.t~.
ON NONPARAMETRIC SEQUENTIAL POINT ESTIMATION OF LOCATION
BASED ON GENERAL RANK ORDER STATISTICS*
by
Pranab Kumar Sen
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1234
June 1979
,
ON NONPARAMETRIC SEQUENTIAL POINT ESTIMATION OF LOCATION
BASED ON GENERAL RANK ORDER STATISTICS*
PRANAB KUMAR SEN
University of· North Carolina, Chapel Hill, N.C.
ABSTRACT
A nonparametric sequential procedure for the point estimation of
location of an unspecified (symmetric) distribution based on a general
class of one-sample (signed) rank order statistics is considered and
its asymptotic theory is developed.
The asymptotic risk-efficiency of
the proposed procedure is established and certain almost sure and
moment convergence results on rank based estimators are studied in
this context.
AMS Subjeat CZassifiaation Nos:
62L12, 62L15, 62G05.
Key Words & Phrases: Almost sure representation, asymptotic normality,
asymptotic risk-efficiency, loss function, moment-convergence, rank
estimates, risk function, sequential point estimation, signed r'ank
statistics, Skorokhod-Strassen embedding, stopping variable.
*Work supported by the National Heart, Lung and Blood Instutute,
Contract No. NIH-NHLBI-71-2243 from the National Institutes of Health.
-2-
1.
INTRODUCTION
Sequential point estimation of the mean of a normal distribution
and its asymptotia pisk-effiaienay have been considered by Robbins
(1959), Starr (1966), Starr and Woodroofe (1969) and Woodroofe (1977),
among others.
Recently, Ghosh and Mukhopadhyay (1979) have studied an
asymptotically risk-efficient sequential procedure for point estimation
of the mean of an unspecified distribution (having a finite eighth
order moment).
For the dual problem of bounded-length aonfidence
intepval for the mean of an unspecified distribution, Chow and Robbins
(1965) have developed an elegant sequential procedure which is asymptotically both consistent and efficient.
Sen and Ghosh (1971) have emp1oyed_
a general class of one-sample rank order statistics for nonpapametpic
sequential intepval estimation of location.
for the point estimation problem.
Nothing parallel is done
The asymptotic theory of nonparametric
sequential point estimation of location, based on a general class of
rank statistics and derived estimates, is developed in the current paper.
The asymptotic risk-efficiency of the proposed procedure is established
and a comparison is made with other procedures.
Along with the preliminary notions, the proposed sequential procedure for the point estimation of location is formulated in Section 2.
Certain moment-convergence results on rank order estimates of location
are also studied in this section.
These results have an important role
to play in the development of the asymptotic theory in the subsequent
sections.
The main theorems of the paper are presented in Section 3 and
their proofs are considered in Section 4.
Unlike the case of Woodroofe
(1977), our stopping vaPiable does not involve a sum of independent
random variables (or martingales/reverse martingales), and hence, a
-3-
different approach is needed here.
The concluding section deals with
the asymptotic relative performances of the proposed and some other
competing procedures and the superiority of the nonparametric appraoch
is stressed in this context.
2.
THE PROPOSED PROCEDURE
{X., i ~1}
be a sequence of independent and identically
l.
distributed random variables (i.i.d.rv) with a continuous distribution
Let
function (df)
Fa(X),
x EE = (_00, (0), a real (unknown),
Fa
unknown
and where
Fa (x) =F(x - a) ,
F symmetric about O.
For nonparametric estimation of
~* ={~*(u),
0 <u <I}
+ ~* (1 -u) =0, V 0 <u < 1),
saOl'e
funation~
n(~
a, we proceed as follows.
Let
be a non-decreasing, skew-symmetric (i.e.,
~* (u)
~
(2.1)
non-constant and square-integrable
= {ep(u) = ~* ((1 +u)/2),
0 <u < I}
and, for every
1), let
(2.2)
where Un. 1· :;:; ••• :;:; Unn are the ordered rv's of a sample of size n
from the uniform (0, 1) df. Further, let "'Il
X =(Xl, ... ,X),
1 =(l, ... ,n)
n
"'Il
and for every real
b, let
X (b) =X"'I1 -bl"'I1 .
"'I1
Consider then the usual
one-sample rank order statistics
Sn (b) =S(X"'I1 (b)} =I~l.='
lsgn(x.l. -b)an (R+.
nl. (b)), bEE, n ~l,
where
for
is the rank of IXl.' -bl among
R+.(b)
nl.
i =1, ... ,n. Then, for every n(~ 1),
S (b) is \
n
.
