On the Serfling-Wackerly Alternate Approach
by
I~yr:~nd
J. Carro11*
Department of Statistios
university of North Carolina
Chape l Hi Zl.J North Caro Una 2?514
Institute of Statistics fIi.meo Series Ho. 1052
February~
* This
1976
research was supported. by the: Air Force Office of Scientific
Research under Grant no. AFOSR-75-2796.
On the Serfl ing-!<!ackerly Alternate· Approach-
by
P4ymond J. Carroll*
Department of Statistics
university of North CaroZina at ChapeZ RiZZ
ChapeZ RiZZ 3 North CaroZina 27514
.e.BSTRACT
For the problem of fixed-length interval estimation of the mean
and median, the limiting distributions of the stopping rules recently
introduced by Serfling and \vackerly [8J are compared to those of Chow
and Robbins [4] as both the non-coverage probability a and interval
length d converge to zero.
The procedures based on the sample mean
differ in the centering constants, while those based on the sample median
also differ in the scaling constants.
Institute of Statistics filimeo Series #1052.
*This research was supported by the Air Force Office of Scientific Research
under Grant No. AFOSR-75-2796.
1
1.
Introduction
The primary purpose of this note is to illustrate the second order
differences (where they exist) betl'.reen two cO:r1peting approaches to the
problem of fixed length confidence :L'1terva1 estimation of the center of
[4J~
syrrnnetry: one due to Chow and Robbins (C-R)
by Serfling and Wacker1y (S-V'V) [8].
The intervals of prescribed coverage
probability 1 - 2a are of the fom
differ; C-R fix a and let d
~
the other recently introduced
(~-d,
Xn+d) but the stopping rules
0, while S-W fix d and let a
show that the C-R approach with stopping time N(a?d)
for their problem in the sense that if m(a,d)
observations with F
c(d)
=1
c(d)
~
if F
1 as d
as both a,d
(1)
~
= ~,
~
O.
known~
N(a,d)/m(a,d)
~
~
o.
S-W
is lIinappropriate i l
is the required number of
c(d)
;t
1 as a
the normal distribution function.
~
0, although
They also show that
0, "mich suggests that the two approaches are equivalent
If the S-W stopping rule is denoted by M(a,d) ,
N(a,d)/i/l(a,d)
~
1
(a.s.) as a
~
0, then d
-+-
o.
TIle consistency property (1) says that the rules approximate the
same quantity asymptotically, but they still may be different in terms
of a limiting distribution.
The asymptotic distribution of N(a,d)
is easily found (see [6]), and in Section 3 and the
distribution of M(a,d)
Contrary to (1), this
(2)
(a)
~~pendix
the asymptotic
is obtained, a reslut of some interest in itself.
pa~er f:L~ds t~at
N(a,d) and I;I(a~d) have limiting distributions with the same
scaling constants as a,d ~ 0; for large classes of a,d the
C-R(S-W) alJIlr6ach has smaller centering constCl.:"1.ts if thepopulation kurtosis exceeds (i5.,less than) 3.
2
In the normal case with d fixed, the S-W approach is preferable
as a ~ 0 if d/o < 2.33 .
(b)
111e asymptotic distributions of analogous rules based on the sample
median ([5], [8]) are also obtained.
The differences are more pronounced
here; both the scali.l1g and centering constants are different.
As an
example of the results, Serfling and Wackerly's modification of Geertsema's
rule [5] is found to be infinitely nore variable than their own as a
2.
~
o.
The Two f'lethods
Let Xl'X2 , •.• be i.i.d. observations from a continuous symmetric
distribution F(x-e) with unique median 8 and with variance a2 •
Let
~(Aa)
ChQ'tlI and
= I-a.
111e two approaches are:
n
- 2 and define
Let sn2 = n -1 I (Xi-X
n)
i=l
Robbins (C-R) [4].
N(a,d) = inf{n: n
;?:
n'(a,d) = inf{n: n
;?:
Serfling and Hackerly (S-W) [8].
lCA)
Let
(A
S /d)2}
a n
2
(Aao/d) }.
