1 ~1is paper is based in part on the author's 1974 Ph.D. dissertation at Purdue
University. TI1is research was supported in part by the Office of Naval
Research Contract N00014-67-A-0226-00014 at Purdue University.
M'S 1970 Subject Classifications: Primary 62G31; Secondary 62G45, 62G70
Key Words and Phrases: Ranking and Selection; Sequential Selection; Tail
Orderings; Stochastic Orderings; Quantiles; Nonparametric Selection.
ASYMPTOTICALLY NONPARAMETRIC SEQUENTIAL SELECTION PROCEDURES 1
by
Ra.Yinond J. Carroll
Department of Statistics
University of North CaroZina at Chapel Hill
Institute of Statistics Hi:[~eo Series #944
August 1974
ASYrl?TOTICALLY NONPARAMETRIC SEQUE~rrIAL SELECTION PROCEDURES I
by
naymond J. Carroll
University of Porth Carolina
SUM~~RY
Let
F(x; e.)
~
ITI, ... ,IT K be
(i
= 1, ... ,K)
K independent populations with distributions
which are stochastically ordered or tail ordered.
The ranking goal is to select the stochastically largest population or the
population with the lightest tail.
Sequential selection rules in the spirit
of Robbins, Sobel, and Starr (lS68) and Geertsema (1972) are proposed and
studied.
The above problems are corollaries of a general TheoreM proposed,
and the solutions are nonpararnetric in an asymptotic sense.
1This paper is based in part on the author's 1974 Ph.D. dissertation at Purdue
University. This research was supported in part by the Office of Naval
Research Contract N00014-67-A-0226-000l4 at Purdue University.
-2-
1.
INTRODUCTION
independent populations with distributions
F (x; e.)
1
6.1 <
~
(i
= 1, ... ,K)
, where
is some unknown indexing parameter.
6.
1
e. indicates that e. is preferred to
J
J
6. , the most fundamental prob1
lem in ranking theory revolves around finding the
6. S
1
unkno~m
population
TI[K]
ai ) = F(x
- 6i ) , then 6i ~ 8 j
6. , in which case the goal would be to select the largest
which is most preferred.
could mean
If
For example, if F(x;
J
location 1: .':ITameter.
Suppose the true (unlmo'Vm) preference ordering is
6[1]~"'~
6[K] •
Using a natural decision procedure; Bechhofer's (1954) indifference zone method
consists of determining the smallest
sar~le
size
nCo)
for which the
proba~ili
of correctly selecting the preferred population whenever
F(x; 6[K])) ~ 0 > 0
is at least as large as a specified constant
o
cl(o, 0)
>
0
is known and
is some distance function.
case, using sample means,
e.J -
(1.1)
(1. 2)
(1.3)
N(o)
= first
p*
For future use, define
6.
1
integer n ~ (b%)2 , where
= f~oo ~K-l
(x +
b)d~(x)
.
p* , where
For the normal means
-3-
If
0
2 is unknown, Robbins, Sabel, and Starr (1968) and Geertsema (1972)
investigate and generalize the natural rule which takes
N(o)
Qbservations
from each population, where
N(o)
(1.5)
= first
integer n ~ (bS nl',"/0)2
2
Sni~v is the usual pooled sanp1e variance based on
and
freedom.
degrees of
K(n - 1)
Geertsema (1972), using translation invariant estimates, derives
asymptotic (as'o
~
0 ) results lV'hich are nonparal'letric in the sense that,
under certain conditions; they hold independently of the underlying distribution
F.
H(o)
is a stopping rule of Chow and Robbins type (1965); in order to in-
vestigate its a.symptotic properties, the follol.ring definition is needec:
DEFINITION 1.1 (Ansconbe (1952))
A sequence of r.v.'s
uniformly continuous in probability if
c (e,o)
:3
(1.6)
p{lyl'l -
where
d.f.
if n
~
V
e > 0 ,
{Y } is said to be
n
> 0 , 3 J(e,o)
0
J (e: ,0)
Yn I
<
ow-n 1 V integers
~:3 1m - nl
<
n c(e,o)}
>
1 - e ,
e and
is a sequence of norming constants such that for some real
F, P{y
n
- e
$
xw- l } ~ F(x)
n
The sample means satisfy (1.6) so that from Anscombe (1952), as
(1.7)
where
and
indicates convergence in law.