IX -bl, ... ,lxn -bl,
I
in b: _00 < b <00 ,
while, under H : a =0 and (2.1), S (0)
O
n
symmetric about 0 and for large n,
has a distribution
-4!,;
~V N(O~
n-2g n (O)/An
~
1)
(2.5)
where
(2.6)
We define
e~l) =sup{b: Sn(b) >O},
8~2) =inf{b: Sn(b) <OJ;
(2.7)
@ =(6(1) +8(2))/2.
n
Then~
n
(2.8)
n
is a translation invariant~ median-unbiased~ robust and
en
consistent estimator of
6~
as
and~
n
-+OO~
(2.9)
where
2/B
2 and B = B(4)~ F) = food
v2=A
dx 4>* (F(x) )dF(x)
(> 0) .
(2.10)
_00
We let
v~
We shall see
= E(6n _6)2
Theorem 2.1] that
[viz.~
mild regularity conditions and
(2.11)
v~ exists «
00)
under very
Theorem 2.2] under additional
[viz.~
conditions~
nV
2
n
-+
v2
as
n -+ 00
To motivate and propose the sequential
the Zoss incurred in estimating
8
'"
Ln (c)=c
,..,
l (8n -6)
where
c
by
2
en
(2.12)
procedure~
suppose that
is
+c 2n; ,..,
c=(c1~
c2»Q.~
.-
(cost per unit sample) are specified constants.
2
the risk (for a given £) is
1
and c
(2.13)
2
=EL n (£) =clvn +c 2n
and we like to minimize this risk by a proper choice of n.
\t(£)
of(2.12) and
n =no (c)
,.., ~
(2.14)~
where
if
c /c
2 l
is
sma11~
then
Then
(2.14)
By virtue
An (c)
,.., is minimized at
-5-
(2.15)
here,
aCE) ""b(£)
means that
a(£)/b(£) -+1
as
c 2/c 1 -+O.
Thus, even
in this asymptotic setup, the minimum risk estimation of e demands
the know1dege of v,
which, by (2.10), depends on the unknown df
B($, F)).
(through the functional
sequential
pro~edure
F
Hence, we take recourse to a
based on a sequential estimator of
v.
As has been mentioned after (2.4), there exists a sequence of
known constants
p{
{Cn,a}
(for any
Isn (0) I ~Cn,a Ie
0 <a <1),
such that
=O} ~a > {ISn (0) I >C n,a Ie =o}·)
(2.16)
1
where
n-'1: n,a -+A1"a /2 as n -+00 ,
1"S is the upper 100S% point of the standard normal df.
(2.17)
Let
then
eL,n =sup{b:
S (b) >C
}
n
n,a
vn
=nA(eu
,n
au ,n =in£{b:
Sn (b) < -C n,a }; (2.18)·
- eL,n ) /2Cn,a .
Then, it follows from Sen and Ghosh (1971) that
(2.19)
V
n
is a trans1ation-
invariant, robust and strongly consistent estimator of
v. Hence,
motivated by (2.14), we consider the following sequential procedure.
Let
nO be an initial sample size and
y(> 0)
constant (to be defined more precisely in Section 3).
be a positive
Define the
stopping variable by
(2.20)
Then,
"-
eN(c)
is the proposed sequential point estimator of
e
and
""
the risk for this estimator is
(2.21)
,
-6-
We are primarily concerned with the asymptotic behavior of
N(£),
@N(£)
and
in Section 3.
A*(£)
cztc l ~O)
(when
and these will be reported
In the remaining of this section, we consider certain
additional results on rank statistics and estimates, which yield
(2.12) and have a useful role in the derivations in Section 4.
Let
Xn, 1 S"'S Xn,n be the ordered rv's corresponding to
X1 "",X ' Then, it is known [viz., Sen (1959)] that for any positive
n
p « 00),
k
(2.22)
Elxl.IP <00 > Elx n,r I <00, V kip Sr Sn -kip +1 .
Also, if 0 <'1. slim infrln slim suprln S 1 - a l <1, then
n-+oo
n-+oo
ElxllP <00 -> lim supElx
Ik <00,
n-+oo
n,r
Further, note that the scores
a (i)
n
for every k ~O.
(2.23)
r
in (2.2) are nonnegative and
~=
nondecreasing (not all equal to 0) and hence,
<j>(u)du >0
(or
A >0).