= n- l
~(x)
i-'i(a,d) = inf{n: n
;?:
;?:
As further notation, let
. 1
1=
Then
tn a/tn ll'h(-d) a.Tld
m(a,d) = inf{n: Pr{IYn-el
I
I{X1·-X :S x}, where
n
Further, define
denotes the indicator of the event A.
Inn (-d) = inf exp(dt) ;"[ exp(tx)dHn ex).
t:SO
-00
n
Xi-~+d
< 0 for some i:sn}
d} :S a}.
110 (t) =
f
exp(tx)dF(x)
be the point at which WoCt) = exp(dt)l:1o (t)
since F has a unique median at zero,
and let to Cd)
achieves its infimum.
Then,
3
n
= n- l i~l exp(t(Xi-5\~+d)),
Define Wn(t)
to W~(t)
= 0,
so that for n
= inf
IDo(Z)
large,
exp(-zt)
t<O
and let
tn(d)
be the solution
= Wn(tn(d)).
mn(-d)
Finally, let
I exp(tx)dF(x).
3.. Sample Means
The asymptotic distributions of the nIles
N(a,d)
and H(a,d)
are obtained here.
The scaling constants are the same but the centering
constants are not.
For many sequences
a,d
-+
0, the value of the population
kurtosis detel1nines the smaller centering'constants.
I
xdF(x)
I
= 0,
2
x dF(x)
natural logarithm.
= 1.
Let
Let A(a,d)
log x both denote the
be the variallce of w(X) , where
= exp{to(d) (y+d)}
$(y)
R-n x and
In this section', assume
- yto(d)E exp{to(d) (X+d)}.
The proof of the following Lemna is given in the appendix.
idea is to represent
log mn(-d)/mo(-d)
as a sum of i.i.d. random variables
with remainder tern and use the techniques of [6].
the case d
Lemma 1.
Let
If, for same
fixed.
!:J.
= -R-n moC -d).
£ >
0, as
a,d
-+
Then, as
°
The basic
a
-+
0,
Note the result includes
4
then
(4b)
Lemma 2.
([6]).
As
a 1d
+
° or as
a
+
0,
(5)
(see Proposition
Now~
1), so that the centering constants
A~/d2
-9.na/t1 in (4) may be replaced by
(and the conclusion that the asymptotic distributions are equivalent
can be made) if, as a,d
+
0,
Note that if dAa _ A~ for some £ > o? then since A~/(-Z9.na)
(4a) holds but because of Proposition S?
H(a?d)
+
+
-00
°
+00
+
1,
if EX4 > 3
if EX4 = 3
if f:.X 4 < 3,
the result being that the C-R approach has slTl.aller centering constants
4
(and is thus in a sense "prefcrab1ell .but lIinapp~o?tiat~") if, EX >3, giving (Za).
Lemrra 3.
ci
(7a)
(7b)
St~pose
X1 ?XZ' .•• are nOrITally distributed with variance
and let n = d/o be fixed. As a + 0,
5
where
2 2
Note that (Jcr!0ST,t1"" 1 as n"" 0 as is predicted by Lemmas 1 and 2?
2 2
but that (Jcp!(JS-iJ"" 0 as n"" 00. In Table 1? the values of this ratio
are presented for various values of n. It appears that for 0 <
2
2
(J~if < (JCR
while the opposite is true if 2.34 < n < 00.
n < 2.33?
TABLE 1
2 ,.,
Values of the quantity (JCR/(J~.
n
0.00
0.05
0.25
0.50
1.00
1.50
2.00
2.25
2.33
.. 2.50
3.00
3.50
4.
2 2
(Jm/as:!]
1.00
1.00
1.01
1.04
1.15
1. 25
1.19
1.06
1.00
.87
.45
.16
Sample f'1edian.
Serfling and 11ackerly [8] also provide a rule M(a?d)
the median.
C'J€ertsema [5] has defined an analogue N*(a?d)
based on
to the C-R
approach? while S-W provide a modification of this? call it N(a,d).
The results of this section are not as satisfactory technically as those
of the previous one? but they do shed insight into the behavior of the
three rules.
