..
On Swanepoel's Elimination Procedures
bv
,
Raymond J. Carroll *
Depa~tment
of Statistias
Unive~sity of North Ca~otina at ChapeZ HiZZ
Institute.·6£ -Statistics Mimeo Series #1074
June~
* This
1976
research was supported by the Air Force Office of Scientific Research
under Grant No. AFOSR-75-2796.
On Swanepoel l s Elimination Procedures
by
Raymond J. Carroll *
University of North Carolina at Chapel Hill
SUMtJiARY
Swanepoel (1976) has developed classes of sequential ranking and
selection procedures which have the elimination property and which asymptotically satisfy certain probability requirements.
e
Under a variety of circum-
stances, this note obtains the asymptotic distributions of his stopping
times.
In the indifference zone formulation ~ it is shown that his procedures
are infinitely more efficient than those appearing in the literature except
possibly in circumstances closely related to the least favorable configuration.
* This
research was suppiJrted by the Air Force Office of Scientific Research
tmder Grant No. AFOSR-75-2796.
2
'1It
Introduction
Swanepoe1 (1976) has developed classes of sequential ranking and
selection procedures (indifference zone and subset selection formulations)
which have the elimination property and which asymptotically satisfy
certain probability requirements.
Under a variety of circumstances, this
note obtains the asymptotic distributions of a class of stopping times
which includes the Swanepoe1 procedures.
We show that his procedures are
infinitely more efficient than those appearing in the literature except
possibly in circumstances where the least favorable configuration obtains.
Elimination rules are of obvious importance in the problem of selecting
the best of k populations because they allow the experimenter the option
of removing obviously inferior populations (compare the non-elimination
~
procedures in Geertsema (1972) and Bechhofer, Kiefer and Sobel (1968)).
Swanepoel 's paper is noteworthy in that it is the first paper to adequately
treat this topic, the most practical one in sequential ranking theory.
However, he does not treat the topic of how many observations the user
can expect to save by using the elimination feature (i. e., the distributions
of the stopping times), and it is to this problem that the current paper is
addressed.
We deal only with the case of selecting the better of two populations
for two reasons.
First, the extension to three or more populations is
straightforward (one of the advantages of studying elimination procedures)
but notationa1ly cumbersome.
Second, the savings available from an early
elimination of the inferior populations when k
and quite striking.
=2
will be' illustrative
This is especially in comparison to the Robbins,
3
In the general problem, we have 1< populations 'IT p ... ,'ITk and
available from 'IT i • One fonns location-scale
equivariant statistics Tin from 'ITi based on XiI' ••. ,Xin (i.e.,
a sample XiI' Xi2 , ...
Tin(~(Xil-e), ... ,~(xin-e))
= ~(Tin-e)
crfn. The assumptions are the existence
equivariant variance estimates
2
of constants IIp •.. ,llk' cr1'
(l.la)
,crk2 such that
-1 n
L y .. + R.m , where Y·1 I , ..• ,Ym· are i.i.d.
1
j=l 1J
zero, variance cri random variables witIl E IXfl/ < 00 .
· - ll·
Tm
mean
) and location invariant, scale
=n
(l.lb)
(l.lc)
(a.s.).
Assume without loss of generality that III S ••• S llk and that it is
desired to select the population with the largest value of lli'
The
indifference zone approach (we consider subset selection in the last
section) assumes llk - llk-l
(Swanepoel chooses r
Let
~ ~
= [1/~],
and defines r
where
( 0]
g be a given nonnegative function.
eliminates population 'ITi
...
_-
+
00
as
~ +
0
is the greatest integer function).
Then Swanepoel Ys procedure
at the nth stage if there is a population
still in contention at the nth stage for l>lhich
~
= r(~)
'IT j
4
e
Sampling continues until only one population is left.
If Wet)
is the
standard Wiener process, and CS denotes a correct selection, Swanepoel
shows
(1.2)
lim tnf Pr{CS}
A+O
~
1 - Pr{W(t)
~
get)
for some t > O}(k-l).
where it is assumed in this paper that
(1. 3a)
Tn
n
= n- l . r1 Y.1
1=
-n , n~R-n
+ R
-+
0 (a.s.), BYI
= 0,
3
BYI2
= cr 2
and EIYll < co.