This insures that there exists a sequence 0 {k } of positive
n
numbers, such that k =min{k sn} for which
n
(2.24)
L{i<k }an (i) >L{i>k }an (i)
n
n
1
and there exists an a: '2 <a <1, such that
n -1 k +a
as
n
[For
n
£
~0
(or 1),
(2.25)
{i< k }) is an empty set, so that
n
the corresponding sum in (2.24) is null. Also, note that fl <j>(u)du +0
as
k =n
n +00 .
{i >k } (or
and hence ~ >0
Theorem 2.Z
n
1-£
insures (2.25).]
Under (2.22), for every
k (> 0),
nO =nO(k) =min{n: n -k +1 ~k/p} where kn
n
'" ·Ik <00 for every n ~nO and
suah that E Ien
-e·
Then, we have the following
there exists an
is defined by (2.24) - (2.25),
lim supEle _elk <00, for every k ~O.
n-+oo n
In partiauZar, if p ~k, then Ee n exists for every
A
(2.26)
n
~1
.
-7-
PROOF:
By (2.4), (2.7) and (2.8), we have
[8(2) >x k ] <==> [S (X k) ~O], [~(l) >X k] <-> [S (X k) >0].
n
n'n
n n, n
n
n, n
n n'n
(2.27)
,..,
Let
for
1 be the rank of IXn,1. -Xn, kn I among Ix n, 1 -Xn, kn I,···, Ixn,n -Xn, kn I,
i =1 , ... , n. Then, by (2.3) and the monotonicity of the scores in
R.
(2.2) ,
Sn (Xn,k) = {-LU<k }an (Ri ) + LU>k }an (Ri ) }
n
n
S{LU>k }an(i) -L{i<k }an(i)} <0,
n
n
by (2.23). (2.28)
Hence, by (2.7), (2.8), (2.6), (2.27) and (2.28), for every
A
en
<X k
n, n
Similarly, for every n
n
~l,
with probability 1.
(2.29)
with probability 1.
(2.30)
~l,
en >Xn,n_k +1'
n
Then, (2.25) follows from (2.29), (2.30) and (2.22) -(2.25).
Q.E.D.
It follows from Theorem 2.1 that unlike the case of -X =n -lLn. IX.,
n
1= 1
k
n ~l,
the existence of Elx I does not necessarily require that
n
EIxll k <00; (2.22) and n adequately large suffice for
E I§n - elk <00.
We need to strengthen (2.25) to (2.12), under appropriate regularity
conditions.
Since
e
is not a martingale or reverse martingale, the
n
usual theorems are not applicable here.
We proceed to exploit the
asymptotic linearity (in b) of n -~2S (b)
for this purpose.
n
First, we note that by virtue of (3.4) of Sen and Ghosh (1973a),
for every
t
~o
and n
~ 1,
1
E{exp(tn-~Sn(O)) IH :
O
2
where An is defined by (2.6).
there exists an n (= nO(c,d),
O
e =O}
1
2 2
sexp{.2 t An}'
Hence, for every c >0
such that
(2.31)
and
d <0,
-8-
p{n-~ISn(O) I >c lognlHO: e =O} ~n-d, '</ n ~nO
Now, we denote by ep(r) (u) = (dr/dur)<j>(u),
assume that the
constants
r =0, 1, 2,
0 <u <1
and
ep(r) exists almost everywhere and there exist positive
K and
O
o«~),
such that
(2.32)
Further, we assume that the df F is symmetric, absolutely continuous
and possesses an absolutely continuous density function
first derivative
0,
flex)
for almost all
x
f(x) with a
(a.a.x), such that
(i) for
defined by (2.32),
sup f(x){F(x)[l _F(X)]}-o-n <00
for some
n >0,
(2.34 )
x
and
(ii) flex)
is continuous a.a.x. and
sup If' (x)
I <00
(2.35)
.
X
Define
B =BCep, F)
as in (2.10) and let
(2.36)
Then, following the lines of the proof of Theorem 2 of Sen (1980), we
arrive at the following:
Suppose that (2.33) -(2.34) holds for some
1" >0
and (2.34) holds.
and an integer
H :
O
nO
0 «4 +21")
Then, there exist positive numbers
(possibly dependent on
-1
1
« 4)'
d* and q
1"), such that under
e =0,
P{wn>n
-d*
(logn)IHO} ~ qn
-1-1"
,'</n~nO'
(2.37)
Now, by virtue of (2.3), (2.4) and (2.7) -(2.8), we have
!.:
~
A
P{nZ(en -e) >logn}=
. P{n- Sn ((logn)/v'ii)
~ole=o}
k
(lOgn) -S (0) +Bvnlogn] +n-Z[S
!.:
= P{n-z[S.