}Ience
0
~
0 ,
-4-
lin inf peeS) = p* ,
(1.8)
0-+0
where
CS
indicates a correct selection and the
This paper
atten~ts
inf
is taken over
nco) .
to generalize the location parameter results to such
preference patterns as stochastic ordering (Lehpann (10S9)) and tail orderinf
(Doksum (1969)).
In Section 2, the sequential rules to be used are
Section 3 motivates and presents the results and includes
presente~.
so~e
applications.
For
i = 1, ... ,!~ ,
Section 4 gives the proofs.
2. STRUCTURE
The initi?,l structure of Section 1 will be assumed.
x.1. 1 ' ... ,x.In
independent observations
and statistics
T. (n) , g. (n)
T.(n)
(2.2)
where
- ll(e.))
1
CJ.
1.
1
g. (n) -+
(2.3)
1.
1l(6 )
i
= lli
CJ (6
and
are formed such that
11
~) (T.1 (n)
(2.1)
are taken according to a distribution
i
o(6 i )
1
= 0i
g.(n)>0
1.
are constants.
distance function will be
(2.4)
(2.5)
6. < 6. <='> ll·
1
~
J
n-+
oo
a.s.
-+ lI(e.)
a. s., and
.)
1
as
<p
1
~
ll. ,
J
a.s. ,
The preference
pattel~
and
-5-
nco) will be as in (1.4), with the obvious replacement of
and
e[~].
For
N.(o)
= first integer
1
i
rn)
for
Since no sinple location parameter structure is assumed, there will be
K independent stopping rules
(2.6)
~(e
= 1, ... ,1(,
we will take
N (0), ...
l
n ;:: (bg
defined by
,1\)1((0)
?
(n)/o)·· (i = 1, ... ,K)
i
~!.(o)
1
observations from
T1· (ll1' (0)) , and select the population with the largest
1T.
1
•
,
form
T. (~J. (0)) .
1
1
Examples
considered in this paper for stochastically orderec families will include
cases where the
T.(n)
1
are sample fleans or sample mepians; the interquartile
range will be used for selecting the population with the lightest tail.
will be assumed throughout that the
REHARK 2.1.
{T. (n)}
1
It
satisfy (1.6).
The stopping rules (2.6) being independent has certain drawbacks.
Eowever, improvement here must await results in the location case, for which
there does not appear to be a nonparanetric rule which eliminates obviously
inferior populations and satisfies (1.8).
3.
RESULTS AND APPLICATIONS
The structure of Section 2 (especially (2.4) and (2.5)) will be
and the goal will be to guarantee (1.3).
assu~ed
At this point, no specific assump-
tions concerning the preference pattern (2.4) will be made; only later ,dl1
such orders as stochastic orderings or tail orderings be used.
The major difficulty encountered in r,uaranteeing (1.8) is the lack of a
recognizable least favorable configuration (see Bechhofer ,(1954)).
(3.1)
Letting
-6-
(3.2)
For
= 1, ... ,K
i
- I
it is easy to see that
lim inf peeS)
(3.4)
where the latter
case,
inf
N. (0) :: N(o)
1
~
n*
is taken over
(i = I, ... , K)
Now, in the location parameter
and
P(~,o)
(3.5)
= Q* (say),
lim inf P(!,O)
_ P(Q.,o)
so that
lim inf P (CS)
(3.6)
~
lim P(Q.,o) = P* •
However, in the cases which are dealt with here, the parameter point at which
pees)
attains its infinum is as yet unknown.
(en, 0)
n
However, there exists
such that
(3.7)
If there were a
p(~n, 0 ) + Q*
n
e such that
and 0
~n
+
+
n
~
,
0
as
since
(3.8)
P (en, 0 ) + Q* ,
n
n +
00
•
-7-
some sort of a continuity criterion for
suggests itself.
P(',c)
Eelly-Bray LeJI1..ma, it is clear th?t a sufficient condition for
(3.9)
1
P{C-n
b[T.(N.
