THE INSTITUTE
o STATISTICSTHE CONSOLIDATED UNIVERSITY
OF NORTH CAROLINA
A SURVEY OF SEQUENTIAL APPROACHES TO THE PROBLEM OF
SELECTING THE BEST OF K POPULATIONS
by
w.u..u.a.m L. Lan.e.
Ve.paJL:tme.n.:t On S.ta.:U6UC6
UY!iVeJr..6J.:t.y On Non:th CevwUna. a.:t Chapel.
Hili
Institute of Statistics Mimeo Sepies # 1079
JuZy, 1976
DEPARTMENT OF STATISTICS
Chapel Hill, North Carolina
A SURVEY OF SEQUEHTIAL .. APPROJl,CHES TO THE PP.OI3LEiJ OF
SElECrH.IG THE &EST OF I< POPULATIm~S
by
William L. Lane1
DepQPtment of Statistics
Universi ty of North Caro Una at Chape Z Hi ZZ
Institute of Statistics Mimeo Series # 1079
~-
,July~
1976
1 Research sponsored by the Air Force Office of Scientific Research under
Contract AFOSR-75-2796.
if
1.
IrrtrCicluc h
2.
".Oll-
O~)
Elininatins: Procedures
/i ,
Concll's ion
f\ SURVEY OF SEQUEi,TIAL APPROf\CHES TO
TilE PROBLH1 OF SELECTIfJG THE BEST OF
1.
K POPULATIOnS
Introduction
'There arise many situations in which a statistician must chose
the 1Ibesti7 one, or the best few, of a number of populations.
1T
1 ? • • • ,'IT Ie denote the
vopulations.
Ie
Let
Different assUMptions will
be made about the 1Jnderlying distributions
F of the different
populations.
For example, a sequential procedure night assume that
il'i
C1 ),
is N(lli'
unlmown.
2
where the param.eters
In general it is asslU"ned
and
IIp ... ,llI(
FEA,
where
C1
2
are
is some specified
.f\.
subset of the space of all distribution f1.U1ctions wi t~ location
parameters
lip i
= 1, •.. ,K.
Denote the ordered II
values by II [11•
:s; ... :S J.I
Typically,
[1:' •
". j
populations with large (or small) values of the paraneters are considered to be t."l-J.e 9;best 9:.
7T[;(] corresponding to
J.I [K]
That is, if
ll[K] > ll[K-l]'
is referred to as the
the population
"best·~
population.
In the case where the populations are nornal, the <ibest 9: population
might then be the
pO~Julation . . v ith
the larsest (or smallest) Dean.
l1e will also see that tIle population with
s~al1est
(or larpest)
variance can be defined as the '·best n population, although this is
not very comr.lon.
Usually it will be assumed tI-lut .8
certain subset
S'
= ell!' .•. , lllC)
pf" the. parar,ete'r~space n.
,%
•
lies in a
The ~ubset
S is
-2-
called the
i1
preference zone!!, and its compleJllent is called the
Hindifference zone.!>
5ists of
In many sittk'ltions, the preference zone con-
eE:S such that
].1 [Ie] ,: ].1 [K-1] ~ 11,
where
111i5
11 > O.
preference zone is used since the experimenter is often only interested in detecting a difference in the
pO~lulations
difference is of a certain wugnitude.
This is in analogy with the
l'!!len the.
power function problem l.'vhere one sets the sample size to detect
differences of a certain size.
where the preference-zone
~11'j.s
fonlulation of the problem,
S is specified, has cone to be known a§
the liindifference zone problcJ!f I~ Note that if
is no unique "best" population.
only interesting if
11 > 0,
fj,
=
0,
then there
TherefoTe, the problem is in a sense
although the specification of this zone
Day be difficult in applications.
TI1ere£ore it is of interest to
examine the asymptotic properties of the procedures to see hm'J they
perform as we let
11
+
o.
lfnen we s:atistica.lly select the best population, we are interestpd ,ill. knowing the probability of a correct selection.
TIlat is, if
"CS"
denotes the eveht of a correct selection, we seek rules such
that
p(es)
~
1 -
a
l:J ~ES,
tlThere
S is some subset of the para-
meter space (possibly the whole parameter space).
j1
(IC] - ]J (IC-l]
:2:
11},
].1[1]
Mlen the
]J
l'~len
S
= {il_dt :
the "least favorable easel? is when
= ••• = ~K-l) = ]J[K] -
~ .
(l'· ,~flr~' in this t~nfiguration, the populations are as
-3-
"close;' together as possible.
to find.
PeeS)
If
or equal to
1 - a
~
Hence the best popula.tion is hardest
1 - a intJlis casc 1 then it is greater t!um
in every case.
In some circw]stances it lnay be
.
desirable to investigate the perfOITlanCe of a selection procedure
as
a
-+-
0,
just as it may be to let
f':, -+-
O.
....
TIlUS the asymptotic
~.
properties of SOT'le of the procedures will be exanined as we let
a
-+-
i
O.
We can carry this concept further by lettinz both
libese~
> 1
-+-
0
I'lore generally ~ one could consider the selection of the
and a-+-O.
r.1
6
populations.
However~
for ease of presentation, we·
will usually just consider the case wl1en m = l~ realizing that
m > 1 can be derived in a
5 iF.ilar
manner.
In general, there are t\,ro methods for selecting the "best"
"non-eliminating procedures'I' and "elir.linating proce-
population:
dures. 11
Non-eliminating procedures sample from each of the
populations
tilat
N = Ilea,
PeeS)
1-
~
f':,)
tilacs.
TIlernnnber N is determined so
at least approximately.
a,
Ie
TIlese types of pro-
cedures and the comparisons between then are the topics of Section 2.
.Eliminating procedm.2Cs discard obviously unwanted populations (for
example, those with 'Ivery small ' : means) as the sm:rpling proceeds.
Elir.linating procedures and their properties are the topics of
Section 3.
A great deal of the pioneering work is due to Bechhofer [2] and
?"
Gupta [10].
Beclli'lofer introduced the Ilindifference Zone Approach"
Jefined eatii~. . In Bec1L~o~r's procedure the nwnber of observations
~
~.
...
.
required from each population is then detennined from the desired
-4-
probability of a correct selection and fYOTIl the least favorable configuration of the parameters.
This is therefore a non-eliminating
procedure.
"
Gupta constructs subsets of the
I~
populations which contain
the best population with a given probability.
For each n
~
1
let
lIn denote a subset of the Ie populations based on n observations
from each population.
In Gupta's
fOnmllation~
the size of the
selected subset TIn is random and could be large.
is known as the Hrandom subset-size" approach.
This approach
It nay be undesirable
In contrast, a nUr.1ber of eliminating procedures we deal with in
Section 3 find the best population with PeeS)
Pr{w
7
h
E
-TI
n
for all n
~
I}
~
~
1 - a,
and satisfy
1 - a.
In Section 4 we give a comparison of the elininating and noneliminating procedures.
The advantages and disadvantages of each
type are explored and comparative asymptotic properties are given.
This essay is by no'means a review of all of the work in
/""
sequential ranking.
The, purpose of this essay is to trace the
~,
developme~~'~ j5equential' ~anl~ing in a broad
map~er.
The concepts
~
-5-
of ranking are presented here (a$ far as possible) in their order
of discovery.
There are? hal-leVer, exceptions to this.
In any
case, it would be inforraative to study the bibliography at the end
of the essay to get an overall view ..of the chronological order of
the concepts mentioned.
J.','
.
"
,'$>
..
-6-
2.
Non-eliminating Procedures
Barr <uid Rizvi [4] consider a Bechhofer type of procedure.
The procedure is generally concerned with ranking the means of a
nonnal population 'with known var:i..:TI1Ces.
Bechhofer designed an
experbn.ent to rank the J11eans based on a rredctenninerl number N
of independent observations fron each population.
The required
mmber of observations is detemined by the indifference zone and
a.
The following theorem is useful as a generalization of this
fOTIi1u1ation of the selection problem.