(0) -Bvnlogn]
n ~
n
n
~
1 .
!.:
1
P{Wn >2Blog n Ie =o} +p{n-ZSn(O) >2Blog n Ie =o}
>
.;:,... q*..n -1-1" , q*< 00, v.
v n - nO'
~ole
=O}
-9-
by (2.32) and (2.37), where we let
p{n~(en
bound holds for
d =1 +T,
- 8) < -log n}.
1
c ="2 B.
A similar
Then, we proceed to prove the
following
theorem 2.2
T
Under (2.22), (2.35) and (2.34) for some
for every
>0..
k 2
lim E{n /
PROOF:
,
k
k
-8I } = VkElzl ,
Ie
n
(2.40)
has the standard normal df.
Z
Let
I (A)
-8)\k=E{ln\e
1:A
2
+ E{ln (8
q >k,
A.
be the indicator function of the set
E{n~(en
For
-1
k < 2 (1 + T),
n~
where
o«4+2T)
n
k
-8)1 I(n
1:2 A
I8n
n
_8)lkl(n~len
Then,
-81:0; logn)}
-81> logn)} =J
(2.41)
n1
+J , say.
n2
by the Holder-inequality,
(2.42)
By Theorem 2.1, for every
E Ien - 8 Iq
<00,
for every
q >0,
there exists an
n ~nq'
n,
q
while by (2.39), for
such that
n
adequately
large,
n k/2 {pen1:2 I8 -8 I > logn)} 1-k/q
n
A
= 0(nk/2-(1+T) (l-k/q)) .
Since,
k <2(1 +T),
by choosing
q
adequately large, the right-hand
side of (2.43) can be made to converge to 0 as
J
n2
+
0
On the other hand, letting
as
n
(2.43)
n
+00.
Hence,
(2.44)
+ 00 •
1: A
Y =n 2(8 - 8),
n
n
we have (on letting
8 = 0)
J 1 =E{ IY ,kI ( IY 1 :0; logn)}
n
n
n
=E{ IY IkI( IY I :0; logn)I(w :o;n- d * (log n))}
n
n
n
+E{IY IkICly 1:0; logn)l(w >n-d*(log n))}
n
n
n
=J
+ J
,
nll
nl2
say,
(2.45)
,
'
-10whereby (2.37) and (2,45), for
J
n adequately large,
k
n12
S (log n) P{W >nn
d*
(log n)}
(2.46)
=O(n -l-T (log n) k) -+ 0
Further, for
IYn I S
as
n-+ oo
logn and w sn -d* (log n),
n
by (2.7), (2.8)
and (2.36),
BY =n-~ S (0) +R • IR I sn- d* (log n) .
n
n
n'
n
Finally, note that
1
In-~ (0)
n
1
I sn~An
1
=O(n~),
(2.47)
with probability 1,
_!.:
(2.31) holds and n 2Sn (0)/A is asymptotically N(O,l). Thus, by
some routine steps, it follows that under HO: 8 =0, for every fixed
k (~O),
logn)I(w Sn
n
-d*
k
k
(log n))}-+ A EIz I
.
(2.48)
Hence, from (2.45), (2.47) and (2.48), we have
lim J 1 =B-kzkElzl k = VkElzl k , V k ~O
n-+oo n
Thus, (2.40) follows from (2.41), (2.44) -(2.46) and (2.49).
Remarks.
~(u)
(2.49)
Q.E.D.
For the particular case of the Wilcoxon Scores (i.e.,
=U: 0 SU Sl), Inagaki (1974) has considered an almost sure (a.s.)
representation for
!.:
n 2(8 n -8)
A
interms of a sum of i.i.d.r.v's.
Our
Theorem 2.2 provides analogous results for a broad class of rank order
estimators.
Following the lines of the proof of Theorem 2.2, we have
A
-1
n(8 n -8) - B Sn (8) = ~,
say,
n
where under the hypothesis of Theorem 2.2, as n-+ oo,
(2.50)
(2.51)
Firther, it follows from Sen and Ghosh (1973a) that under
HO: 8 =0, {Sn(O), n ~l} is a (zero-mean) martingale sequence and under
conditions less restrictive than (2.33) -(2.34), there exists an n >0,
,
-11-
such that under HO'
A-lSn (0)
where
= Wen)
W={W(t), t £[0, oo)}
+ O(n~-n)
a.s., as
n
(2.52)
-+00,
is a standard Wiener process on
[0,
00).