(1.3b)
In Section 2, we investigate the stopping time
T\T
as
6. -+
0 (r
-+ co),
relate this back to the selection problem in Section 3, and discuss subset
selection in Section 4.
5
'It
Asymptotic Distributions
= w(Xi )
To begin with we will vlrite Yi
on the original sample
~'Xz 1
••••
It
to emphasize the dependence
is not surprising that since
the PT{CS} is related to the Wiener process
N
is also related to that of W.
W~
the distribution of
The proof of Theorem 1 gets its inspira-
tion from work of Lai (1975) and, while self-contained, depends heavily on
the approx:imations of Swanepoel.
Theorem 1 Suppose that g(O) > 0 and that g is a nondecreasing continuous
function for t
~
(2.1z)
t
0 satisfying
-!.:
2g
(t) t
~
(2.lb)
as t
00
+ 0 and as t
t
00.
t- 3/ 2g(t)exp{-g2(t)/32t}dt < 00.
0+
Then~
1
for t > O~ if r~(~+~)
!.:
r2(~-A) +
Co
(0
~
Co
~
cI <
oo~
+
max
+
H(t)
=
+
O.
!.:
If r2(~+~)
00),
O~s~t
Pr{N/r > t}
Pr{N/r>t}
Pr
min
O~s~t
(W(s)-g(s)+sc1/a)
~
0 and
~V(s)+g(s)+scO/a)
~
0
Proof of Theorem 1 Let a > 0 be arbitrarily small.
+
cl '
6
!'lax (Tn~;ag(n/r)crn/n+(ll+Ii))
Pr{N/r>t}
n~[rt]
=
1
min (Tn+r~g(n/r)crn/n+(ll-li))
n~[rtJ
max
O~s~t
=Pr
~
0 and
~
0
tT[TS]-T~g{[Ts]/r)cr[rs]/[TS]+(ll+Ii)) ~
0 and
1
min (T[rs]+T~g([rs]/T)cr[rs]/[TS]+(ll~Ii)) ~ 0
O~s~t
max (Wr(S)~cr[rs]g([rs]/r)/cr+[rs](ll+li)cr-lr-~)
~
min (WT(S)+cr[rs]g([TS]/T)/O+[TS](ll-~)cr~lr-~)
~ 0
a~s~t
Pr
a~s~t
0
and
Since WT => W, O[rs]/cr -+ 1 (a.s.L and g is continuous, the Theorem
is proved-if r ~2 (1l+A) -+~, while if r ~ (1l+1i) -+ c l ' r ~ (ll-A) -+ Co ' the
last probability must converge for fixed a > 0 to
max (W(s)-g(s)+scl/cr) ~ 0 and
H(a,t)
= Pr
a~s~t
min (W(s)+g(s)+sc%) ~ 0
a~s~t
Since
H(~,t)
is continuous in a, lim Pr{N/r>t} :;; H(O,t). Hence, it
suffices to show that as r
limits,
-+
~ and
a
-+
0, if r~(ll±A) have finite
7
We only show that
= Pr{
G(a)
max
(Tn - r~g(n/r)an/n + (~+~)) > O}
lsnsra
as the other case is much easier.
nO such that
~rlthout
pr{a~
< nO
Now, for all
for all n
~
E
+
0,
> 0 there exists
1} > 1 - E.
Taking nO
=1
loss of generality, it thus suffices to show that
G(a)
= Pr{
max
(an-1Tn -
k
l' 2g (n/r)/n
lsnsra
+
(~+~)
)
> O} + O.
Now'
G(a) s Pr{a~ll:~1/J(Xi) - Wen) ~ (r~g(n/r) - n(~+li))/4
+ Pr{a~lnFn ~ (r~g(n/r) - n(11+li))/4
1
+ Pr{W(n) ~ (r~g(n/r) - n(~+li))/2
We can make the substitution
1
n(li+~)"" nc
for some 1 S n S raJ
for some l's n s raJ
for some 1 s n s raJ
~
l /r:2 and note that for 1 s n s ra
1
r~g(n/r) - n(~+li) ... r~(g(n/r) - ncl/r)
~ r~(g(n/r) - acl ).