Theorem 2.1:
Let T1 , ••• , Tv1.. be K inde-
(Barr and Riz"i)
'pendently distributed .random variables with continuous distribution
functions
F(t, ii')'
1
ll. . < II . =>
. 1
J
P(T ·
1
(when the FCt, Ili)
:'S:
t) ;e: P(T;
1-' i
:s:
t )
and let T[i1
Let Il [1]
:'S: . . . :'S: lJ [I~]
denote the ordered
denote the random variable from the
1-'[
I]. 1hen for b
is a non-increasing function of
"
t
satisfy this property, then F is said to be
distribution with parameter
v
for all
J
stochastically increasing.)
values of
satisfying the folloHing: '
i:: 1, ... ,K
<
K,
and a non-
decreasing function of ]l[IC-b+1P··· ,1l[K]·
",.....
TIle approach Barr and Rizvi advocate is to taJce a sample of
,
;t1
..
size r.r from the'",i,\<\;! 1 •.distribution.
Then the statistic T.1
is
-7.
~
computed~
where the distribution of T.
1
is as in Theorem 2.1.
The
procedure then ranks the values of Tp i
"r
= 1__ '(,l.Q''''\~
that the population with the largrcst
T.1
.
is the population with
the largest value of IIp Le. , .. tho
l~estl1
and asserts
population.
The sample
size N taken from each population is detennined by the condition
pees)
~
1 -
when OES.
a
H~
To detennine
where
Barr and Hizvi considered the preference zone
is a function satisfying
tjJ
that peeS)
is a non-increasing function of
a non-decreasing
fUI~ction
up to select the
of llC(]'
tjJ ( P
[J}
Thus, if F is
>
O.
, .•• , llr··r·
1."j
l~\.-
Thus
and
1Ve consider the
peeS) achieves its infimun for
P r',l'
.'..1
is continuous in 11,
Further, it
then for b
<
K
This n
-00
blO\in,
(2.1) can be equated to 1 - a
and solved
is then the number of observations N to be taken
from each population.
f:,
ll\
inf
6ES
for n.
~l.r]
(Note that the theorem is set
= tjJ ( II [lj) ""
can be shown that if FCt; 11)
(2.1)
111eorem 2.1 says
best populations for b <.K.
b
case b = 1 for sirrq)licity.)
eE S when
1/J (11) > 11.
In tJlis case,
Consider the case where
tjJ (11) = J..l + f:"
,,,here
-8-
If the populations are nonnal with mean
2
cr, then (2.1) becomes
(2.2)
inf
=b
P(CS)
8ES
(<1>
J
vn
(t +
("K-1
~
~
]1.
1
and eomllon knovm variance
"h-l d ¢(t) .
{1-T(t)}V
fj,)
cr
-00
is the ClDllulative distribution-function of the stardard normal
,
distribution).
Thus, for
and b
8ES
= 1,
(2.2) gives us
00
(2.3)
P(CS) ;:::
J
v-I
epA
(t +
rn.~) d
ep (t)
cr
-00
l<lith equality if ]1[,...,] = Il..,.,. = ... = Jl rr:_'71 = 1l','l - b.. For
i.t..J
l ....... ~
..... J
2
"define c::: fj, /1/ , where h = h(K,d) is the solution of
fj,
> 0,
00
J
epK-I Ct +h ) d t1J(t)
=1
-
C~
•
-00
Then trom(2. 3) it is easily seen that peCS);::: 1 - ex is satisfied
.' 2
?
for jJ such that cr ::; elL That is, N = lnf{n: n;::: cr"/cL
]~e~hhofer
[1] has tables of
h
The assUJilption that ~cr2
for various
and~~.
a
is knm·m is very restrictive.
Robbins,
Sobel, and Starr [12], hereafter denoted n85, consider a procedure
cr 2
where
tion
'IT
i
is assurr..ed tmknoiAm.
They make the assumption that popula-
2
has distribution Nelli' cr ),
i = 1, .... ,K.
that no fixed· sample size procedure will l,vork for all
/..
is unknown.
The procedure ;-;5:3
it in a fOrTIIUla
.
~ugge~t
is to estimate
arial~g~us to (2.3)~ Forn;:::
2
It is know:n
SEQ
"
cr'"
when cr
2
and use
the estimate of cr
2
~.
.-:-:~.
"
If:~
:''f'"....
~ ,1 :
.
i
.,,}
~f
that is most natural to use is the pooled sample variance
a
2
n
1
= (K(n-l))-
r I
j=l i=l
n
K
_
2
(X .. - XJ,Cn)) ~
lJ
"
x.. denotes the
where
I
an2 -< en.
!
X.] (N)
el
I
;::
popJ
1'!e sequentially sample from'
1.J
n th
.
state we compute a f res h estlmq.te
N is the least odd integer
After salirpiing stops ~ if u
x. eN)
1.
IT. th
1'".ne proceuure
1
th.en t en11.na
. t.es samp 1·lng at..... the
2
a.
t
!
i=.l
1
·
eac11 popu1at1.on,
and at t!le
stage, where
'I
= I x. ·/n.
J
an2 0 f
observation from the
n
ulation and where }C. en)
~l
I
i til
lJ
for all
i
=1
9 ",
,K~
n
~
'Tth
1"l
5 such that
is the smallest
such that
j
then we designate
TI
as the
u
The sampling terminates '\\Tith an odd number of K-
best population.
tUples, of observations in order to simplify the computations of the
probability distribution of N.
{N
= n}
Zen) = (XI(n)""~XK(ll))
is independent of the event
01hich is dependent only on ( n2) and since the selection
Since
~(n) ~
procedure depends only on
P(CS)
(2.4)
then for all
t:,
> O~ Q.dt
we have
In PCCSIN=n)P(N=n)
=
= 01
'i' P (CS)P(N=n)
n
00
1>
r
1,.-
1
J
(t + ::1.'"t,
-----) d <p(t) • P(H=n)
(J
-00
This
proc~ure
.
"
~'~\ft
has
. severalfpleasant 'properties.
_
Let n
*
denote
J
-10-
the smallest integer greater than or equal to
obtained by the knmlJn
Then as
a,s.
~
fJ. -+- OJ (i)
and
(/
case (the case discussed by Bechhofer).
n * -+-
EVn '*
(iv)
(12/c which would be
(ii)
00,
N+
3..5. ~
00
I"
'
.
Slnce
Furtner,
-+- L
N/n* -+- 1
(iii)
N~/
!l (1
-+-
Ii
h aImost
surely, it follows frOm (2.4) tl1at
.
00
lin inf peeS)
(2.5)
~
f
0
K- l (t+h)d
~(t)
::: 1 - a .
-00
The above results also hold without the assumption of normality if.
a < (1 2
<
Another interesting result is dlle to Simons [14].
00.
In
the analogue of" fixe<.1 width confidence intervals assuming normality s
Simons shows that for all
E(N) - n
'ET.1S,
and
fJ. > 0
1;
0 < (J2 < "",
5.
~
the !lcost.of ignorance n of
(J2
is small.
He therefore have
that this sequential procedure is both consistent and efficient and
that the cost of igorance of (12
is small.
to see if it is approxinlate1y true tha.t
It is still inportant
p(eS);:: 1 - a.
RSS developed a recursive scheme for finding
p ((J s l\ ,m) •
D(N):::
of
I)
peN
for all
=
2 m+l)
=
'They find that the expected sar.1ple size
I
T.l P((J) fJ.jm) + 1 and. give a table of the exact values
m~l.
and EN when a:::: .05, J' = 2,3,4,5. TIle values n =
2
l' h
I12(J 2/ fJ. 2 wuc
*
Z were known are a"so
1 ·inc1ud ed
"\,VallI '
..a b e usee1 J..t
"~
'.;--
.
f
(J'
11
in the tables a.11,d '~r~ cOIJIParedwith EN.
"l:
The values of n
range
I.
-11-
from 2 to SO.
n
1:
K
EN
2
20
40
60
80
A portion of this table is given below.
=2
5.02
19.87
40.12
60.24
80.23
v -
.•'. -
.99539
.94236
.94751
.94879
.94921
5.01
20.40
40.53
60.56
80.57
K ::: 4
EN
B
7
.J
.99'763
.9964;3
.94921
.94961
.94974
Ie
5.10
20.73
40.76
60.77
80.77
.99840
5.00
20.62
40.63
60.70
80.70
=5
EH
.94911
.95009
.95012
.95011
As the table shows, the results are very good.
....
S
.99379
.95068
.95066
.95049
.95039 ."
