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A-A178 145 REGENERATIVE SAMPLING AND MONOTONIC BRANCHING PROCESS 1/1~
(U) SOUTH CAROLINA UNIV COLUMBIA DEPT OF STATISTICSS D DURHAN ET AL. MAY 86 TR-i1S AFOSR-TR-86-0429
UNCLASSIFIED AFOSR-84-0i56 F/G 12/1 NL
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11. TITLE (Include Security Classicaoni 61102F 2304 A5Regenerative Sampling and Monotonic Branching Processes
12. PERSONAL AUTHOR(S)
Stephen D. Durham and Kai F. Yu13& TYPE OF REPORT (13b. TIME COVERED 114. DATE OF REPORT (Yr., M., Day) 15. PAGE COUNT
Technical FROM TO 1986 May 2716. SUPPLEMENTARY NOTATION
17 COSATI CODES 18. SUBJECT TERMS (Continue on reverse if necessary and identify by block number)
FIELD I GROUP SUB. GR ' Sequential design; play-the-winner, branching process,I martingale, estimation, consistency and conditional
hypothesis testing.
19. ABSTRACT (Continue on reverse if neceosary and identify by block number)
A regenerative sampling plan is proposed for the sequential comparison of twopopulations having positive integral response. It is designed to be both an extension andan improvement of the play-the-winner rules for binary trials in the sense that a muchwider variety of responses is allowed, the fraction of inferior selections approaches zero,and the play-the-winner rule is contained as a special case. Almost sure convergence andmoment convergence in the pth order is studied for the fraction of inferior selections andfor a maximum likelihood estimator of the mean response. A conditional test of hypothesisis given for the binary case."' DTIC
ELECTEJUL 2 5 1986
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REGENERATIVE SAMPL ING AND MONOTONI C BRANCHING PROCESSES
by
Stephen D. Durham and Kai F. Yu *
University of South CarolinaStatistics Technical Report No. 118
62 LO5-5
DEPARTMENT OF STATISTICS
The University of South CarolinaColumbia, South Carolina 29208
0i APPrOved forP'abl10 role,.e
d'tIbitIt
- W ' S XW u- w r v 'r.rr..r.-_r r .. :. - - . -. .--
'.4
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%," REGENERATIVE SAMPLING AND MONOTONIC BRANCHING PROCESSES
by
Stephen D. Durham and Kai F. Yu*
University of South CarolinaStatistics Technical Report No. 118
4 62L05-5
May, 1986
Department of StatisticsUniversity of South Carolina
Columbia, SC 29208
,.-
Research supported in part by the United States Air Force Office of Scientific
Research and Army Research Office under Grant No. AFOSR-84-0156.
'TF '1( TS .FTr FE, (.RC (AFSC)S ".. T 7 L TO D I C
. . 11 rc pot hi:; oi,-,, reviewed and is
fo" puhlic rclc1;c IAW AFR 190-12.
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Abstract
A regenerative sampling plan is proposed for the sequential comparison of
two populations having positive integral response. It is designed to be both
an extension and an improvement of the play-the-winner rules for binary trials
in the sense that a much wider variety of responses is allowed, the fraction of
inferior selections approaches zero, and the play-the-winner rule is contained
as a special case. Almost sure convergence and moment convergence in the pth
order is studi d for the fraction of inferior selections and for a maximum
* olikelihood estimator of the mean response. A conditional test of hypothesis is
given for the binary case.
Accesioil For
NTIS CRn,&I
DTIC TABUrianro,A, cod ,rjJ U stifcJto.
... - ... . .. . . . .. .. .. ... . . .By .. .
Dist, ibuio~i0 A~, m,,I .ty Codes
Dist A", ." p,:CA l
,?..,a
U''=
Section 1. Introduction
The problem of sequentially sampling two populations with unknown means so
that the sum of observations is maximized has been formulated by Robbins
(1952). For the binary case (success/failure trials), play-the-winner
strategies have been shown to produce better results than a random selection of
populations, in the sense that the fraction of inferior selections approaches
the constant q/(qA+qB) where o < qB are the failure probabilities (Robbins
(1952), Zelen (1969), Wei and Durham (1978)). A randomized play-the-winner
plan has been used for the assignment of patients to treatments in a controlled
clinical study of a potentially life-saving medical procedure because of its
tendency to put more patients on the better treatment (Bartlett et al (1985),
Cornell et al (1986)).
The purpose of this paper is to present a sampling procedure in which the
fraction of inferior selections approaches zero, in general, whenever the ob-
served response is a positive integer-valued random variable. The main idea is
to generate new samples on the two populations according to the cumulative
response observed on each, as is done with the play-the-winner rules, but modi-
fied so that the sample sizes for the two populations are independent. The
successive samples then correspond to the generations of two independent
Galton-Watson branching processes and the attendant limit theory applies.
.9 Based on the observed successes on the two populations with a binary response,
a conditional test of hypothesis is given along with explicit bounds on the
p-.;er function. While other methods for dealing with the binary trials exist
in which the fraction of inferior selections go to zero (Bather (1981)), they
dl,) not seem to have as tractable an inferential structure.