Since g is increasing and g(O) > 0, we then obtain
(2.2)
for a
sUfficiently small.
Thus we can replace the left side of (2.2)
by its right side in Pz' P3 and P4' The rest of the proof follows
from Swanepoel' s equations (8) - (12) which we will sketch. First note
that from Jain, Jogdeo and Stout (1975)
8
-!.:
+
Si.lJ.ce x 2 g (X)
00
as x
-+
O? we have
Pz ~ pr{(a~lL~~(Xi)-w(n))/~ ~ a-~g(a)/16 for some lsn~ar}
s Pr{(cr-1L~~(Xi)-w(n))/~ ~ a-~g(a)/32
+
-+
for same n~l}
Pr{(cr~1_cr-1)L~~(Xi)/n~ ~ a-~g(a)/32 for some n~l}
(as a
0
P3
-+
0).
~ pr{cr~l~Rn ~ a-~g(a)/16 for some n~l}
-+
0
(as a
-!.:
Finally? since a. :ZV1T(a.t)
0).
-+
is a standard Wiener process and applying Ko1mogorov i s
test (see Ito and rtKean (1965) ~ page 34) we get
!.:
P4 s Pr{W(t)
~
r 2g(t/n)/4 for some O<t<ar}
= Pr{W(t)
~
g(t)/4 for some O<t<a}
-+
0
(as a
0).
-+
Corollary 1 Suppose that Theorem 1 holds and in addition, that
g(S)S-l+€
(2.3a)
(2.3b)
g(s) (s logllog sl)
-!.:
2
-+
0 as s
-+
00
-+
00
as s
-+
O.
Here log is the natural logarithm. Then H(t)
cl > 0, H(t)
Co = c1 = 0,
-+
0 as t
-+
00.
for some ~ > € > O.
-+
1 as
If (2.3b) holds when s
t -+
-+
00,
0 and, if
then if
9
lim inf B(t) > O.
t-+oo
The proof of Corollary 1 is a consequence of the Law of the Iterated
Logarithm for the Wiener process (Breiman (1968) ~ page 266). The conclusion
is that 1 - H(t)
is a distribution fwction only if c l > O. This has
important consequences in Swanepoel's procedure (r = l/ll) when
= ll,
1..l
consequences which are discussed in deta1l in Section 3.
While it is clear that the proof of Theorem 1 breaks down (N/r ~ 0)
1
if r~(lJ+ll)
+
00,
if r~ll = 0(1)
it is not clear why this should be so. The key is that
(true in all our applications), then r~(lJ+ll)
that r~(lJ-ll) ~
00
-+
00
implies
and one begins to make correct selections with probability
one and the problem becomes that of a one-sided stopping rule as opposed
to the original two-sided problem.
;,;
LeJImJa 1 Suppose r2(wll)
r~-E(l..l+ll)
+
00,
-+
1.
00.
Then Pr{CS}
+
1.
If for some E > 0,
then M(lJ+ll)/r~ ~ ag(O).
Proof of Lemma 1
Pr{CS}
+
(11 fixed)
It
is clear that N +
This means that Pr{N;tM}
+
00
(a. 5.) implies
0, where
Since Tn + 0 (a.s.), crn + cr (a.s.) and "tt1/r ~ 0 by Theorem 1, an
analysis along the lines of Chow and Robbins (1965) yields
~g(Wr)cr~/M(lJ+ll) 4 1. Since Wr.4 0 and g is continuous, the
proof is complete.
Proof of Lemma 2 (lJ
+
0). By inspection or ,by Swanepoel vs Theorem 3.1,
10
PrUncorrect Selection}
Now 1 for all
~
Pr{-T
n
k
~ l' 2g (n/r) a
In + ('1-"6)
n'"
n~l}.
for some
€ > 0 there exists nO > 0 such that
Pr{a~ ~ nO for all n~l} ~ 1 - €, so to prove Pr{CS}
-+
1
it suffices
to show
-1
Pr{.an Tn
Setting nO
al
-+
= 1,
aZ +
0,
00
~
k
r 2g(n/r)/n + (V- 6)/nO for some
from the· argument ..q-f
Theore~
n~l} +
O.