From (2.50), (2.51) and (2.52), we conlcude that the Skorokhod-Strassen
{ V -1 n(8A
embedding of Wiener proeess holds for
hypothesis of Theorem 2.2.
n
-8)
}
under the
Sen and Ghosh (1973b) have also condidered
an a.s. representation for one-sample rank order statistics.
special case of their theorem, we have under
Sn (0) = I~1= l</>* (F (X.1 )) + ~*,
n
As a
HO: 8 =0,
n ~ 1,
where, under conditions less restrictive than the ones in Theorem 2.2,
n-~~; =O(n-n) a.s.,
as
n
-+00
(for some n >0).
(2.54 )
From (2.50), (2.51), (2.53) and (2.54), we conclude that
n-~In(~n -8) -B-II~=1</>*(F8(Xi))X-+() a.s., as n -+00,
(2.55)
and this extends Inagaki's theorem to a broad class of signed rank
statistics where
</>*
need not be bounded.
3•
THE
MAIN TIIEOREMS
A
The asymptotic behavior of N(£), 8N(£) and A*(£) (as c 2/c l +0)
will be studied in this section. The results to follow depend on
through
c =c 2/ c l
only, and hence, for notational simplicity,
A
we let
0
A
c l =1, c 2 =c, N(£) =Nc ' 8N(c) =8 ' A*(~) =A~,
Nc
o
An ()
=Ac • Then, we have the following.
c (c)
~
o "'"
Theorem D.l Under (2.22) and (2.33) -(2.34) for some
no (£) =nc
~
y>O
[in (2.20)], as
cS
and
1
<'4' for every
c +0
(3.1)
(3.2)
.
·.
-12-
Theopem'3.2 Undep (2.22), (2.35) and (2.33)
os(4+21")
-1
-(2.34)~
,1">1; 1+2y<1",
when
y>O,
(3.3)
whepe y is defined in (2.20), then
lim (A*/Ao ) = 1 •
c+O
c c
(3.4)
It may be remarked that (3.4) asserts that the pisk involved
in the proposed sequential procedure is asymptotically (as
c + 0)
equal to the risk of the corresponding optimaZ fixed-sample size
procedure.
Thus, for all
F and
satisfying the hypothesis of
C/>
Theorem 3.2, the proposed sequential procedure is asymptoticaZly pisk-
f11 C/>(U) Irdu <00
efficient.
In particular, (3.3) demands that
some
and this is true for the Wilcoxon, normal scores and the
for
°
r >6
r
other commonly used rank statistics and is less restrictive than
exp{t(j>(u) }du <00
(for some
t >0), as employed in Sen and Ghosh
o
(1971) for the confidence interval problem.
The asymptotic normality of
(no)-~(N
_no)
c
c
c
depends on the
asymptotic normality and uniform continuity, in probability, of
~
{n 2 (V -v)}.
n
~
.
For Wilcoxon scores, the asymptotic normality of n 2(V -v)
n
has been studied by Jure~kova (1973) and her treatment holds generally
for bounded and continuously differentiable score functions; however,
for unbounded scores, this remains as a challenging open problem.
The following theorem presents the impact of this on the asymptotic
normality of stopping times.
Theopem
3.3
Suppose that in (2.20)
Y
loP
>-2' Nc /n c ~ 1 as
c + 0 and
n ~2(vn -v)/f3-:;r-N(0, 1),
whepe
{ n2lvm-vnl
~
} ~Oas
p
sup
0+0,
m: Im-nlson
(3.5)
13 is a finite positive numbep. Then, as c +0,
( n0) -~ (N -n0)
c
c
c
---rr N( 0,
f32/ v 2) •
(3.6)
..
-13Note that (3.5) may be replaced by the weak convergence of the
1
{n-~(vk -v)/13: k sn}
partial sequence
to a Gaussian function.
The
proofs of the theorems are presented in the next section.
4.
We let
n
o
c
PROOFS OF THEOREMS 3.1, 3.2 and 3.3
= [c
-!.:
(see (2.15)), and, for every
2V ]
0 <E <1,
we
let
n
lc
=[c
-1/2(1+y)
where, we choose
c
]
n
'
0
so small that
definition in (2.20),
and
=[(l-E)n]
c
2c
nO sn
lc
n
<n
o
3c
2c
(4.1)
= [(1 + E)n ],
c
<n 3c '
Then, by
with probability 1, so that
Nc <::n l c '
(4.2)
as, for
n
n
~
2C
~O
.!.:o
' C"ll -v sc~C (1 -E) -C"llO"'" -EV.
