof is now complete.
We may observe that the weak convergence of the process {Wn(a)} can be
established along the lines of Jurecokova [10] under less stringent conditions.
However, we need here a stronger result as given in Theorem 4.3.
We shall now prove that when F(x) is synnnetric about 0 and I n (u) is
specified by (b) of section 2, then, T~ = n(Tn-~)' n=1,2, ... forms a martingale
sequence with respect to a non-decreasing sequence of a-fields
"n defined as
follows.
Let c-n = (c(Xl), ... ,c(Xn)) and R
de-n = (Rn1, ... ,Rnn), where the Rare
no.
fined in (2.1). Tn is the a-field generated by (c~~
,R ). Then, obviously Tn
t in n and we prove the following theorem to be used in proving the "unifonn
continuity" (to be explained in section 5) of the estimates 8L,n and ~U ,n .
I
I
.-
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.I
-15-
If In(u) = EJ(Uni ), (i-l)/n<u::.i/n, and if F(x)+F(-x) = 1 for
all real x, {T~,Tn} forms a martingale sequence.
lHEOREM 4.6.
Proof.
By definition in (Z.Z),
(4.38)
Under the hypothesis of the theorem, c(Xn+l ) is independent of Rn+l,n+l (see
e.g. [7], p. 40), and hence,
(4.39)
R
1
n+l
.
E{c(X )J
(n+l,n+l)IT} = Z(n+l) .L In+l(n{Z) = ~En+l'
n+l n+l n+2
n
J=l
where En+l is defined by (Z.lO).
(4.40)
It is easy to see that
- = n-I,L. n lEJ(U .) = II J(u)du = ll, for all n.
E
1=
n1
0
n
Also, given En' R +1 can either assume the value R or R +1, with respective
n ,0'. n+l-R
Room
oo
conditional probability n+l and nB. Hence,
(4.41)
R
n+1-R
R
R
R +1
E{J (n+l'O'.)IT} =
00 J
(00)
00 J
(00)
n+l n+Z
n
n+l
n+l n+Z + n+l n+l n+2
after using the fact that by definition of In+l(n~Z)' l::.i<n+l,
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
-16-
(4.42)
Hence, from (4.39) through (4.41), we obtain that
R
2
no.
E(Tn+1 ITn ) = (n+l) -1 { 1=
. n1c (X.)J
1 n (n +1) + ~~}
(4.43)
n
1
= n+1 Tn + 2(n+1) ~, n=1,2, ...
This implies that for
T~
= n(Tn -~) ,
(4.44)
Remark.
q.e.d.
The theorem may not hold when In(u) is defined by (a) of section 2.
However, if J (u) is convex, then it can be shown by the same technique that
{nTn } forms a sub-martingale sequence with respect to Tn even when I (u) is
n
defined by Ca) of section 2.
Also, if F(x) is not synnnetric about 0, the
However, as E >0, it follows by the same
ntechnique that {nT } forms a sub-martingale sequence with respect to Tn when
n
martingale property does not hold.
JnCu) is defined by Cb) of section 2.
5.
1\fi.I\lUt1\lCl~"J\7'Grvtrlm'l:lV'IJ\N'(N'G.
We do it in several steps.
First, let us
prove the following 1ennnas.
LPM4\ 5.1.
For every s(>O), there exists positive constants c
(1
(2
) and c )
s s
and a sample size n , such that for n > n
s
- s
(5.1)
(5.2)
~
P{n
ceu,n-e) A
(1)
> cs
a /_A/2BCF)
'l: ~
T
(2) -s
(log n)2} -< c s
n
.
I
I
.e
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.e
I
-17-
Proof.
We only prove (5.1), as (5.2) follows exactly on the same line.
Now
= P{8 L,n < 8-n-~T(J..{£
-A/2B(F) - n -k2Cs(1) (log n) 2}
A
where T(l) = ~ + ~-~T /2 + o(n-~). Let us define T = T (X +n~AT /2/2B(F))1 )
n,a.
a.
n
n -n
a.