Robbins? Sobel,
and Starr conjectur.ed, without proof 9 that the minirr!Uill value of
8
is attained in the computed range n* = 2, •.. ,80.
It miGht sometimes be desirable to base the selection of the
best population on statistics ,"hich are more robust than sample means.
For one thinl.r 9 in fixed sample procedures an inflated varia'tJ.ce due
P{CS)
to non-robustness will caQse
<
1 -
at
and in sequential
procedures, a large variance typically causes an inDrdinately large
saJ~lple
....
size.
Geertsema. [8] considers this problem [md studies the
aSYi~lt)"cotic properties
these new procedures eyJiibi t.
'DIe procedure is
to take a suitable number N of observations frora each population
and select
as the i1bestll population if u is the smallest inu
teger such 'that T (N) ~ Ti O,j)/.... 'if i = l~
~K, where T (N) is
i
u
an estimator of j.l.1
based
on' 'IT • • only. It is assumed that Ti(n) has
1 ·
TI
0
•
•
Has'Ij;:Dto'tiG~lly
.... a."'nomal distribution as
/ ...
•
~
"
.
.
••
with mean
j.l.
1
-12and variance
(i/n.
(This
the observation Xij
0
2
is not in general the variance of
within each population, as it ':Jas in Robbins,
Sobel, and Starr vs procedure.)
= ~[K]
S = {i€Q:. ~[K-l]
that
First suppose that
0
2
It is further assUJncd, as before,
- ~}
is Imown.
asymptotical1Ynormal implies that
The fact tratT (n)
is
i
asymptotically, inf peeS)
~
1 -
.
a
i
for an
iE:5~ FEA.
'l11en Geertsema proposes letting 1·1
n ~ n
variable) equal the first
being an estimate of
2
such that n ~ h2S~/6.2,
52
n
K populations and no beirrg
0 , 1
based on all
Using this
a positive integer.
chosen.
0
(the stopping
This is thus seen
cedure with o~ replaced by
Nt
the best population is then
to be a generalization of the RSS pro("'2
n'
o
i\fter the following theoreI'l another
example will be given.
We would like to look at the asymptotic:·properties of these
procemlres.
To this end Geertsema gives the following set of assump-
tions.
Asstll'1ptions 2.2:
:I..-
- e.) = 0 z.(n)
Ca) n.!? (T.(n)
111
zi(n)
+ 0(1)
a.s.
as
n-+
oo ,
is a standardized average of i.i.d. random vari-
ables having finite second moment and
0
> 0
is a
constant depending only on F.
2 a. s.
Cb)
lim 52 ::
(c)
The?~t. {~2 NCA). :'4 i O} is uniformly integrable.
n
}it -
..
\\There
.-'?it
0
.. '
as" n -+
00.
,.
-',.:
.. -
-;-
·""f'
~
-1311hen these assumptions hold Geertsema proves the following theorem.
\
Th.eorem 2.2
For all
eES
,.Fld\~
and
(2)
monotonically as
'/ ..,
lin 1:1 2 Nell) := 11-o
a.s.
(3)
lim /:,,2 E
(4)
lin inf PeeS)
(1)
N(lI) +00
t
ao s.
0;
-
'f
ll~(j
;:::
lI+
,
/
--
Netl)
if
.;'"
-a
1
as
tl +
O.
TIlis theorem shows that the rrssUff!ption of non:18.1ity is lUliiecessary
(J.nd widens our choice of estimators of the
use the median as a.T). esti!i1ate of the
That
is~
Let
s~:= ~
let
V
n (K
K~)-l
itl
(Xn~a(n)(i)
1.
ex~le,
let us
instead of using the r.1ean.
T. (n) = median of population
l
For
ll· •
after
7f,'
1
n
observations.
- Xn,b(n) (i))2 where, Xn,l(i)
Xn , 2(i) ~ .ee ~ X,u,n (i) are the order statistics of the i U1 sample,
i := 1 g e.o?K 7 a(n) and ben) are appropriately chosen sequences of
im::egers, a.71d
¢l (K
a
Geerstema shows that \'lith the above
) ::: 1 - a.
assmptions 2.2
definitions
~
..
~
Theoren 2.2 hold with a~
hold~
and thus the results of
?
= 4 r- (O)?
with F' (0) := f(O).
The last non-cl:minating procedure to 1:e
Hackerly [18].
outlineJis due to De
The procedure is based on larB;e devi2.tion theory
instead of the central limit theo-ry of the other non-eliminating
procedures.
The procedure thaJ;, Fad:erly constructs is appropriate
for choosing the best population when
:"
a"
are given as
a.
Fixing
.~.
tl
a
+
09
and asyrn.totic results
• 11
~
0." . fOJ :fLced
and letting
a
+
tl
instead. of as
tl -+
0 for fixed
0 may he'Dore relevant when discri-
~
It
-14laination is desired.if
b.
is .KD.ovm to be of sufficient magnitvde.
In this case the samllest po~si:>le
is desired.
0,
Wackerly presents his theory in a general form., but for
"
simplicity I le.r.ill give his procedure
when it is based on sample means",_
")
In general?
is assUTIled tmknmm and
a""'
tinuous and syrunetric.
Further it is aS5~ed that
where ~={~d6: ]J[K_r":Vfn'-lIL
For each
.......
J
~~
X.. -
t
i
t
1J
X.1 (n)
a > 0
let
.
F(n/2) < 1,
N = N(a, 11)
L"'\..J
denote the smallest integern
(1)
F is assUl'ied to be con-
< -
such trAat
M2
for some
i
= l~ ... ~K
atld some
j
= 1 ~ ... ,n
. and
r.
r
-
,\
".
l'
~
r.~
I
where
!i1
n
esti.'11ates the tmlmown
(-11)
<
constant
1.
The procedure is then
to .sample fro:rr, each popu12tion Il(a, 11)
....
times CL'1d select as the best population
tl~e
population with the
largest mean.
We will
N(L~~ h)
n01l,7
compare this pr0cedure '\o1i til the
be the RSS stopping fille and let
sample size
n
such that
N(a s 11)
be the sf'laHest
an :::'$~P an (1l1:> ... ,]..lK) 2 a.
.'
shows that
nss procedure. Let
.:,)
~
Wackerly
I
-15-
E (1\J(a» ll))
e
+
1
1,·1 (a , ll)
whereas
?
lim
-)-
..
9:'b. fu m (-ll)
- 4
t..
a-)-O
.. ',
0
Therefore?
(2.6)
,
lEI
.l..
a+O
If F is assurned nor£ID1~ then (2.6) does approach 1.
Further,
E (N(a, L\))
(2.7)
lim
1l+0
lim
a-)-O
e
e
In a paper by P.. Carroll [6] it is
limit (2. 7) is 1 as
. N(b.) h)
1
+
E (N(ll, h))
a ~ 0
and then
ShO'IJffi
t: +
O~
that al though the
that
have different limiting distributions as
N(a, b.)
111,lX
(a»
a"1.d
d)
+
O.
·\
-16-
3.
Eliminating Procedures.
None of the procedures mentioned in the last section had the
capability to discontinue sampling from a particular inferior populationbefore sa.'TIp1ing terminates.
An elimination procedure, some-
~
'
tITieS called a V1fully sequentia.l procedure:', will often make a
substantial savings in the total m.lT'lber of observations taken.
.
Of some interest to the area of eliminating procedures is the
concept of a i:selection sequence. II
For each n
~
1
let
II
n
denote
a subset of the K populations based on n observations from each
population.
(TIn was also defined this 1,.ray in the introduction of
this paper.)
We Hill now give a formal definition of a selection
sequence.
TI
Definition:
P(IT
is a seZection sequence if
n
K
EO
TIn for all
where
for all
Ct
n;;::
1)
;;::
1 - a
is [T,lven.
The first procedure that 1-JiII be discussed is due to Hoel [11].
It is similar to an earlier procedure due to Paulson~ both assuming
c;2
is knmvn.
In Hoel' s procedure 7 the selection of the best pop-
ulation depends upon the observations through a sequence of
statistics
t 1)
..
en) .< n=1,2, ... ,
~ilich
are defined to be fun.ctions
of the first . n observations fron populations
is,
V
;'v ·.v
X)
f ( A'I""'"'-'
t 1)
.. ()
n::::
..