Now, by (2.16) -(2.19)
and proceeding as in (2.38), it follows that, for
~ ~
{
Pn(t1
~ A
{
Pn
(e
j .
u,n
L,n
}
-e»logn=O(n
-l-T
n
adequately large,
(4.3)
).
-e) <-logn } = O(n -l-T ),
(4.4)
so that by (2.17), (2.19), (4.3), (4.4), (2.36) and (2.37), we
conclude that for
n
p{ Iv
n
adequately large,
l
;<:EV} s3q*n- - T ,
-vi
Hence, by (4.2) and (4.5), as
P{Nc sn 2c } s
\L
n
lc
{3q*n -l-T}• = 0 (n -T )
snsn
P{N
c
>n} =P{k
n
(4.5)
c +0,
1c
2c
= 0 (c T / 2 (1+ y )) ,
In a similar manner, for
q* <00.
b Y (4 • 1) •
(4.6)
~n3c'
<c-~(Vk +k- Y),
1
~P{n <c-~(V
V k E [nO' n]}
1
+n- Y)} =p{vn >c1l-n- Y} .
n
!.:
!.: 0
Y
= p{V - v > c ~ - c ~- n - }
n
c
sp{
IVn -vi
>n}
where by (4.1),
n >
(4.7)
o.
,
"
r ,
•
-14Thus, by (4.5) and (4.7),
P{Nc >n3c } + 0 as c +0. Since E(> 0) is
o
arbitrary, by (4.6) and the above, p{ IN /n -11 >d + 0 as c +0.
c c
Also, by (4.5) and (4.7), I >
peN >n) = 0(n3-T) +0 as c +0, and
n-n 3
c
o
c
0
hence, E{Nc I(Nc ~n3 c }}/nc +0 as c +0. Also, E{Nc I(Nc ~n2 c )}/nc ~
(1 -E)P{N
~n2
by (4.6), while by (4.5), (4.6) and (4.7),
E{N I(n
)}/no can be made arbitrarily close to 1 by choosing E
c
2c
c
3C
small. Hence, E(N /no ) +1 as c +0. A similar proof holds for
c
c
<n <n
} +0
c
c
E(N /no)k + 1, V 0 ~k <1.
c
c
This completes the proof of (3.1).
Now,
it follows from Sen and Ghosh (1973a) that under
H : 6 =0, {Sn(O)} is
O
a martingale sequence, and hence, using the Kolmogorov inequality (for
submartingales) and some standard analysis, we obtain that as
lim{
max
n-~Is (0) -S (O)I} =0,
0+0 m: Im-n I~on
m
n
in probability.
By (2.39), (2.47) and (4.8), we obtain that as
lim{ max
n~ 11~ - I} = 0,
0+0 m: Im-nl~on
m n
e
n too,
n+ oo ,
in probability,
so that (2.9), (4.9) and (3.1) insure (3.2).
(4.8)
(4.9)
Hence the proof of
Theorem 3.1 is complete.
To prove (3.4), we make use of (2.15), (2.21) and (3.1) (i.e.,
limE(N /no ) =1), and hence, it suffices to show that
c+O
c c
k
1
A
lim(vc 2) - E(6 -6)
N
c+O
c
2
= 1
(4.10)
Now, by Theorem 2.2, for every k E (2, 2 + 2T),
n
E{(§N -6)2I(N ~n2 )} =I 2c E{(e _6)2 I (N =n)}
'c
c
c
n=nlc
n
c
n
~In:~ (EI~n _6I k )2/k(p{N =n})1-2/k
c
lc
n
.
e _6 Ik) 2/k (P{Nc ~n 2c }) l-2/k
2c E I
Ln=nlc
n
:c;; [\'
=(0(ni~k-2)/2))2/k(0(cT/2(1+y)))1-2/k
=0(c(k-2)(1+T)/2k(1+y)).
[by (4.1)]
(4.11)
[by (4.6)]
f
-15Since, by (3.3),
~ >0,
(1 +T) >2(1 +Y),
while for
T =1
1
2~ - n
1
(k -2)/k ='2 + 8 +4~ -2n > '2 for
n >0,
(4.11) that by a proper choice of
+~
n > 2~
k £(2,2(1 +T)),
and
k =2(1 +T) -n,
we obtain from
under (3.3),
k 1
I'
2
lim(vc 2) - E{ (eN - e) I(N :;;n )} =0.