-n
A
-k
where 8=0, without any loss of generality, and let Tn = Tn (X-n+{n 2(Ta. /_A/2B(F)
l:+ c(l) (log n)2)}1 ), Then, by theorem 4.2, with probability
s
-n
> l-k(2)n-s,n~(T -1 ) = B(F)C(l) (log n)2 + O(n-~(log n)3). Hence, it suffices
-
s
n n
s
to prove that with probability> l-O(n- s ), IJn IT _T(l) I < B(F)c*(l) (log n)2,
n n,a. s
where c*(l) < c(l). We now write a = n-~AT /2/ 2B (F) , and define H (x), H (x),
s
s
a.
a
n,a
F (x+a) as in section 4. Then T = f J( ~l H (x))dF (x+a), and it can be shown
n
nOn
n,a
n
by some standard computations as in section 4 that T(l) = f J(H (x))dF(x+a)+o(n-~).
n,a. 0
a
Hence, it is enough to show that for every s>O, there exists an ns such that
00
00
(5.4)
p{n~lfOOJ(
~l
o n
H (x))dF (x+a) - fooJ[H (x)]dFa(x) I > c*(l) (log n)2}
n,a
n
0
a
s
< c(2)n- s , for all n > n .
-
s
-
s
Now, we rewrite the first term on the left hand side of (5.4) as
(5.5)
00
n
n 2 f J( +1 H (x))d[F (x+a)-F(x+a)]
o n n,a
n
k
1
+ n~
ro {J[n~l Hn,a (x)]-J[Ha (x)]) dF(x+a)
00
Then, by lenuna 4.2 and (2.7) we have,
= 11+1 2 (say).
I
I
.e
-1800
= In~l
(5.6)
oJ [Fn(x+a)-F(x+a)]J' [n~1
Hn,a (x)]dHn,a (x) I
R(a)
I
I
I
I
I
I
IIJ' [ nil]
-< -L
n+ 1 L.~.n1IF
n (X.+a)-F(X.+a)
1
1
n+
1=
-< O(n
(4.29) through (4.37), it follows that
Ie
I
I
I
I
I
I
I
where
I
K l '-1
n n+1
110g n) .::7'1""+
nT.l 1n+1'
-1
where R~~) = Rank{lxi-al: IXl-al, ... ,lxn-al}. Again, on the same line as in
(5.7)
•e
-~
I
I12 I = E s (n -~ (log
E
S
is some fixed positive munber.
LEMMA 5.2.
(5.9)
This completes the proof of the lemma.
For every s(>O) , there exists an ns ' such that for n
P{B(F)n~ (8" U,n-8" L,n)/ATex. /2 = l+O(n -~ (log n) 3 }
(5.8)
Proof.
n) 3/2 ), with probability>- l-O(n -s ),
By
~
~
ns '
l-O(n- s ).
virtue of lemma 5.1, we have with probability> l-O(n- s ),
8-n-~{(ATex./2/2B(F)) + c(l)
(log n)2} < 8L < eu
s·
,n ,n
Hence, the proof directly follows from theorem 4.3 •
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
I·
I
-19-
A
-k
A.
{8 L,n} and {8U ,n} are uniformly continuous with respect to {n 2}
LEMMA 5.3.
in the sense that for every
such that as
and n (both positive), there exists a 6(>0),
£
n~,
(5.10)
(5.11)
Proof.
P{
~
•
A
sUI?
1m (eu ,-eu )1> n} <
In'-nl<6n
,n,n
A
WT1te n (8
_
) - (n/n')
~
A
IJnT (8
•
~
A
,-8) - n (8
-8).
L,n L,n
L, n , n
L
s
theorem 4.3 and (2.13), we have with probability ~ l-O(n- ),
rn
,-8
£
By lemma 5.1,
1
(eL,n,-eL,n) = ~(Ara/zlB(F))[(n'/n)~-l]
+
B-l(F)[n~{Tn:"il
(X -81 ) - T ,(X ,-81 ,)}]
-n
n -n -n
+ 0(1).
By lennna 5.1, theorem 4.3 and (2.13), we have, with probability ~ l-O(n- s ),
rn (eL,n,-eL,n)
1
= ~(Ara/zlB(F))[(n'/n)~-l]
+
B-l(F)[Il1{Tn(~n-8!n) - Tn '(!n,-81n ,)}]
+ 0(1).