".
n 1
. .\fln ,-A)'l"""')'n
it·:l\\
,
.
..
11'.
1
and
1I'j'
That
.
,
,
.-1
.,
[
-17-
where in a given problem the function
+
-'-n
is chosen to indicate
differences between the populations in a reasonable ""Jay.
(For
example, if the populations are nornal and we wai1.t to select the
population with the largest
mea~,
a sensible choice of f
n
might be
n
L
K=l
Now let
1
t
I
and
aSSUTile
that
(T .. (n))
ii,".
1)
1J
T.. en)
1)
has a joint probability density ftmction
depending on the pa.rameter
8;'"
T· .•
1J
Further, let
(3.1)
e·
,
Then Hoel proposes that we call the best population the population.
1T •
1
for which
t
I
(3.2)
(
If more tlXUl one population satisfies (3.2) ,then we arbitrarily
I
one as the best population.
and
T
= 11 [lq
-
In the nonnal neans probler:
T
ij
pic~
= lli-llj
1111.is. the best population is the population
II [K-1]
\-lith the largest mean, which is 'Hhat He desire.
'~,-
Hoel uses the indifferencC? zone approach, therefore requiring
that
p(eS)·~
1 - a
whenever
i ?'"'~
T
*
Hoel further 3.ssumes that
there exists a monotonically decreasing fllllCtion 11 such that
tot-' )\)
....
-13-
(3.3)
1" ••
J1
= h(T 1J
.. )
(In the normal means example h(t)
Define
= -t
and assume
than or equa.l to
'r.
satisfies (3.3).)
is a fixed constant greater
Also define the log-likelihood. ratio as
It is now possible to define Hoel's sequential selection procedure.
At stage one take a single observation from each of the K
populations and calculate
If for some
i
l!..l(T.; (1)) ~ log {(K-l)/cd= a
1.1
we eliminate
'lT
from population
as the Hbest!?
j
from consideration.
'IT"
1
0)
(That is, lJe stop sampling
If only one population remains we declare it
Otherwise we go to the next stage and take a single
observation from each remaining population.
He then use the sane
elimination rule again,using i z(T ij (2)) instead of il(Tij(l)).
He continue in this manner until there is only one population left,
and then we call this pOplllation the best.
the populations are
eliminated~ 'the
/"
n for which min l (To. o' (n))
j11~
population.
It
0'.:
?-
1J
i~Oeasy.·to
If at some stage all of
population Tr.1
present
at stage
.
is maximum is declared the best
.
4#
show that this procedure satisfies the
o.~
-19-
error requirenent provided that the procedure tenninates with
probability one.
Note
procedure~
t..~at
changing the value 1'1 clearly changes the selection
Hoel remarks that
i.l'1
(TO~
truncated when
PeeS) - (I-a)
(i.e.
*
O = h(T ).
all the applications he studied., the procedure is'''-
where the only
'TIlliS, the restriction
mends that we choose
restri~tion
*
1')
~~
::
. case,
or 1'1
T
' 11
..
constant.
= 0
= T*
1'1
> T.
and there was less over 'protection
i
is smaller)
l' ,'1
'1
~
if 1'1
!~
1
~
l'
= (t*
+
,)/2,
l'
~
1'*, where 1'*
seems practical.
:"
which indicate that this choice of
1'1
is that
and
S110WS
The calculation of
Hoel reCOlTl-
simulated results
is superior to either
in the normal means case.
an.d he T·11.) = 0 =
= heT 11
.. ).
T*
and
and
i:
T
In thenomal means
is a specified
then is a
In general the choice of h might be more difficult.
sin~le,matter.
TIllis, this
procedure is not as general or easy as it first appears.
For the norma.l means case, it can be sho\-li'Il that Hoel' s procedure
reduces to the following:
Eliminate
1f.
J
if
;:,
:;:'-
',:
":":.-
.'
""'f'"
~i
-20-
rt
.;f
,,i
sampling is terminated and the popu.laticn with the largest mean is
chosen as the best.
It should be noted that this procedure lends itself to the
problem of choosing .the nonnal pcJpulation with the srlallest
variance.-~
(It is also possible to fOTIl1Ulate non-eli.rninating precedures in such
.
a manner.)
o [1]
That iS 9 if the ranked
0'2
are denoted by
222
~ 0 [2] ~. .. ~ 0 [K] ,cb.ooSG as the best population the one
with the srrallest variance
0'[1]2.
TIle measure of distance beb''leen 7f [i] and 7f r ~, that Hoel
2
2
i ~ j. Thus ~ the error requirement
considers is o[iJ 10[j]
becomes
~
P(CS)
n = 2,3}...
m
1n
L
:::
&l1d (iii)
2
10 [1]
2
~
*:
T.
Hoel defines for
X. ·/n
j=l
1J
S~1n = j I=1
(ii)
0[2]
the following:
X.
(i)
if
1 - a
T.. (n)
1)
When populations
tilen to elirilinate
2
(X .. - X. ) len-I)
1)
2 2
= S.1n
IS.In
and
7f.
1
1
log
7f.
]
are compared,
fID~I's proce&~Te
is
if '"
7f.
liTo
en-I)
1m
{ 1/T,
~
+
1/T..
(n)
J)
a .
+ liT .. (n)
1J
.
A series of I'iIonte-Carlo trialsArere perfonned by Hoel to compare
tilis procedure to a fixed sa.'Tlple size procedure cue to Bechhofer
-t
~•
..,
. II
l2]llfhich we i-ITill n~2,!() ..:lrt9 here.
t>
$
-21-
= 3,
= - 5,
aad with the parameters in the least favorable confit"Uration
2
=
V[2]
0
'1
2 ) ",oe_
I'
1
= , *(1)]
2
V[3]
= 1.5( - (,*
+ ,
*)/2)
Above,
i!
•
0
' " ,
E(5)
E(S')
. 75
.90
.95
.a60
.930
.962
35.9
52.3
63.J
13.3
1e.7
19.6
.99
. 996
"'Jj....,J
22.7
32.6
56.5
(
23.3
--
.. , ...
.).) • (,
,J
sa~l[lles <ilJG.
are the number of
S
sample case.
j·:Otice t)1at
E(5)
for
S'
are vres€nt.
E(5)
>
E (S')
for
1 - ex = .90, .95, .99.
the mnnber of stages
1 -
=
(X
Thus ,,'ith
.75
but that
in the usual
a
Further, netice that
TI1is is due to the fact that no real "bo;d" populations
If
2
~Iere
0[,"
;.1 J
\,ould be closer to
2E(S)
l2.rge, one HOtlld expect. tlmt
tha~
3E(S).
ta
The second eliIn.inating pTGccdure to be
Swa.."10poel ami Vent.er [17].
or Hoel' 5 procedure
Jistribution function
consi~erec.l
is due to
Unlik::: either the elir,1inating
procedure~
it is anon -par2:Eetric proceduP3: based. on ranks.
It is not assumed th:tt
,I
•
is the nunber of sta;;es for the fi;,ec
r<l:1ge, the elird.nating procedure is better.
E(S) '" ::;qi·]).
.75, . 90, .95,
PV% tne ":Ol101&.g results for
::(N)
ana
[(5')
(X
,
of Hoel's procedure, and
<
1 -
,CJ (roC)
vv
1 - ex
I
and
=
.99,
Le.
'1
1. 5, 2,
i(
(
'0
= 1,
Hith
0
t'
2
is knoHl1 as with Hoel's procedure.
of the populations is ass;.:med continuous
but llllknown with locatioil paTarlet·c~s ~i~ i ::;: l~ ...
square integrable density
£~
:-l(
?.nd "lith
1his procedure is thus potentially
.
~
•
,
• r.
The
,e,
-22l:lOre generally applicable tl1aIl Ibel' s nnd- :jtiJ.l
t~_le
advantas:c
of eliminating inferior populations.
As before, let
equal the nth observation from popula-
X.
:m
..
Suppose
and suppose all observations are independent.
tion
7T
tfLOO
that III
i
~ ll,n
~ llZ ~
Define
~,
-,."
I
- X.
Ja
Y..
1Ja
(i)
, l.)..
(ii)
=
and (iii) c (u)
1
II [i]
{
=
-
ll[j]
0
ti < 0
1
u
:::
-
ll·
1
p.