1 we. see that as'
the above probability is
Fix a p a Z so that the 0(1)
tern: above is arbitrarily slnal1.
Then
the probability becomes
(as
k
since r 2 (lJ.-6,)
consider M.
-+
'Thus,
00.
Pr{CS}
-+
l' -+
(0)
1 and as before it suffices to
It will be helpful to denote the dependence on lJ by
M(lJ).
We first want to shmv that
this l-rould imply that
TM(lJ/ (lJ+fl.)
~
0
10gzM(lJ) I (M(lJ)lJZ)
~ 0;
and hence that
Since M(lJ) 11' ~ 0 1 this argument would then yield (lJ+A)M(lJ)/r~ ~ ag(O).
kp
We have already shown that M(l)/r 2 ~ ag(O) since (l+A)M(l)/r~ ~ ag(O).
Define nl
so that
= (lJ+A)
~-€1/2
, where
€l > 0
is very small.
M(nl)
~
M(l)
11
Since log!M(n1)
~ logzr~
this gives
log zM(nl) P
---""2'-- ~ O.
M(n1)n1
k p
~-EZ/2
n1M(nl)/r 2 ~ ag(O). Now let nZ = (1J+b.)
.
Hence~
Then M(nZ) ~ M(n1)
so that
Pr{2M(nz)n~ ~ ag(O)r~(1J+b.)
One continues the process ~ picking n3
choosing
3/4-E +E /Z
2 1 } ~ 1.
= (1J+b.)
3/4-E 3/Z
and carefully
E1~
EZ' ••..
We note that for technical reasons the approach of Lemma 1 cannot
handle the case
this.
~
+ b.
= r -k2(log2r),
and we can see no way to get around
However, in obtaining the asymptotic distribution of N, only the
result of Lemma 1 is necessary.
1
Recall again that if r::2(il-b.) ~
00, we
are essentially dealing with a one-sided stopping rule, and it was this
that enabled us to show M(1J+b.)/r~ ~ ag(O).
(2.4)
From now on, let us assume
n~(a~-a2) has a limiting distribution F and is tmi£onnly continuous
in probability (Anscombe (1952)).
Theorem 2 Assume Lemma 1 holds and that as
Further assume that 1jJ
Defining
is bounded and that
x~(r~1J) = ~ag(O)/(1J+b.), we have
12
e
!.:
(1l+6)M(1l)/r 2
Proof of Theorem 2 Recall that
it again suffices to consider : M.
+
Since Pr{CS}
ag(O).
Then
(M/r) -~g(M/r)aM - (6+1l)Ivf2
= OVr) -~(g(M/r)-g(o))aM +
Now,
riM ~
~M
so that
00
-
(Mlr) -~g(O) (aM-a)
(r/M)~g(o)(aM-a) ~ (~g(o)a
Now, from (2.4)
- (6+11)M)/M'.l + opel).
and Anscombe (1952),
~(a~-a2) and
1
hence M2(a -a)
M
have limiting nomal distributions.
1
1
n
g(O) (aM-a) {(r/M)~ - (ll+A)r{2/ag(O)} 4
Thus,
o.
Similarly,
which shows that
For the opposite inequality, note that
Then~
since ljJ
.
By assumptlon,
is bounded,
.h 2 2
~'l-(~-aM_l)
= opel),
obtained, completing the proof.
so that the opposite to (2.5) is
+
1,
13
To summarize this section, we have fomd the asymptotic distribution
of N for the two population indifference zone selection problem in the
two cases
k
r2(~2-~1+~) +
k-E
cl > 0 and r 2
(~2-~1+~) +
00.
These turn out
to be mathematically different because the latter case reduces to a oneIf ~(~2-~1+~)
sided problem.
+
0"
partial results have been obtained.