(0 )) Il n
~(S~)) ~ z} ~(z)
+
1
•
as
n
~
(i
00
0*
By the
~
p*
is
= 1, ... ,K)
This fact, together wit!1 the continuity criterion, suggests Theorem 3.1.
the sufficient condition (3.9) depends only on the
tions, the rest of the paper will use the
and
~eneric
one-di~ensional
terms
cince
distribu-
T(n), g(n), N(o) •
11 (S)
This intuitive continuity criterion suggests that
close to
F(x;S)
should he
is srall. so that
F(x; SO)
(AI)
(A2)
For each
8 and for all
0
e:
>
0 • there is a d
such that
The other continuity conditions needed are that (2.1). (2.2), and (2.3) hold
uniformly for
THEOREf43.1
Y
>
(A3)
0 , c
>
jl(8)
close to
jl(8 ) ; this is summed up in
0
Suppose that for each
0 , such that if
n >- J
P8{lg(r) - 0(8)1> o(S)H
o<
H
<
e:
SO' e: > 0 ,
and
1(1
>
0 , and
ze:R' , :1
Ijl(S) - jl(8 ) I ~ y , then
0
n ~
for some
is such that
0
±
J} ~
H)2 -
11
e: , where
<
cf4
.J ,
-e1 2
Pe {n / IT(m) - T(n) I
(AS)
'Then, i f
RE~~RK
Q*
3.1.
>
for some
cr
m
1m - nl < en}
3
is compact and (AI) and (A2) are true, (1.8) holds.
TI1eorem 3.1 is nonparametric in the sense that it holds indepen-
dently of the underlying class of distributions
class satisfies (AI) - (AS) anQ
REMARK 3.2.
S E
is compact.
Q*
Although the proof of
{F(x;e)}, as lone as this
TIleore~
3.1 depends on the rules (2.6), the
Theorem is a fixed-sample result and thus can be verified in individual cases.
Q*
compact is used to guarantee that a subsequence of
(e , 0 )
n
n
converges;
this condition does not appear to be serious in practice, since most measuring
devices are finite.
REMARK 3.3.
It is quite easy to see that the conditions of Theorem 3.1 suffice
for the goals (i) selecting the
selecting the smallest
~(e[l])
i"
largest
~(e[K-t+lJ), ... ,~(e[K])
; (ii)
; (iii) as selecting a restricted subset (see
Santner (1974)).
LEMMA 3.1.
(Location Parameters)
hold, then (1.8) holds and
LEMMA 3.2.
F
(x;
(~,cr))
Q*
If
F(x;e)
= F(x
- e)
need not be compact.
(Location Parameters with Unequal Scales)
= F [x-~ 1
,where
j
(j
and (2.1) - (2.3)
cr£!(
compact and
~ER.
Suppose
Then, i f (2.1) - (2.3)
hold, (1. 8) holds.
The following Proposition 3.1 is adapted from Geertsema (1972) and
Bahadur (1966) and follows in a similar manner.
-9-
PROPOSITION 3.1.
Let
c
>
0 and
0
<
a
1 • and suppose
<
F(x)
has a bounded
second derivative in a neighborhood of the Q.th population quantile 1;, with
l 2
1/2 /2, and
1
F' (1;) = f(1;) > 0 . Let bn = fna
- -' - cn / /2 and a = [na] + cn
n
to be the
define
[an]th
and
(bn]th
order statistics.
Tr.en
It is clear that some sort of ilclosenessll condition must be placed on the
underlying distributions in order that the continuity criteria (AI) - (AS)
hold.
One convenient Lipschitz type condition is
(3.11)
eO' a numbers
For each
Be
+
1
that for all
and
Ce
1 as
+
(Sample Quantiles in Stochastic Orderings)
tributions
F(x;e)
T(n)
= X[na]
and
holds for all
LEMMA 3.4.
F(x; e)
x
Suppose the conditions of Proposition 3.1 hold.
and
as in (3.10).
y
Then, if
~*
are stochastically ordered, that
is the sample variance.
Suppose that
, (1.8) holds.