J
?: 0
Swanepoel and Venter then defined the following rank statistic~
first discussed in a paper by Sen and Ghosh [12]:
n
R.. '
1JUll -
if
'l1ms,
IY ..
1JY
R..
(•.. )
IJlLrL
1)
1J
= 13=1
I
- , 1)
.. I
Iy··
1JY
and if
-n
(, .. )
c(IY ..
-
1]U
,··1
1)
is maxiInized when
IY ..
- , ..
1) •
y = u~
then
R..
1)(3
is minimized when y = u,
- Tijl
1)
1JW.Y
(T .. )= n,
1J
then
...
= l. Otherwise R is sonewhere in between 1 and n.
Finally, "define
zl.'-in
r
(T .. "') ::: n-
o~
0
u : ; I}
<
n
\'
u~l
1J
where {In(u):
{J(u):
1
u
<
I}
c(Y..
1JU
- T· .)3 (R.. '
1J
n 1Jl
(T- .)/(n+l))
- 1)
,
is generated by a score function
in the fo116wing'way:
~
-~
'i,
'1''''''
I
~'Ii"
I'
fl"
-23-
= J(i/(n+l)),
In(u)
(i-l)/n < u ~ i/n,
It is further asstwled that
IjJ(X)
i
= 1, •.. ,n
= lp-l((1+u)/2),
J(u)
is an absolutely continuous
0 ~ u < 1 where
defined on
df
.
(-00, co)
satisfying
.
(b)
Let
=1
~(X)
Ca) *(-X) +
~i(X)/[l
I
- y(X)]
is non-decreasing for X ~ Xo
~
o.
{c } be a sequence of positive real numbers depending on
n
the sibrnificancc level
peeS) ~ 1 -
to find
Ct.
CiS
n
These
cn is must be chosen so that
Swanepoel andV0:lfl:.er use a series of inequalities
which' will
1 - peeS)
(i)
Ct.
~
~
o;r
1 1'.
work~
the first being (Bonferroni)
perro1- eliminates
lItis inequality is fair for snaIl K, but as
is much over correction.
Cii)
2K P(1T 1
o
Secondly, since
eliminates
0
K gets large, there
...
~
o
1
~ PC (Z. fr (_T;o) -~2'-- )?'C for some n~l)
v
LiIl
n n
J
1 A
1= \.
~ ';:::Y
Lr eeZ.L",J.
~'. . (T 1]
.. ) -~ J ) ~c . for some n~l)
n
n
£.
.1.
... l.,.
then S"\'\lt:!Jlepoel and Verlt.er state that l>ihen
(iii)
thenP (CS)
~
1 -'~Ct.~ Next ~'!aTI:ovtfS inequality is used, thC:1 an
fJI
)~
•
-24-
inequality using an approxims.tion of cosh and then three more inequalities, one to
siJ'~plify
the ap?Toxirmtion of cosh, the next to
ll~rkovi
minimize the e)q,Jression frOI:l
thcnseliTcs,
to derive the
s inequality, and the final one
The
arrived at are
(3.4)
where
n
-1
n
I J~
(i/(n+l)).
i=1
cn 's and,
One wonders hmf much over correction there is using these
at least for
X large, it
rl1..!~)t
be considerable.
could be fOlma which were
cn 's
!-rreatly'iluprove the procedure if
It would no doubt
not dependent on so many inequalities, or at least on better
bequalities,
It is possible to show thelt
Z.. (T .. )
lJn
1J
1
and syilf;1etric about -2
irldependent of FEl (X)
J
n
_1..n
I
h..n.s a distribution
-
J n "ihere
. _
J (l/n+l).
i=1 n
= n'
For the ranks procedure, at stage one ~ tal:e a single observation
from each of the
,,'c
Z"l
(-T
1.1
h
lIinere
K populations and
]1 ;
'~, -
i. *-"'J
'* - 2-1 -=J~?-. c 1 '
Zijl (-T)
deration.
cOill1~lte
1:
j.l p' _ 1
~
_"l..
1"'?-
•.1.:.:.;.
T
> O.
then ,-e?}1TIrnate
TI·
l.
If nopopulatiop rCLlains, then the
the
V(Y -1
..L..
If for
_
~
"""
~)
50)11.C
statistics
].
from further consiTI •
].
for which
-25l:~~
'*
ZiJ'1 (-L)
is naxinized. is declared "'best;>, a':j,:.:1 if only one
Jrl
population remains it is declared best.
At stap;e
observation is taken frcPl each of: the rcrii2ining
a sinele
l:.
popi_~lations
and the
statistics
2.re conputed fo:c all of the rC::laining pairs.
E for SOli1e reuEini.ng
1[ ~
-~
,
.;;J'
:. n
?: C '19
1
is eliYtdnated and. the procedure stops or proceeds in +'
. . ne
s,lnf~
n;1.nner as after stage 1.
'::lis proced.ure is based on the follo\,.rinp: t'JO facts:
o: ..
CJ
f'),
..)
(i)
(ii)
T ';r ::'j.l [';] -
1.:.!;.,
..!..
Z.. (1' .. )
l.Jrt 1J
T..
1J
j.l
r':-]
J..-~'"
is non-increasin:=: in.
i
for all
and .j =>
':f"~
Z.-.o'
)
L·n (T.In') ;:: Zl.·~".Tl(-T
__
thilt
..
-,
L· v
(
LIl
SY:ili:ictric about
means that liJe
there is a sr-Jall ?Tobability
b..
l
('~
4)
J •..
give us
not an approxima"ce, bLit
ofe1ir:dnati:ni'~
- a.
illL
can adjust
f;,S
' 1
[ i~J
)
lJ
so that
en s
C
is
n
1 .f= " . ,
!s ne.<-J.nel...
lvith Eoe1 1 s p::.C'cedure ~ this is
1"'~
exa.ct?"T8sult
but possibly conservative; lower bound.
, 11
1[.
ttlEO
\.L··
0
the SEmse of heinz a tn.IG?
-26-
Swanepoel and Ventel" then define a. procedure br:.sed on
means.
[Ix.1)·1
The procedure as Surles th..'it
<
mK~
00
I
for all
i.
X.In -
Jr.
In
1J
n
:::
)1
1
L x.:.m
and Z.. (T .. )
lJn
Ie"'l
rihe procedure is the
T· •
except that J
X.In
(~efine
TIlen
....'1 ...
ij )
E(X
n
S<ll'1C
as the
is replaced by 0 and the c
n
9
sa~ple
:::
~.
1
Ii
:::
1.)
ranks procedure
are defined
S
differently.
T~
T1 be the total nUTilber of observations required for
the ranks and means procedures respectively . .I'm. asymptotic relative:'
Let
r!"t
,
efficiency of the ranks to the means procedure (using the least
('
. 1e parameter con:ngurat1.on
C"
).
J:avorao
J.S
ill
llhere
lim
-!=
0\+0
ET
SF J(X)
2
(J
-2 SF J.. (2~)
tJ.
~
is a knmvn function of X.
If we let
tJ. + 0 9
then
~
the expression reduces to the Pitman efficiency.
J(u) :::
to
th~
'1'-\ (1+u)/2),
then the A.R.E. of the ran1~ procedure compared
means procedure is
equality only if
F
Thus, if
o
is
~
for all distributions
1
Fe' with
Thus .. at least in this sense, the
llOTITial.
ranks procedure is better than the me,,,n.s procedure.
The next type of eliminating procedure to be studied is due to
Swanepoel llild Gcertsena
[15]~
[16].
T11is procedure is designed to
select the nonnal population lJith the largest mean among
/
ulations.
Various assunptions
pOlJulations ~ includ~n~. tl;e case
necessarily equal.
J
.
~
..•
·•.
~
I.
.~
ar~
....
K pop-
naqe about the variances of the
whe!e
they a.re unY.nmm and not
111e possibility of unequal vaTi81lces is something
"" I'
-2'7-
none of the other procedures allow.
1;U1cn the variances are different t1:cre is some question about
\'ihat should be meant by the Fbest' pOV.llation.
Swanepoel a.'1.d
Geertsema contend that in practice the Variances lonll probably not
~vhich
differ significantly, in
case the usual definition of i1Jest n
population (largest or smallest mean) should still be realistic.