6
Letting "n.i
Y
denote the ith order statistic in a sample of size
n, define
t.."
bn = max{l, [n/2 - cn 2 /Z]}, an
•
1
Zn(c) = n"2(~,a - ~,b )/c
n
n
Z
N(a,d) = first tnne n that Zn (c)/n
= n-bn+l
Z Z
4d /Aa .
N*(a,d) = first time n that ZZ(A
)/n ~ 4d Z/AaZ •
n a
Define M(a,d)
as in [8].
~ = -~
If Tn
~
is the sample median, set
2
109(4(F(d) - F (d)))
~ = -~
109(4(Fn (Tn+d) -
F~(Tn+d))).
LelTil!la 4. Asstmle that Zn is unifonnly continuous in probability (see
[1]). For the S-W adaptation of Geertsema's rule, as a,d
For Geertserna's rule as d
Lemma 5.
B(F,d)
+
+
0,
0,
Define
-2
t
= l - 2F (d)) 2 (ZF(d) (I-F(d))) {F(d) (l-F(d))-f(d) (l-F(d))/f(O) "
+(f(d)/Zf(O))Z}.
Then, as a
(9)
+
0,
7
Lemma 6.
If
(j~R denotes the asymptotic variance
denotes the asymptotic variance in (9), then as
A few points need to be emphasized.
is tmifonnly continuous in probability.
d
as
-+
0, while (9) holds only as
a
-+
a.
-+
O.
in (3b) and
a,d
-+
07
(j~w
a~R/a~'1
-+
First, we do not know if
~.
Zn
Second, (Bb) holds only as
The difficulty with (8b) is that,
0, the necessary appeal to the representation theorem [2] cannot
be justified.
The difficulty with (9) is that, as
d
-+
0,
n
-+ 00,
the
asymptotic distribution of the process
(1 - 2F(d)) (Yn(t+d) - f(d)Yn(O)/f(O))
(Yn(t)
= n~(Fn(t)-F(t))
is tmlmO\vn.
TIlese are interesting tedmica1
problems and more work is needed.
In surrrnary, LeITIlT'as 4 - 6 make it likely that
more variable and N*(a.,d)
N(a,d)
is infinitely
is less variable than H(a,d), although more
technical work is needed to confinn these impressions.
Note again the
difference in the centering constants.
It is worthwllile to mention that these results extend readily to the
ranking problem ([9]).
ERR AT ASH E E T
page 8
a)
The statement of Proposition 1 should be
Proposition 1.
Unifonnlyas d
-+
09
n
-+
00,
page 9
b) The three lines immediately above Proposition 2 should be deleted.
8
APPENDIX
The proof of Lemma 1 is long and
tedious~
but the basic idea is
as stated in Section 3. lle first begin with a lllJrrlber of propositions.
Let 0, a be the customary "big oh, little oh".
UnifoITIly as d
Proposition 1.
n
-1 tl
tn
(d) - t o
(d) = n.
1
+ 0~
n
+ oo~
(
) - -X + O(n -1 10gzn)
to(d)X.
lIO
n
(X.+d)e~)
1=
Proof:
It is a simple calculation to show that tn(d)
d + 0,
n
+
+
As in Lemma 3. Z of [8] ~ we see that for
00.
(a.s.).
0 (a.s.) as
n sufficiently
large
I (x.-x
+d)eA~{t
1
n
= n-
B*
~l
= n -1 '1
n
Now~
n
1
A*
n
A* = A(l)
On
n
A(l) = n- 1
°h
t
'1
1=
- +d) 2exp{y (d) (X. -x
-+d)
}•
(X. -X
In
1=
A(2)
On
+
(d) (lG-X +d)}
n
o·~
+ A(3)
n
n
In
where
~
n
X. eA~{t (d) (X.-X +d)}
1
0
1 n
(d)(d-Y~) -1 n
L
i=1
=e O n
L
. 1
{X. exp(to(d)X.) - EXexp(to(d)X)}
1
1=
1
+ EXexp(to (d) (X+d)) {exp(-to (d)~) -l} + EXexp (to (d) (X+d)) .