2c
c+O
c
c
Similar1y,as
E{(e
Nc
(4.12)
c+O,
-e)2 l (N
~n3 c )}:;; (L>
n-n
c
Ele
3c
k
_el k12 / (P(N
J
n
c
~n3 c )) 1-2/k
=0(c(2+T)(k-2)/2k),
so that noting that by (3.3),
T >1
(4.13)
and then taking
k =4,
we
obtain from (4.13) that
k
lim(vc 2)
c+O
-1
A
E{ (eN
2
- e) I(.N_ ~n3 )} = O.
c
c
c
(4.14)
Thus, to prove (4.10), it suffices to show that
k
lim(vc 2)
c+O
-1
A
E{ (eN
c
2
- e) I(n
<N
2c
<n
c
3c
)} = 1.
Now, by Theorem 2.2 and the definition of
lim (vc
k -1
2)
dO
while,
P{n
2c
<N
c
<n
3c
} + 1
as
c +0.
nO,
A
{
c
(4.15)
we have
4 k
E( S
- e) } 2 < 00 ,
nO
c
(4.16)
Hence, to prove (4.15), it is
enough to show that
k -1
A
A
2
lim(vc 2) E{[(SN -S ) l(n
<N <n )} =0.
2c . c
3c
c+O
c
n0
(4.17)
c
Let us then write (as in the proof of Theorem 2.2)
k
A
Bn 2(S
so that writing
:;; ER 2 l(n ~A
Ie -e I
n
n
_k
- S) = n 2S (S) + R ,
n
n
n
(4.18)
2
2
k A
2
k A
ER =ER I(n2le -el:;; 10gn) +ER l(n 2 lS -si > 10gn)
n
n
n
n
n
2
2
kA
I
:;; logn) +2B E{n(S -e) l(n 21s -s > 10gn)} +
n
n
1 2 k A -e I. > 10gn)}
2E{n -Sn(e)l(n2ISn
through (2.47) (where we take
A
and then proceeding as in (2.42)
k =2
and
T >1 + 2y, Y >0), we obtain
"
J
.
-
., "
-16-
by some standard steps that
ER 2
n
= 0(n- 1 - y )
(4.19)
'
so that
(4.20)
Further, for
-1
N
n 2c <Nc <n 3 c '
C
:5:n
-1
2C
k
=O(c 2).
Hence, by virtue of
(4.18) and (4.20), it suffices to show that
lim{(nO)-lE[(SN (0) -S (0»2I(n
<N <n )
3c
2c
c
c
c +0
c
n0
c
Since, under
H :
O
e =0,
(3.1),
<N
<n
P{n
2c
c
3c
}
{Sn(O)}
1
-r
as
(no)-l\,n~c
c
Ln-n
:5:
n
c+o
and (2.31) holds, we have
Ie
=0]
c
(E[(S (0) -S
n
2c
=O]} =0. (4.21)
is a (zero-mean) martingale, by
O
(n )-lE[(SN (0) -S (0»2I(n
c
oc
z <N c <n 3c )
c
Ie
n0
c
(0»4 Ie =O])\P{N
3C
:5: (nO)-1(\,n
E[(S (0) -S (0»4 Ie
c
Ln=n 2C
n
nO
c
=n})~
=0]J~(p(n2 c:5:Nc
:5:n
3c
»~
c
(n~)-1[~:~2CO(ln -n~I»)~-l
:5:
o
= 0((n3c -n 2c )/n c ) ,
where by (4.11),
small by choosing
(n
3c
-n
2c
E so.
(4.22)
)/n
o
c
~
2E
and this can be made arbitrarily
This proves (4.21) and the proof of Theorem
3.2 is complete.
To prove Theorem 3.3, we note that by definition in (2.20),
c-~N
c
:5:N :5:C-\V -1 + (N -l)-Y)
c
N
c
c
whenever
N >n '
O
c
(4.23)
1
Thus, noting that
n~ . . . vc-~
we have from (4.23), for
N >n '
c
O
f
-17-
as
c
to.
Hence, (3.6) follows from (3.5) and (4.24).
5.
Q.E.D.
SOME CONCLUDING REMARKS
Under the hypothesis of Theorem 3,2, the sequential procedure
is asYmptotically risk-efficient.
It may be noted that
depends on the score function through
V
=A/ B,
by (2.6) and (2,10) and are functions of
$
where
A and B are defined
(and F).
To make this
depence clear, we denote
(5.1)
where
defini~g
A$2 =
$*
as in before (2.2),
11 <p 2 (w)du
and
B$ =
o
f°fx
(5.2)
$* (F (x)dF (x) .
_00
Thus, if we have two different score functions say,
the corresponding optimal
0
11.