Thus it suffices to prove that
(5.12)
Define
p{
T~
suI?
In'-nl<on
rn
as in theorem 4.6.
ITnC!n) - T ,(X ,) I > nI 8=0} <
n -n
£
•
Then, routine computation yields that
E(T~) = 0, E(T~2) = nA~/4, where A~ is defined by (2.10). As {T~, Tn} forms
a martingale sequence (cf. theorem 4.6), by using the Kolmogorov inequality
for martingales (see
Lo~ve
(1965, p. 386)), we get,
I
I.
I
I
I
I
I
I
I
Ie
-20-
(5.13)
1
Put t=nln~ and note that ~ = A2+o(1).
Then,
(5.14)
Since, kIn
~
0 and 0
(5.15)
P{
~
Tn+k < ll, we get,
Sup
n<n'<n+[on]
rn
IT (X )-T ,(X ,)
n -n
n -n
l>nle=O}
Proceeding similarly when n-[on]<n'<n, we get (5.12).
< £/2.
The proof of (5.11) is
quite analogous.
We may observe that (5.10) and (5.11) remain true whether (a) or (b) is
I
used in defining Tn(!n) in (2.2). This follows immediately from (2.15).
I
:£..FM.iA 5.4.
I
I
I
I
I
••
I
lim P{
rn (8L
n -e)B(F)
NZ
+ '[0./2
~
x} = <P (x), where
<P
is the standard
J1:"+<lO
normal d.f.
Proof.
See Sen [13].
Define ni(d) = [v(d){l+(-l)i£} + l+(zl)il , i=1,2. We next prove the following lerrnna •
LEMMA 5.5.
Proof.
Ln-n
_00
Ln~2(d) P{N(d»n} < 00.
2
(d) P{N(d»n} = Ln-n
_00 (d)P{Ir(8U,r -8L ,r »2 Ird, for all r=l, ... ,n}
2
~ Ln=00
n2 (d)p{rn(8u,n-6L ,n»2 rn d} .
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
I·
I
-21-
Since for n~nZ(d), Zd
In ~
and as by lemma 5.Z, 11J1(8U
ability ~ l-O(n- s ) as
that
00 P{N(d»n}
l n=n
o
<
-
Jl"+OO,
l
Zd InZed) ~ [ATa/Z/B(F)] (l+E'), where E/3<E'<E/Z
,n -8L ,n)-ATa /Z/B(F)I = O(n-~(log n)3), with probit follows that there exists some n~nZ(d) such
00
-n O(n -s )<00 for s>l.
n-
Proof of the main theorem.
The lerrana follows.
0
It follows from lenuna 5. Z and definition of N that
N is finite a.s. for all d>O and is a nonincreasing function of d.
proves that EN<oo.
Lerrana 5.5
The remaining statements of (3.3) follow from the definition
of N and Monotone Convergence Theorem.
definition of v(d).
We obtain (3.4) from lemma 5. Z and the
(3.5) is a consequence of theorem 1 of Anscambe [1],
1erranas 5.Z, 5.3 and 5.4.
To prove (3.6), consider
(5.16)
where E extends over all n<n (d), E over all n such that n (d) <n<n (d) and
l
Z
l
Z
1
E over all n>nZ(d). Since lim v (d) =:00 and lim N(d)/v(d) = 1 a.s., for every
3
d-+o
d-+o
£>0, there exists a value of d, say do, such that for all O<d<do '
P{n1 Cd)
~ NCd) ~ nZCd)} ~ P{ I~~~ - ll<d~
l-n, n being arbitrarily small.
Hence, for d<d ,
-0
(5.17)
v -1 (d)E n P{N(d)=n}
1
Also, for all n I Cd) ~ n ~ nZed),
(5.18)
Finally,
~
(l-£)P{N(d)<n (d)} < n(l-E).
IvTcrr - 11
I
< £, and hence,
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
I·
I
-22-
Since,
\)(d)~
of (5.19)
as d-+O, using lenuna 5.5, the first tenn on the right hand side
-+0 as d-+O.