Asymptotic Comparisons
The rule proposed by Geertsema (1972), l,,,hich like that of Swanepoel
is essentially nonparametric but does not possess the elimination feature
and assumes cr 2 = cr 2 = cr2 • reduces for k = 2 to
1
20'
where 8(P*) are constants defined for general k by
I
and if k
= 2,
<llk-l (x
+ (3 (P*) ) d4>(x) = P*,
S(p*)2 ~ -4 log(l-P*)
as P*
+
1. One shows that, indepen-
dent of ~2 - ~l '
Let us first assume that
~
r2(~2-~1) +
dO
(0 < dO ~ 00)
and that, as
is done by Swanepoel, r = l/~. Then Lemma 1 and Theorem 1 show that,
for any E > 0, N/r l +E = ~l+~ ~ O~ leading for fixed p* to the conclusion that the elimination rule is infinitely more efficient, Le.,
N*/N
4
00.
14
4It
In support of this conclusion, Swanepoel has found a function g*(t)
given by
for which as p*
if p*
+
1, N*/N ~ 4 if 112 - 111 = 8, while N*/N ~ co
1 and then
+
(112- 111)/8
(r
= 1/8)
co.
(112-11l)/8~
It is in the case
tion rule
+
+
0 that the comparison of the elimina-
with Geertsema 9 s rule is not clear; this is due to
the fact that 1 - H(t)
is not then a probability distribution function.
One might conjecture as in Lemma 1 that
1
(112-11l+8)N/r~ has a limiting
distribution, but we have been unable to show this in the least favorable
configuration 112 - 111
= 8 = r -1
•
Now suppose, however, that we define r
= ([1/8])2.
Then, since
1
112 - 111 ~ 8 the case r~(112-11l) +0 is impossible and there are only
two important cases
(i)
and (ii) r~(l1Z-lll)
+
!.:
r 2 (l1Z-111)
co.
dO
(1
~
dO < co), i.e., 112 - 111
us that N/r ~ LlZN has a limiting distribution and, as Ll
+
0,
Pr{N*<N} ~ Pr{8~ > f3(P*)O'~}
max
+Pr
2 ~V(s) - g(s) + S(I+do)/2O'~) ~ 0 and
O~s~13 (P*) 0'0·
min
(W(s) + g(s) + s(do-l)/2O'~) ~ 0
. O~s~13 (P*) O'~
As O'~ becomes large, this approaches
1 - Pr{-g(s)
and when one recalls that
~
d08
In the first case, Theorem 1 and Corollary 1 tell
+
~
W(s)
~
g(s)
for all 0
~
s < co},
15
Pr{W(s) s g(s)
= p*~
for alIOs s < oo}
we see that the elimination procedure N would have high probability
of beating the noneliminating procedure N*
configuration.
as before.
even in the least favorable
In case (ii) above~ a simple analysis shows that N*/N ~
Similarly~
for fixed 1.1 and
6.~
as
p*
-+ l~
TN -+ 0
00
(a.s.)
so that
~
r 2g(N/r) -+ (1.1+6.)
N
IZ aO •
suggested above ~ this leads to N/ BCP*)
Using g*
as p*
N*/N
4
-+
I?
00.
N*/N
4
4 as as
p*
-+
a5 (1.I+6.f 2
4
1 and then 1.1/6.
-+
We note in passing that for this choice of
00
l' ~
so that
we obtain
the correct
selection probabilities are asymptotically bounded below by P*.
•
Truncated Subset Selection
~lanepoel
proposes a subset selection rule (see Gupta (1965)) III
cases where at most
l'
observations can be taken.
Here 6. = 0 and
sampli..t'"1g proceeds as in the indifference zone problem with the exception
that at the r th stage sampling is discontinued and all remaining populations
are declared llbestVi •
Ilia inf Pr{CS}
~
Swanepoel shows
1 - Pr{W(t)
~
get)
for some
0 < t < l}(k-l).
r~
Theorem 1 holds for
0 s t s L while Pr{N>r}
= O?
.: and both Lemma
I and Theorem 2 hold as stated.
Acknowledgement The author
thartl~
Professor Swanepoel for sending a
preprint of his paper? and for some interesting discussions.
16
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mow,
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ITO~
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