A(c l , c 2 , A$l' B$l)
c
$1 and $2'
then
are
and
A(c I , c 2 , A$2' B$2) ,
(5.3)
and smaller is the quantity, the better is the corresponding procedure.
Hence, the relative efficiency of the procedure based on the score
function
$2
with respect to the one based on
$1
is
e($l' $2) =A(c l , c 2 ' A$l' B$l)/A(c l , c z ' A$2' B$z)
= (A$IB$2)/(A$2 B$1)
(5.4)
and this agrees with the (square root of the) classical Pitmanefficiency of the rank tests (for location) based on the score
function
$2
with respect to the score function
define 1jJ(u) =1jJ* ((1 +u)/2),
0 <u <1,
A~
= f(ff/f)2 dF
Hence, if we
where
1jJ*(u) =_f(F-I(U))/f(F-I(u)),
then
$1'
0 <u <1,
is the Fisher information and by (5.4),
~,
"
.c- ..
(tl
'
-18e(~, ~)
= p(~,
=
~nd
~)
(5.6)
(5.6)
(to~*(UN*(U)dU)/Al~)
the equality sign holds iff
~* .e~*
•
Thus,
~
is an optimal score function.
For the procedure based on the sample means and variances,
considered by Starr (1966) and Ghosh and Makhopadhyay (1979), the
corresponding
is
(5.7)
Thus, the asymptotic relative efficiency of the proposed sequential
procedure with respect to the normal theory procedure is
e(~, N) = aB~/A~
(5.8)
.
In particular, if we use normal scores (i.e.,
the standard normal df), then, (5,8) is
equality sign holds only when
~
1
F is normal.
~*(u),
for all
the inverse of
F,
where the
This expa1ins the asymptotic
supremacy of the normal scores procedure over the parametric procedure.
Even for Wilcoxon scores, when
and it is usually >1
F is normal, (5.8) reduces to
for distributions with heavy tails.
1~
(3/n)'2
For both
these scores, conditions for the applicability of Theorems 3.1 and 3.2
hold, while Jure~kova's (1973) theorem insure (3.5) for the Wilcoxon
scores.
~
.978
t
-19REFERENCES
[1]
CHOW 1 Y.S. and ROBBINS 1 H. (1965). On the asymptotic theory
of fixed-width sequential confidence intervals for
the mean. Ann. Math. Statist. ~R1 457-462.
[2]
GHOSH, M. and MUKHOPADHYAY 1 N. (1979). Sequential point
estimation of t-e mean when the distribution is unspecified.
Commun. Statist. SeT' A. ~, 637-652.
[3]
INAGAKI,
[4]
JURECKOVA, J. (1973). Central limit theorem for Wilcoxon rank
statistics process. Ann. Statist. 1, 1046-1060.
[5]
ROBBINS, H. (1959). Sequential estimation of the mean of a
normal populat:i,on. PT'obabiUty and Statistics
(H. Cramer Volume), Almquist and Wiksell, Uppsala 1
pp. 235-245.
[6]
SEN, P.K.
(1959). On the moments of the sample quantiles.
Calcutta Statist. Assoc. Bull. ~. 1-19.
-[7]
SEN, P.K.
(1980). On almost sure linearity theorems for signed
rank order statistics. Ann. Statist. ~, in press.
N.(1974). The asymptotic representation of the
Hodges-Lehmann estimator based on Wilcoxon two-sample
statistic. Ann. Inst. Statist. Math. £2, 457-466.
v
.;
[8]
SEN 1 P.K. and GHOSH, M. (1971). On bounded length sequential
confidence intervals based on one-sample rank order
statistics. Ann. Math. Statist. i£, 189-203.
[9]
SEN, P.K. and GHOSH, M. (1973a). A law of iterated logarithm for
one-sample rank order statistics and an application.
Ann. Statist. 1, 568-576.
[iO]
SEN, P.K. and GHOSH, M. (1973b). A Chernoff-Savage representation
of rank order statistics for stationary $-mixing
processes. Sankhya3 SeT'. A. ~~, 153-172.
[11]
STARR, N.
[12]
STARR, N. and WOODROOFE, M. (1969). Remakrs on sequential point
estimation. PT'oc. Nat. Acad. Sci. USA 2~, 285-288.
[13]
WOODROOFE, M. (1977). Second order approximation for sequential
point and interval estimation. Ann. Statist. ~, 984-995.
(1966). On the asymptotic efficiency ofa sequential
procedure for estimating the mean. Ann. Math. Statist.
~I, 1173-1185.
lit.'
,..