A(Z)
.n
n
= _n- 1 On
x . L1
exp{t (d) (X.-X +d)}
1=
t
= -~ e 0
(d)(d-~)
0
n-
1
1 n
.L
1=1
n
{exp(to(d)Xi ) - Eexp(to(d)X)}
- Xu Eexp{to(d) (X+d)}{exp(-t o (d)Xn)-1} -
Xn
Eexp{to(d) (X+d)}.
9
1
11
= On-\'L
i=l
= de
exn{
t 0 (d) (X-1 -::rn +d)}
.~
to(d)(d-~1)
n
+
-1
n
2
- 1
1=
{exp (t( (d) X-) - EeXf.l (to (d) x) }
)
1
(lJjeJ~{to (d)(X+d)} (exp (-to (d)X".) -1)
=0
Since EXexp(to(d)(X+d)) + dEe}9(to(u)(~C+d))
-}~Eexp{to(d)(X+d-~)}
this
r:;ea.'1S
A* = n
n
1
o
and
1
+ O(dn-~2Clog2n)-'2)
= -1\t
+ dEexp{t (cl)(X+d)}.
(a.s.L
that
-1
n
I
- 1
)..=
(X1 +c1)exp(t o (d)X1-)
o
-
x.. .
," (-~2
1)
+ J dn (loezn) 2
(a.s.)
~
H
ana hence that
Now ~ by carrying out t:le Taylor expansion (10) one nore 1]laoo as in Lerrrna
and going t;1rough the above steps once ElOre ~ cne completes
the proof.
Proaf:
The key to the expansion of to Cd)
lies in the relation
a = B(X+d)exp(to(d)(X+G.)).
10
By using the facts
EX
= EX3 = 0,
EX
Z
= 1,
one will obtain to (d)
by
progressively adding terns to the Taylor eX'pansion of exp(to(d)(X+d)).
moe-d) = l'.lo (to (d))exp(dto (d)L where
Now,
110
(t (d))
= 1 + t 2 (d)EX2/2 + t 04 (d)EX4/24 + o(d4).
00
Thus,
6 =
-~
__4
4)
m0
(-d) = -to(d)d
- (t 2 (d)/2 + t 4 (d)EX'/Z4
- t 04 (d)/8 + o(d)
,
0 0
so that using the expansion for to (d)
Proposition 3.
Uniformly as d
fine-d) - moe-d) = n-
1 n
L
0,
-+
yields the result.
n
-+ <Xl,
{~(Xo) - E~(X)} +
i=l
-1
O(n logzn)
(a.s.)
1
and
m (-d)
n
10.g mn ( -d) -- (dZ/m (-d))n- l L
o
. 1
1=
o
{(X~-1)(1+d2(1-EX4/3))/4 + d(X1o+X31o/3)/Z
1
+ d2(X{-EX4)/24} + o(d4) + O(n- l log 2n)
As
a
-+
0,
log(m (-d)/m (-d)) = nn
0
Proof:
(a.s.).
By
1 n
L1
{~(X1o)-E1HX)}/mo(-d) + O(n
-1
10g2n )
(a.s.).
1=
0
a Taylor expansion_
exp(tn(d)Xi )
= exp(to(d)Xi )
+ (tn(d)-to(d))Xi exp(to(d)X i )
+
(tn(d)-to(d))2x~ exp(Zi(d))/Z,
11
(11)
exp{ -tn (d) (d-~)}Inn (-d)
=
{n- I I
. 1
1=
{eA~(t0 (d)X.)
1
1
- E exp(to(d)X)} + E exp(to(d)X)
n
i~l {Xi exp(to(d)Xi ) - EX exp(tO(d)X)}
+
(tn (d)-t O(d))n-
+
(tn(dJ-to(d) EX exp(tO(d)X)
1
+
O(n- I (10 g 2n))}
n
= n- . L1 {exp(tO(d)X.)
- E exp(tO(d)X)}
1
1=
- d(tn(d)-to(d))E exp(to(d)X)
+
+
E exp(t0 (d)X)
O(n- 11og2n)
(a.s.).
NeVI,
exp{tn(d)(d-~)}
= exp{dto(d)}
{d(tn(d)-toCd))-tn(d)~}exp(dto(d))
+
+
001-11og2n)
(a.s.).
}/fuUtip1ying the two sides of (11) by this last expression yields the
expansion for Inn(-d).