P[N (d)=n 2 (d)]
2
~
Using the same argument as used in the lerruna 5.5,
P{N(d)
~
n 2 (d)}-+0 as d-+O.
6.
(ARE).
(3.6) is now proved.
Q.E.D.
SUppose we have two bounded length
confidence interval procedures A and B for estimating the median of a symmetric
population by means of an interval of length
~
2d (d>O); if NA and NB denote
the stopping variables and PA and PB the coverage probabilities of the two
procedures A and B respectively, then we define the ARE of the procedure A
with respect to the procedure B by
(6.1)
provided lim PA = lim PB and either of the limit exists.
d-+O
d-+O
Let s and c stand for the procedures suggested by us and that by Chow and
Robbins [4] respectively. Using (3.5) and (3.6) of theorem 3.1 and the
corresponding results of [4], we get under the assumption that cr 2 =Var(Xl)<oo,
(6.2)
The above is the Pitman-efficiency of a general rank order test with respect
to Student's t-test.
In the particular case when J(u) = ~-1(12u), where ~ is
the d.f. of a standard normal variable, (2.5)-(2.7) are satisfied and it has
been proved by Chernoff and Savage
(1958) that a 2 B2 (F)
~A2
for all d.f.
F with a density f and a finite second moment, equality being attained when
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
I·
I
-23-
and only when F is a nonnal (0,0 2) d. f.
Hence, in that case, the ARE of our
procedure with respect to the Chow and Robbins procedure (see [4]) is
~l,
equality being attained if and only if the parent df is normal.
In the particular case, when J(u)=u, esc = 120 2[
00
I
f2(x)dx] 2 and
-00
this includes Geertsema's [6] e(W,M) expression as a particular case of the
sequential procedure suggested by us.
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
I·
I
-24REFERENCES
[1]
ANSCOMBE, F. J. (1952). Large-sample theory of sequential estimation.
Proc. Camb. Phil. Soc. 48 600-607.
[2]
BAHADUR, R. R. (1966). A note on quantiles in large samples.
Math. Statist. 37 577-580.
[3]
CHERNOFF, H. AND SAVAGE, 1. R. (1958). Asymptotic nonnality and efficiency
of certain nonparametric test statistics. Ann. Math. Statist. 29
972-994.
-
[4]
CHOW, Y. S. AND ROBBINS, H. (1965). On the asymptotic theory of fixedwidth sequential confidence intervals for the mean. Ann. Math.
Statist. 36 457-462.
[5]
FARRELL, R. H. (1966). Bounded length confidence intervals for the p-point
of a distribution function, III. Ann. Math. Statist. 37 586-592.
[6]
GEERTSEMA, J. C. (1968). Sequential confidence intervals based on rank
tests. Unpublished doctoral dissertation, University of California,
Berkeley.
[7]
HAJEK, J. AND SIDAK, Z. (1967).
New York.
[8]
HOEFFDING, W. (1948). A class of statistics with asymptotically nonnal
distribution. Ann. Math. Statist. 19 293-325.
[9]
HOEFFDING, W. (1963). Probability inequalities for sums of bounded random
variables. J. Amer. Statist. Assoc. 58 13-29.
,
[10]
[11]
v
v
Ann.
,
Theory of Rank Tests.
Academic Press,
,
JURECKOVA, J. (1969). Asymptotic linearity of a rank statistic in regression parameter. Ann Math. Statist. 40, in press.
,
LOEVE, M. (1963). Probability Theory (3rd Edition) Van Nostrand, Princeton.
[12]
PURl, M. L. (1964). Asymptotic efficiency of a class of c-samp1e tests.
Ann. Math. Statist. 35 102-121.
[13]
SEN, P. K. (1966). On a distribution-free method of estimating asymptotic
efficiency of a class of nonparametric tests. Ann. Math. Statist.
37 1759-1770.
[14]
SPROULE, R. N. (1969). A sequential fixed-width confidence interval for
the mean of a U-statistic. Unpublished doctoral dissertation,
University of North Carolina.