This is probably true in many cases ~ but one still wond.ers whether
tilis matter
C~l
be brushed off so lightly.
If one has reason to
believe that the variances do differ significantly, it might he proper to change the definition of ,Cbest\: to one which might fit the
situation (for example to eliminate those t",ith large variances as
,>-•.••
bad~
.and then select the population with the largest mean from this
reduced set of populations ) .
Swanepoel and Geertsema's procedure is built upon the results of
nobbins (1970) in cormection with confidence intervals.
procedure defined by Swanepoel and Geertsema is as
l~
= N(a,
6)
be the first integer n
Purther
has been eli.minated.
tions such that
•
.
1.
h(t~n)
= Letn )n
= (1
+ a )2 j2
t
'
...
1-,.~
e'r
2
fol1o~AIS.
Let
such that all but one population
~rI l'
1.
.••
1 'IT
K be
= 1,2," .. ~K.
_I(
.'1-0""""".'
" 1 pOlmla~l=_
AsSUlne, without
and that
f.ll
1 ' ... 10K are all unkl1()l.vll.
(i)
.
is
'IT.
loss of generality, that
2
2
(J
-
Tile sequential
Let
-1
where
satisfies the following
eq~qtion:
-23-
iJcte that
h(t 9n) t ""
hCt~n) -+
as
a
0 as
for fixed
+ ""
t
and tx-',at
for fixed n.
0
+
n
In addition let:
11.
I
(ii)
[md
i = 1, •. o))I(
V
J\. • • ,
i=l 1J
=
(iii)
W..1lere
[n
J· f ·1
population
n
2 k
i{(X' Q -Xia ) - (XJ.Cn)-Xi(n))} ]2,
6=1
Jp
I-'
-1 '\'
an..dX . . is the j th observation from
1J
~i'
i
= 1, ... ,K.
Stage Olle of the procedure is to take one observation frrnn
each population. Ear n;,?: 2,
stage n
iS~.to
take one observation
from each population not eliminated by the n_l st stage, Le.,' the
populations in
after stage n
(3.5)
IT
nif
l'
TIle stopping nile is as follmvs:
'IT •
J
X. (n) - h(t,n) H(i,j ,n) + ll,
Y. (n) ;,?: max
J
retain
1
iE~ n- 1
where i f j,
and where
II >
O.
,.>I
Swanepoel and Geerstema prove that their procedure is a selection
Furthermore, if J.IK
limEN/(-41og a)
a+O
=
> )Jv_~'
,'. J..
then
Y[K-l] ,
....:.,
~ Y[l~-l]
,
where y [1] .~ Y[2] ::; .
H
.'
arranged in increasing order, with
are the values of Yl"" 3YK-l
-29-
Finally, \ihen
it can be
shm~1!1
that
P, ('J. T.. <
00"' ::: 1
)
.~ 0
i';ext S"Janepoel and Gcerstema sho\'1 that it is possible to conE TIn for all n~l) ~
K
111e stopping role defined
struct selection sec;.uences which·satisfy P(1T
l&
,.
1 - a.
for aU p,wameter configurations.
in (3.5) with
IJ?
/). > 0 reMains a selection sequence Nith f1:::
is the stopping variable obtained by letting 11::: 0,
o.
If,
it is
shown that
(3.6)
(a)
N' <
and
(b)
peN' < (0)
Thus for Swanepoel
assume
fj,
>
00
if
a.s.
>
~l
< ...
<'~K-I ~ ~K
0 for other con.figurations.
lli1d Geerts~la9s
procedure, it is unnecessarj to
o.
As a final note of comparison, Swanepoel and Geerstema remark
that their procedure and that of Swanepoel and VentoI' are asymptot~cally
equivalent.
Therefore;; if it is kn01J.;'.that the popuiations
are nomal ~ then S\Aranepoel and Geerstenw. 1 s procedure is both cas ier
. .:t
to use and IIlore general in "b.1.e sense that it is not necessary to
assume
th~t
Further~
the
vari~~ces
of the different populations are equal.
Swanepoel and Geerstena's procedure is much more general
than hoel is;; since Hoel D.5sumed the variances known and equal.
the distribution of the popll.}ation
C2.rmot ":'2;.
assunec.
If
D()Tlr.a::',
. _..
_-_ ..,_._,----
-30-
V(~§"!'f~'S. procedure and if (J2 is Imov,rn, Hoel' 5 procedure may be used.
For Swanepoel and Geertsemais proced.ure with K = 2~
")
2 1 \'
_.'\ 2
11,2 ( J
,lr n = an = ::- dX-.l'. - X ; - (Xl Cn) -X) (n))J . Suppose there was a
Z
11
1
1
• .L
'"'
Then a nZ »(52 and it will be
n
2
a , Since the stopping TIlle for Ie = 2
outlying observation at time n r
,)
a long time before
•
\J
is to stop when
N.
an outlying ob?ervation iI.nplies a. large
Therefore Swanepoel
and GeertsenmVs procedure lacks robustness,
The final paper I will
discl~ss
is by R.J. Carroll [7] and deals
with ,a: robust generalization of Swanepoel and Geertsema's procedure.
Asymptotic distribution and expected sample size results are also
given.
From the sami)le XiI" "
statistics
"'1 'n
l
~Xin
V)
= "'(V
1 A'l~···,A'
1..'
In
?
taken from population
i
:=
1,
It
0
•
,1(.
generalization of the RSS procedure, Section 2.)
'ITp
£0,,'111
(See Geertsema's
It is assumed that
the }.t,1 are the lind. ting r,leallS of popUlations
't.
The sequential
'IT.
1
selcctionpro~edure Carroll
suggcsts is to eliminate
the first time n that
... "!
~.
(3.7)
In (3.7),
T. - T.
In
In
~ h(a~
n) an (i?j) n
- t:..
a~ (i~j) is an est:i.rlate of tf.te variance of
kl'
n l (T.]n- Tin - ()lj - }.t,)) and
1. .. Al.:.
111e constant
f:"
{ h(a'?n)} 'is a sequence of constants.
.
is deternl1ned by
'",'"
\\;.
••
ltJhere again it
-31-
is assumed without loss of generality tllat J.ll
s;
J..lZ ::; ...
J.lK
is
if in (3.7)
the correct urumown ordering of the
above He let T.
HI
s;
1
_ 1]-'2
= X.1 (n)
then vie
get Swanepoel and Geertsema 1 5 proodure.
One main coneenl of the pa}}3T is cOTITfJuting the
a?ymptotic
distributions and expected value of the stopping rule.
s i!;Iplicity 1 consicier the case of two populations.
For
To conpute the
asymtotic distribution of the nu;rrber of observations from any
l)opulation as
a -r 0 ~ it then suffices to consider
(3.8)
= inf{n:
H (d)
a
- d}
1_
-.}
n,,--IT
1.vhe1'e
:::
11
(ii)
11~ )
d - A + (J.lz
30.
and
It is assuraed that
o - a 2)) are jointly asymptotically
(n;i ,-,in'. n~(2
n
nonnal and are both lmifonaly continuous in probability (this concept 9
.
~
'f
and a 2 - 0 2 can .De
n
n
expressed as averages of i.i.d. randoTIl variables plus a small
due t.o Anscombe [1], basically means that
remainder term).
It is also
aSSl.Uih?d
that
Using these assunp'cions the £0119win£. theoren is proven.
'Ii
.
¥
'(
.
'.
i
a + 0 that
Suppose as
Theorem 3.1:
.(
a.s.
(a)
h(a, H (0))
a
(b)
h(a, N (d)) - h(a, l-l (d) - 1)
a
a
+
00
->
a.s.
0
l-
(c)
,
'TIwn, as
Cl
-d.\Vd)!;;-Hi~(cl)(~'i
3..'1d there is a cons tant
•
(oh(a,
(3.10)
-
[-j
a
Ad (F)
Z
(d) -d(o;j (d)"i)/20 )
C1
__
woulti ;.lal,e
Ad (F)
0
:for which
(d)) - d 1/' (,i))L H(O, A_j(F))
a
C
,. .:nown +1
a2
were
L -len
i
If
independent of
d.
<l n!i(0;_oZ)/ zo 2
;/i Tn
It is interesting to note t~1at estiDating
in (3.10).