Thus,
(m (-d) -TIl (-d)) exp{ -dto Cd)} .
o
n
= n- 1
n
.r {exp(to(d)Xi )
1=1
+ 0 (n
-1
-
E exp(toCd)X) - toCd)Xi E exp(toCd)X)}
log 2n).
We have
4
4
(d )
L (to (d)X. ) K/K!
+
0
M0
(t ) 0
= E exp
( t (d)X) = 1
0
+
d2/2
exp (t (d) X.)
o
1
=
K=O
1
+
d4 (~ - EX4 /8)
12
This leads to
(~(-d)-mo(-d))exp{-dto(d)}
= d2n- 1
I {(X~-l) (1+.(1-EX /3)d )/4 + d(X.+X~/3)/2 + d2(X~-EX4)/24}
i=l
4
2
1
+ O(n
-1
1
log zn) +
1
1
oed4)
and completes the proof.
If A(a 1 d)
Proposition 4.
~
I
(1JJ(x) - J1JJ(y)dF(y))2dF (XL then
A(a,d) = d4 (EX4-1)/4 + o(d4) as d ~ O.
A(a?d)
= Var(1JJ(X))
= exp(2to (d)d) (r'1o (2to (d)) - r'l~(toCd)) + t~Cd)I'1~(toCd))
+
NO\'J, simple Taylor expansions show that
f1 ( t
0
2/2 + d4 (1-EC4/4)/2 + o(d4)
Cd))
=
1
+
d
o
p~(toCd)) = I
+
2
4
d + d CS-EX4)/4
+
oCd4)
r1o (2to (d)) = 1 + 2d Z + Zd4 + o(d4) ,
which? l'lith a few computations, yields the proposition.
Proposition
5.
.
For all
£
> 0,
Z
(Aa - C-2~na))/A~a ~ 0 as a ~ O.
2dtoCd)Il~(toCd))).
13
Proof:
Aa
It is well known that
= _~-l(a)~
Thus,
A2/(-2~na)
a
-+
1 as ex +,0.
by L'Hospital's rule applied to
A -£
(H(a)) a.
-+
Proof of Lemrna 1.
if
1
£
Since
A;/(-2~na),
> 0, so that
By the nature of the stopping rule,
M(a. ,d)
stops
the first time n that
It is possible, using ted1niques in [6], page 298, to neglect the excess
ten1S so that as a
Since
1.
-+
0, from Proposition 3,
M(a,d)/(-~na/ll) -+ 1
Letting N
(a.s.), this completes the first part of Lemma
= H(a.,d), note that
(1
so that as a,d
-+
-
b. -1 log (TIt+ 1 (-d) /mo (-d) ))-1
0, M(a,d)(-~na/b.)-l
-+
1 (a.s.) if
(12)
Once (12) is shown, the proof will be completed by once again using the
14
technique in (6] together with the expansion of log (llh(-d)/mO(-d))
2
as well as Proposition 4. Now 6. = O(d ) and log (n71 (-d)/m (-d)) =
o
-1
o(M 10g 2r,'D, so that it suffices to show
d-
(13)
2 ,-11
. - -+ 0
11
ogl';
( a. s. ) .
A sufficient condition for (13) to hold is that for some
7,,,-1+£
d - "1,
a: ~ d
Now, as
0,
-+
0
> 0
( a.s .).
1,1 (a, d) ~ - tna , so that
d-7_
-M(a.,d) -1+£
Proof of Lemma 3.
-+
£
~
d-2 (-Qna) -1+£
By Lemma 1, since 6.
= n 2/2
-+
O.
and
(-2tna)/A~
-+
1,
we have
2
= 4A(a,d)/n 6
where
crS1J~
where
cr~ = (EX4 -1)/d 2 = 2/n 2 . In the normal case A(a:,d) =
1 - c- n
2
(-2tna -
- n 2e -n
A~)/Aa:
2
-+
Proof of LeL'Ina 4.
L
m ...., cr[2/2, as
n
-+
.
By Lerrnna 2,
,cornp~.eting the proof because, by Proposition 5,
o.