Suppose the distributions
2
T(n)
~(e)
Define
is compact, and (3.11)
~(eO)
in some neighborhood of
(Sample r1eans in Stochastic Orderings)
tl'ueifF(o;O)
Suppose that the disth
be the a
popu~(e)
are stochastically ordered and let
g(n)
such
x,y,
LEMMA 3.3.
lation quantile.
~(e) + ~(eO)
is the sample mean, and g en)
= F- l [F(~(eo) ; eo) : e) (l11hich is
is synunetricJ, and that for all
8
0 ,
-10-
(3.12)
is bounded in a neighborhood of eo'
If
(3.11) holds and
Q*
is compact,
then (1.8) holds.
LEMMA 3.5.
(Interquantile Ranges in Tail Orderings)
Suppose that the
F(x;8)
are tail ordered and it is desired to select the population witr. the lightest
tail. If o < a < 1/2 < a < 1 , and ].11 (e) , ].12 (8) are the a l st and
l
2
nd
v
and let g(n) be
quantiles of F(x; e) , set T(n) = X[
(12
na 2J - A[na I ]
the obvious estimate based on functions of the type (3.10). If Q* is compact and (3.11) holds for all
x,y
in some neighborhood of
].11 (e)
(and
].12(8) ), then (1.8) holds.
4. PROOFS
DEFINITION 4.1.
Let
= [ba(e)/a]2
Ml(a,e) = [ba(e)(l -
M(a,e)
(4.1)
M (0,8)
2
Recall that
M(a,e)/N(a)
PROOF OF THEOREM 3.1.
~
=
[ba(e)
(4.2)
+
1 a.s. under
H)/0]2
F(o;e)
as
a
It is sufficient to prove (3.9).
and (A3) , one sees that d 00' J,
1].1(e) - ].l(8 0) 1 ~ y
(1
E)/a]2
and
y
such that if
~
0 .
Then, by using (A7.)
°
~
00 and
-11-
Since
bO(El)/o(M(o,El))1/2 = (bo(El)/o)/[bo(El)/o] , one can easily show that
(4.3)
IPa{bo-1[T(M(O,B)) - "(a)]
Choosing
00 and
y
<z ±
o} - o(z
± 0)1 < < ,
small enough that
1M.1 (o,El)
- r'!(o,El)
I
(i = 1,2) ,
< cH(o,El)
from (A3) and (4.2),
(4.5)
How,
(4.6)
Pe{o-lb[T(N(O)) - )J(El)) :;;
Z} -
Pa{o-lb[T(M(O,El)) - )J(El)) :;;
:;; 2€ +
,
~
I •
Pe{ho~lIT(N(o)) ~
and
IN( 0) /H ( 0 , a)
T(M(o,a))
-l
i I :;
2€ + PEl{bo-1IT(m) - T(M(o,a))
for some
Similar ly,
Z+
m
3
Im/M(o,a) -
a}
I~
a
c}
I~
a
11 :;
c}
-12-
By choosing
0
small.
0
$
0 ' and
0
Ill(S) - ll(8 0 ) I
y , this means from
$
(4.3) that
and hence that (3.9) holds.
PROOF OF
L8~
to
3.3 from Bahadur (1966).
Len~a
3.3.
For convenience, assume that (3.11) holds; one can extend
Since
(AI) and (A2) follow from (3.11).
(A3) follows from the probability integral
transformation. (3.11). and (A2).
For (A4). assuming
and
z positive.
ll(e o) I small. since F-l(F(X; eO) ; e)
Ill(S)
~(Cez)
(4.10)
-
£
$
Pe{O(8)-ln
l/2
(T(n) - ll(e))
n
large,
is increasing,
$
Z}
$
~(Bez)
+ £
•
(AS) follows from the integral transformation, (3.11). and Proposition 3.1.
PROOF OF LEr~~ 3.4.
Assume that
ICe - 11
$
IB e - 11 •
and let
g2(n:s)
be
F-l(F(X ; eO) ; e) •...• p-l(F(X ; eo) ~ e) .
l
n
Then, from (3.11). one can show that
the sample variance obtained from
(4.11)
where
G
n
~
c > 0 almost surely.