L
C1
is theasymptotic varia:1ce of
where Ad (F)
o
2
On -
2
by
?
0 .... =
2
on affects
O:s-
1'Ihich
'fhis me,ms that estimating
theasynptoti.c distribution of H (d)
by a statistic an cham~es
......
0.
from the case
an
= o.
TIle [oHm.ling choice of
11 (ex, n)
.~
A =
a
¢
-1 (I-a). b-.fA
a
'.
•
is suggested
h(a, n)
(3.11)
IIDere
....
+ 0
(oh(a ,N (d))
a
(3.9)
i-! (el)/n (d) :...,. 1
a
a
for which
na (d)
There exist constants
,-
l"
"
.~
+ 1
a
ami
.•
.
C'2:
O.
Carroll gives the
-
,
.. _;~~IJi~-~~~~~
-33Following corollary to Theorem 3.1.
Let h(a, n)
Corollary 3.1
nCL (d)
= (A
c:
0'/<.1)2.
be defined as in (3.11) and define
Then as 0.-+-0,
(N (d) - n (d))/(2J!i (d)/d)
(3.12)
a
a
CL
-l
N(O, Ad(F)).
.
further ~ if d~
2
a-+-O in such a "(,\ray that (log d)/A a,
h-
of nl? T.
n
0,
then (3.12)
asymptot~c~1r.{",ri,~nce
replaced by the .
still holds, with Ad (F)
-+-
".'.
•
• d
.'.....",\.......;
•• '
."D(F)
I.
.
TIle proof of Corollary 1 is infomative md makes use of
the standard arguments in the literature.
of the case
=C
c
POT
simplicity, the proof
c~)
The gen.eral case
is given.
follows in a
similar way.
Proof:
Fro~
(3.2) one has
(N
~ (d)
-1fT.
.
~r~a (d) -1)
This inequality is a standard and flh"1damental one, which l , as t-Jill
be seen below, is often used to derive llconsistencyl? of the stopping
rule.
T'nus, since T a~. 0, "-we have (even if d
therefore since
n
bf/A.
a. a.
This shows that na,(d)
+
-+-
0)
1
="(~acr/~)
..
)i\
•
2
suffi~s
even if d
+
o.
(For
"
-34-
the case
t-
c
n (d)
Cl
By silnplifying (3.10) of Theorem 3.1 with the above
suffices.)
0,
it is more diHicult to Sh0101 that this
the corollary is thus proven for
h(Cl, n),
~
c
O.
.
The next choice of
COr011<11'y 3. 2 :
Suppose
where b
•
~
as
~
n -1?{ (t n)n
a + O.
~
1 + (ccil 0) ~ .
F:.:ytl1er let
l}
'£!c,
a
Cd)
~
where
~
+ a!Cl
TI
I
-1
CL
i - (arc tar. a)!n
and
'TIlen roa - + 4
is motiv<1.ted by the nOIT:1C.l case
h(Cl, n)
h
()
,CI,n
.
c
> 0, ::-
ClIZ.
lor; t/log b,
'TIlen
Hext Carroll finds moment conditions en
0rtn
vlhicll [:lk'lrantee
that
E(N (d)jn (d))
(3.13)
6>
a
as
0,
Cl
a.
+
Cl
1+6
+
Only the caSE;
1
i\ ~ 2
is cons i'lered , and it is
~
pointed out that t[:e case
J:Ja.'{
:.J -
1;-)
o2(i J')
n
'
.
Let
K > 2
is handled by Jefining
I,j ~ ;-j(Cl,d) ~ in[{n:
h 2 (a, n)o2
n
<
o~ ~
n
n cl 2 }.
If
then by de:"inition. and by (3.3) one arrives at the
Cl
/.
two centradictory conditions
J-J (d) > l1(a, d),
•
r
I
"
-35-
and
11lus
TZr1
- T111
h(a,
<
~1)
1~
(1q/I:j'2 - d <
It £ol1ows that
N",(d) ::;; M(a, d).
o.
E(N (d)/n (d))1+6
u.
(f:l(a, d)/n (d))l+0
a
a
is unifon::ly. integrable.
a
-+
if
1
Carroll proves the
following two propositions using a theorem by Bickel and Yahav (1968) .'IIere r equals the number of finite moments of an for n large.
....
Proposition 1: If r > 2(1 + 0) and h(a, n) is as in Corollary
._
3.1, then (3.13) holds.
Proposition 2:
If r
~
2(1 + 0)
and h(a, n)
is as in Corollary
::; . ~ then (3.13) holds.
The assunptions of ll.leOrer.l 3.1 hold for nany choices of
T
n
including triIilmec1 rJ.eans, ?:·-estir.1ators \Ilith preliminary scale being
the interquartile range, and Hodges-LeI1r.lall estimates.
Finally, it is interesting to note the possibility of defining
an eliminating Wackerly rule froIT!. Carroll's procedure.
be done in the next section.
TI1is 1,.,rill
-36-
'.
4.
Conclusion
This section will deal
~Jith
comparisons of the eliminating and
nOIl-eliHmatmf:; procedures discussed in t;',is essay.
In studyin[;
,
the lJerfonlJD.nce of rar>j;iq, procedures, one of the following two
assuHptions is often
(1)
j,~ade
(~..
(but see
le~st
The parameters are L'l. the
or
...
,
3b)):
favorable configuration, i. e.;,
the pammeters are in the "equal
(2)
parai:teter cDnfiguration", which assumes all fe-pubtions to be the saEle.
In both of these cases "bad" populations are not present and this
should be kept in mind \Jhen cor.rflUring el:ininatinz procedures l'lith
non-eliJ.Jinating procedures.
First, Swanepoel ;ind Venter's proce-l.ure will be cOfilpared l\1ith
•
GeertseIr.a' s, and. thas with the
nss i1roceJure.
Let
T denote the total m.n.1ber of observations required fOT
2
Gecrts~'aa's procedure and let T
denote the total r~m~er of obser-
I
vatior.s required for
S~mnc;J081 rc;~~.
the indifferer,ce ZO:1e
'1(s.:c;" sproceduTiJ.
'Then outside
I1[K] - 11[1(-1] ;;, 6, it cm be shown that
lim
(4.1)
(1-+-0
and ,,2
v
. I
F
evaluate<1 and 1'/h'Ore SF,.r
.:'
(4.1)
ex )is
n(12/ p 2
(4.2)
..
u~
-'
•
"
.~.
...
Expression
a ]mown fur..ction of
is difficult to eV::Jluatc
tenus to
\Vhich can be
~n
ge!1eral, but as
!'>
-+- 0, (4.1)
-37-
Here C2 is also a constant depending on F Hhich can be evaluated
in an easier manner.
S\Ilanepoel and Venter evaluate (4.2) for
various choices of F and l',rith J(u)
=u
(Wilcoxon case).
For
eXaITI'.fl1e, with F Eon1al
(4.3a)
lim ET,,/ET
0.+0
1
"
?:
.
4.
They also reIYJark that with J(a)
(4.3) still holds.
= cp-l((1+U)/2)
Thus, even though we let
and F n.onnal
!'> + 0"
the eliminatin?
procedure appears to be significantly better than the non-eliminating
procedure.
S~mnepoel
and Geertsffina
,Bechhofer~s procedure'
shot~ ~Ie as}~JPtotic
efficiency of
(0'2 l<n01'm) relative to their procedure is
(4.30)
TIIb expression is ,:::; 1 and thus Swanepoel and GeertseJTll1' s pro-
cedure is better in an asymtotic sense.
l
I
A more interesting comparison that Swanepoel and Geertsema IT'.ake is
'"
:to the F~S procedure, where a 2 is not ass1Jf1ed knmm. Swanepoel
mId Geertsema found the relative efficiency of tileir procedllre to
Robbins, Sobel, and Starr's as a
+
0 in the least favorable
configuration
/'"
~ ~[K-l]
• 11
- ~[K] - ~ .
$
-38-
•
i\5 remarked before th,) '1S;" ]~;'(:o,t:ic
expected sa;aple size of
S\'I;~71eflOel
and Geertsema' s procedure has its maxllJ1E:'< (outside the indifference
zone) in this conER,ura tion wherc,as thea3/;";:'cotis expected sanple
size in ;l.obbins, Sobel, and Starr I s procedure does r.ot depend
the specific configuration.