In the notation of Block and Gastwirth [3] with
2
2
00
Zn -+ E, (a.s.) by [2] and
IS
~ = 1/£(0). Now, N(a,d)/(~/2ka)2
where
Neglecting excess tenns,
N(a,d):::::
1 (a.s.), where
+
ka = d/Aa .
~(a,d/(2ka)2 so that
L
Since m'" cN(a,d)'2/d ,.. ct;/4k ' the ill1ifonn continuity in probability
a
guarantees that as a,d + 0,
GeertseJTl.a I S nile basically replaces
the above, as
d
Proof of Lerrrna s.
+
c
by
It ; by steps similar to
a
0,
By once again neglecting excess over the boundary,
I:!(a,d) - (-,Q,na/L'.) :::::
'"
(-.~na)(L'.L'.r~
-1
'"
(L'.-~.
Then, by expansions of log (l+x) and F(Tn+d) - F(d),
1
'"
n~(L'.-L'.n)
=~
1
(
~ log 1 +
{F (T +d)-F(d)}{l-F (T +d)-F(d)}j
n n
2n n
F(d)-F Cd)
= (1-2F(d)) {2F(d) (l-F(d))}-l r/i(Fn(Tn+d)-F(Tn+d)+Tnf(d))
By the representation theorem [2], if Yn(t)
+ 0(1) (a.s.).
1
= ~(Fn(t)-F(t)),
then
ri2(L'.-~n) = (1-2F(d)) {2F(d) (l-F(d))}-l(yn (Tn+d)-f(d)Yn(O)/f(O)) + 0(1)
Now,
M(a,d)/(-,Q,na/L'.)
process Y
n
+
1 (a.s.); denoting II
= M(a,d),
(a.s.).
since the empirical
is weakly convergent tmder ra."'ldom sample sizes [7] and
16
T~1 +
0 (a.s.)
~
this yields
so that
Proof of Lenna 6.
The ratio of the variances is
O~R
A~/ 2d3(f(O))~
0sw
AaB(F,d)/2~
--r::
Since
F(d):: ~
+
df(D)
+
d2fi(O)
+
d3~i(0)
+ o (d
3) , we have
Also,
(1-2F(d))2 :: 4d2f 2(O)
2
+ o(d )
F(d)(l-F(d)) = ~ - d2f 2(O)
2
+ o(d ).
Two more Taylor expansions yield
fed) (I-F(d))/f(O)
(f(d)/2f(0))2 =
=~
+
d(~~~~~
!<{l + 2d(f' (O)/f(O))
so that
Hence, as
d
+
- f(O)) +
A~
2 I 0SV1
2
oCR
+
~
';<;.
2
d2(~~t~~-
+ d (2fcW +
- 2f' (0))
(~(~~))2)
+
2
o(d )
2
+ a (d )},
17
ACKNOWLEDC.afENf
of
The author wishes to thank Professor Wacker1y for sending preprints
[8J~ [9J.
REFERENCES
[lJ
A.11.scombe~ F.J.~
(2]
Bahadur, R.R':I A note on qua,'1tiles in large samples, Annals of Mathematical Statistics 3?~ (~Jne 1966), 577-80.
Large sample theory of sequential estimation, Proceedings
of the CaJnbridge Philosophical Society, 48, (October 1952), 600-7.
J
[3]
Block, D.A. and Gastwirth, J .L':I On a simple estim.ate of the reciprocal
of the density fmction, Annals of Mathematical Statistics, 39,
(June 1968)~ 1083-85.
[4]
ChOW,
Y.S. and Robbins~ H., On the asymptotic theory of fixed-width
confidence intervals for the mean, Annals of Mathematical Statistics,
36, (April 1965) ~ 457-62.
[5] Geertserna, J.e., Sequential confidence intervals based on rank tests,
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[6] Gut, A':I On the Doments aTld linit distributions of some first passage
tL~es, Annals of Probability, 2
(April 1974), 277-308.
J
[7]
Pyke~
R., The weak convergence of the empirical process with raTldom
sample size~ Proceedings of the Cambridge Philosophical Society,
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[8] Serfling, R.J. and VlackerlY:l D.D., Asymptotic theory of sequential
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