With arbitrarily large probability under
-13(4.12)
where
Gn*
c * > 0 almost surely.
-+
follOi;T from (4.12).
(AI)
The proof of
fo11o\'JS
from (4.11) and (A2) and (.'\3)
(1\4) is somewhat involved.
It is possible
to show that, if
then (A4) 1tJill follow if it can be shown that V e: > 0 ,
SUdl
that
n
(4.14 )
~
Pa
Ie - eol
and
N(e)
{Iii I!- I
o
n i=l
(E.1 (e)
< nee)
imply
- 1) (X.
- eo)1
1
>o}
It turns out that (4.14) is true if for any sequence
with
n.
J
-+
and
00
ej
-+
J
11
:<;;
IB e - 11 ,
convergence criterion
n (e)
e.
(nle l ) , (n 2 ,
e2), ...
n.
J
L
i=l
(il.
1
(e .) - 1) (X. - eO)
J
1
converges to that of a random variable putting all mass at
IHi(e) -
0 , 3 T-l(e) ,
eO ' the cistribution of
-1/2
n.
(4.15)
<
e>
if
(Lo~ve
Elx i
-
eol2
o.
C'
,Ince
exists, one !'lay use the degenerate
(1963), page 317) to achieve the result.
The
proof of (AS) requires an extension of Anscombe's (1 0 52) proof that the sample
mean satisfies (2.6), and uses (A4), (3.12), and Yolnogorovis Inequality.
PHOOF OF
LH~\1A
3.5.
Again, as in
Lcr~ma
follow in a manner similar to the proof
cerived from Proposition 3.1 and
t~e
involves (3.11), the monotonicity of
3.3, assume (3.11).
o~
(AI) - (1\3)
Le!"lr-1a 3.4, although
expression
F-l(F(x;
~or
are)
g(n:e)
is
The proof of (A4J
en) ; e) , and decol'lposing t~,e
-14probability space into the
4
sets where
-x [net.]
1
i = 1,2.
REt~.RK
-
~.
1
(8) >
n or
< 0
for
US) fo11o\l1s from LeJ"!'.ma 3.3.
4.1.
~~te
that a stochastic ordering was not really
or 3.4 except to insure that (2.4) makes sense.
use~
in
Le~na
3.3
One could also combine Lemma
3.1 ",lith Lel'll1las 3.2 - 3.5 to extend the set over t'!hich the infinuJ11 in (1.3) is
taken.
ACKNOHLEDGEMENT
I would like to e:q>ress my deepest appreciation to my advisor
Shanti S. Gupta for his advice and support.
-14-
REFERENCES
[1]
Pnscombe, F.J. (1952).
Proa. Camb. PhiZ. Soa.
[2]
Bahadur, n.R. (1966). A note on Quantiles in large samples.
Statist. (37) 577-580.
[3]
Bechhofer, ~.E. (1954). A single-sample nultiple decision procedure for
ranking means of normal populations "Ti th Imo"m variances. Ann. Math.
Statist. (25) 16-39.
[4]
Chow, Y.S. and Robbins, E. (1965). On the aSYf.1.ptotic theory of fixedwidth sequential confidence intervals for e,e !Jean. Ann. Hath. statist.
(36) 463-467.
[5]
Doksum, K. (1969) . Starshaped transformations and the power of ranI:
tests. Ann. Hath. Statist. (40) 1167-1176.
[6]
Geertsema, J.e. (1972). Nonparametric sequential procedures for selecting the best of k populations. J. Am. Statist. Assoa. (67) 614-616.
[7]
Lehmann, E. L.
York.
[ o" ]
Lo~ve,
[9]
0.obbins, E., Sobel, r:., and Starr, f':. (1960) . A sequential procedure
for selecting the largest of k means. Ann. Hath. Statist. (39) 88-92.
[10]
Santner, T.J. (1974). A restricted subset selection approach to ranking
and selection problens. To appear in Ann. Math. Statist.
N.
(1959).
(1963).
Large sample theory of sequential estimation.
(L!-8) 600-617.
Ann. Math.
Testing Statistical Hypotheses, John Fiiley, New
Probability Theory, D. Van Nostrand, Princeton.