Let
T
Oil
deilote the number of
observations aeeded. by the p.ss procedure anel S1-Ianepoel and GeeTtse:na' 5
Theil it nay be shm,'ll t.l1at
procedure res?ectivf:}y.
lim ETfEl'.,1:.. =(lI/a)2 flog (I+2(II/a)2)
(1+0
,; I
at least for
~
o.
<
A further advantage of t;1C eliGinating procedures is the fact
•
that all' of t11c;:1
pees) " 1 -
~lentioned
for
(l
I'(IC] - 1'[1(-1.]
procedures (except
:.ls:.~ 'l·/i:(yt:.c:~lly.
(except for Carroll's) guarar.tee that
Ii],]
Tn
!l+O
11
property that
Gccrtsema's procedure.
variable
:
H!Jile the nor.-eliminating
EecJh~ofer's) satisf~
l'Ioreover,
configurations,
"il,
with
(a)
I~'
(b)
peN' < "") >
< co
(~OCS
"" a.s.
not
hol(~
~
replaced by
a.s.
O.
-
for all
(l
only
paranete~
for Swanepoel and
In Section 3 we defineci N'
j'j
11
=
P(C,,) "
as the stopping
The results Here (sec (3.6))
if
I
•
a.'1d
0
for all other para-neter configurations.
,'\5 a final cO).1parison it win be snm,n that ir. Carroll's ['ro-
ceuurc the
'- ,
"
and~
d· can be specified in such a 1,\!ay
•
'.
. ,.
"
-39-
thnt an eliminating equivalent ofWackerly's procedure is obtained.
K=
In Carroll's procedure let, for
2~
(4.4) (i)
(J
n
(ii)
=1
d = -£n(I1 (-li))
0
(iii)
1~
h(a, n) = -in a/n~i! .
and (iv) :
Then, the nUTilber of observations for
K = 2 for Carroll's procedure
is p from( 3. 8)
1~
. Net (d) = inf{n:
Tn;;:: h(a, n) (Jn/n'? - d}
.],,-
--tn a/n: .
=
-L~0n (-li)) ~ -
inf {n:
= inf { n:
n
'I-
:p
>
-fu
(~
in a/n}
}
-.en(~ (-1I))
or
Recall that the stopping rulert.\\' = 1T.w(a , d)
each a > 0,
thp
s~.tC.:l1est
of Wackely was, for
integ<tr 'n such that
=
-40-
" 1,)• - Xi(n) < -M2
(4.6) (i)
SOIre
i
= 1,
a..TJ.cl some
J
:::
for
1\.
1)
0
Cl
:¥
Ie
l?oolljll
(ii)
and
W}leTe condition (4.6) (ii)
is the same as (4.5) above from
Carroll's procedure 8nd where condition (i) from 4.5 is true a.s.
as n
-+
00.
na (d)
ofCarr~ll's
An adaptation
"here
Na (d) = inf{n:
(4.7)
x..
- X.1 (n)
1)
n {n:
i
',r.·
'. , J
1e·.n.
=
·1,
Cl
$.
SlI~~
"'1 ,.L
}, ( a, n)
. , a ,
n
n
is Hac1~erly's rule.
Net (d)
-+
Net (d)
as
<
an<l
SOTJe
-+
oo~
j = J.~ ..• "Jno}
d are defined as in
ffild
That is,
n
-612 for some
l\,/d)
=N\J(a, J).
then for large n,
(4.4)~
then (4.7)
Further, since
nO. Cd)
~ !\/a, d).
It should. be noted that for Wackerly's rule to be a special
casc~ it is necessary f'rom Theorem 3.1 to have
as
a-+
O.
Wackerly
ShOl'lS
that
h(a,
Na (d))
-+
00
-41-
(4t
which ir.iplies
lim h(a, N (d)) +
a~'
00.
a
111e other conditions of TheorcJu 3.1 follow similarly, We thus
expect that the adaption to Carroll's rule (4.7) is an improvement
over Hackerly's in the sense that it is in a more general frarnewqrk,
In
conclusion~
most of the infonnation at the present tir,lc
points to the supremacy of eli:m.inating rules over non-eliainatiYlg
rtlles, in tenilS of sarrrple size and generality.
The fact that the
distributional requirements of the eliminating procedures are as
general, as a group, as those of the non-eliminating procedures
indicates that it is -advantageous to use one of the eliminating
procedures, the particular one being used depending on the circuDstances.
A topic of further research would be a c0P.1)ronise of the
two techniques.
That is, el:i.r:1.invte certain populat.ions
~md
allow
them to come back later, if :further observations indicate a possible
error.
There are still many
be investigated.
of sequential ranking left to
fao~ts
..
•
'I
i
•
•
-(,2-
BIBLIOGHfl,prpf
[1]
ANSC(l;JBE~
F.J. (1952).
Estrr:mtion.
Large Sample 'Eleory of Sequential
Proc. Comb. PhiZ. Soc. (48), 600-617.
[2]
BECtliiOFErt, R.E. (1954). A single-sample nu1tiple decision
procedure for ranking r,leans of HOTInal papulations with known
variances. Ann. Math. Statist. (25), 16-39.
[3]
DECHHOFlm., R.E. ~ KIEFER, J.~ and. SOBEL~ r.'i. (1963). Sequential
ranking and identification procedures. Univ. of Chicago Sta"
tistioaZ Research i'4onog1?aph Series.
[4]
BAm::.? f..APT. D.n. y and P..EVI, ~'LH. (1966). An introduction to
ranking aIid selection procelhlres. J. Am. Statist. Assoo. (61),
MO-64G.
[5]
CANTON, A.11.P. (1975). Literature review of certain topics in
selection and ranking of populations. Department of Bio-
statistic Joctoral vTrittcn examination.
Carolina at Chapel Hill.
University of nort11
[6]
CA,"UlOLL, lLJ. (1976). On the Serfling l1ackerly alternate
approach. Institute of Statistics j"lineo Series #1052.
University of Horth Carolina at Chapel Hill.
[7J
CPRROLL~ R.J. (1976).
A class of sequential ranking a~d
selection rules with elimir:ation.· Unpublished manuscript.
[3J
GEERTSEf'}A~
J .C. (1972). Honparali1etric sequential procedures
for selection the best of K populations. J. Am, Statist.
Assoc. (67), 614-616.
[9]
GOVnIDAP.,ft.JDLlf~ Z..... (1974).
Academic Press, flew York.
SequentiaZ Statistical Procedures.
[10] GlWTA, S.S. (1965).
,"
On some multiple decision (selection and
TechnometrioB (7)~ 225-246.
ranking) rules.
[11] FOEL, D.G. (1971). A nethod for t~le construction of sequential
select~on procedures.
A~n. F1ath. Statist. (42)~ 630-642.
[12] ROBBINS, H., &1BEL 1 II. an(r"'STAI~R1 N. (1968). A sequential
procedure for selecting the largest o·E K means. Ann. Math.
Statist. ,(3Q):, 38-92.
,$
;;' ,\1I
.. '
[13] SEN, P.K. ~ 'and GHOSr:~ H. (1971). On bOll.'1ded length sequential
confidence intervals based on one-sa-mole rank order statistics.
Ann. l'viath. Statist. (42), 139-203. "'
...:
-43-
•
[14] Sll<OIJS, G. (1968). On the cost of not mowing the variance
lvhen IJ'.aking a fixed-width confidence interval for the mean.
Ann. Math. Statist. (39)9 1946-52.
[15] Si1AHEFDEL, J.N.H. and GEERTSEMA, J.e. (1973). Selection
sequences for selecting the best of Ie nOTr:lal populations.
S. Afr. Statist. J. (7), 11-210
[16]·S1'!ANEPOEL, J.W.r-I. andGEERTSHilA, J.e. (1975). Sequential
procedures with eljs!lination for selecting the best of K
nomal populations. lJnpublished manuscript.
[17] S'IANEPOEL, J.W.H. and vmrfI:m, J.B. (1975). On the construction
of sequential selection procedu.res. lJnpublished rranuscript.
[18] lJACI<ERLY ~ D.D. (19'75). ful alternative sequential approach to
the problem of selecting the best of l( populHtions.
Tech.nical Eeport No. 91 9 University of Florida.