I..
2
Section 2. Regenerative Sampling with a Positive Response
A sequence of stopping times called generation points are defined for the
observations on each population independently of the observations on the other
population. For the population i = A,B, let RRR.. be independent
and identically distributed (i.i.d.) random variables taking positive integer
ivalues having a common mean, mi. The Rk correspond to the observed
responses on population i. Beginning with an initial sample of size ui, a
positive integer, the sequence T1 of generation points are defined by( un
11T i
T U + R R for n > 1.
n+l i 1 + Tn
Note that the generation points are defined separately for each population
and the detailed specification of the order of selection is left open. It will
be seen that the observations between g, eration points
0 i(2) 0 = u
(2Z i = Ti - Ti for n > 1,
n n+l n
form the generations of independent Galton-Watson branching processes for
i = A,B, regardless of how the samp'es are ordered.
The sampling scheme with random sampling order within each generation
may be visualized as an urn model:
Two urns are given; Urn I, a sampling uw, and Urn II, a holding urn.
Initially, u balls of type A and uB balls of type B are placed in the
sampling urn. To begin the first generation of sampling, a ball is drawn at
random from the sampling urn and its typr note.1. An observation is then made
:.
3
on the population indicated and the response, R > 1 is recorded. A total of
R balls of that type are then placed in the holding urn. The process is re-
peated until the sampling urn is empty. That is the end of the first genera-
tion of sampling. To begin the second generation, all the balls in the holding
urn are placed in the sampling urn and sampling begins anew. The analysis to
follow is based on the ball populations at those points where the sampling urn
becomes empty. They are the generation points, Tn
Theorem 1: Assume mA > mB. Then as n tends to infinity, the fractions of
inferior selections
- . TB:(n
n n-. (3 )B
-"*'" A ZB
'.5n n
approach zero with probability one.
Proof. Temporarily suppressing the population superscript i for A,B let
Zn = Tn+1 - Tn be the nth stage sample size. Then Zn may be expressed as
Xn +'".+ X n where Xk -RT+k are iid with R1 and independent ofn-l' ,n
Zn1l n > 1. Thus Z represents the nth generation of a Galton-Watsono-l n
branching process initiated by Z0 = u ancestors and having offspring distri-
bution equal to that of R (Harris (1963)). The generation points
Tn~ Z +..+Z are the cumulative progeny up to the nth generation. Itn 0 n _
follows that the expected generation size is
um if m< ER
(4) EZ m EF~ {4 n ICO'-. if M = O
and the average sample number (ASN) is
eq.-
4
nu if m= 1
(5) ASN =ET= uif < and m > 1
O if M=
To prove (3), we first note that by assumption mA > > 1, we can choose an
integ:pr M so big that
.6) m* E n A > mB"[R (
Let R* F I and define T ,Z in a similar manner with u, = uA
Then for all n > 1
(7) T" *.T**(7) n - n"
Z* ZBz zB
Next, it is wcll-know .n that (n, n>l] and (-, n>l} are martingales.n -n -
m , ZB , B
Z zBnn
Since E -- = uA < c eand E wuB < ro by the nartingale convergencen An
in* nB*B
Z ztheorem, -- an n converge with probability one to random variables W*
B n
and WB respectively as n tends to infinity. Furthermore, since R* < M,
W is a strictly positive random variable with probability one. In view of
Sn+l 1 n , !(8) -* z .i
=n+l m, 0 n-j /nn A
T /Im, converges to T /(m,-l) with probability one as n tends to infinity.
Similarly if B > 1, T, /mn converges to WB/(mB-I) with probability one and
if = , TB =nuB. Next, choose X such that X c(mB,m,); then by (7)
Sn.ro.•
5
(9) n < , B
. n n n nT;+T" T T
"'- ii ( n nMB
M': T T"n[ n-m mB n
n nB
M*
which goes to zero with probability one as n tends to infinity. This also
implies that ZB = o(nA ) = o(Tn + ZA ) with probability one sincen nil n n
TB ZB
(10) n+l n
n+1 n+1 n+I n
As a result, it follows that
Z B ZB
IzA + Z B TA T + zB
n n n+l n n
converges to zero with probability one as n tends to infinity.
The following corollaries show that favorable comparisons need not be res-
tricted to the same generation points on the two populations.
d dCorollary 1: If mA > mB for some positive integers dA and dB, then
TBndB
TA + TBnd nd
A B(12)
TB~Tnd B (n-l)dB
A A B BT -T + T -Tnd (n-l d ndB (n-l)B
- C.I..
".. A _ A T
"--n A (-O AN% Tnld
6
converge to zero with probability one as n tends to infinity. In particular
i f minB< iA=w then for any positive integer d,
TBnd
T+TBn nd
K T B TB
(13) TB ~ TBn (n+l)d nd
converge to zero with probability one as n tends to infinity.
Corollary 2: if A >m for some positive integers dA and dl then as
n tends to infinity
T. nd ndA O B /mA
(14)B A~B )/TTA ) E d
,B nd (n+l)dA nd A MB /mA
with probability one.
Corollary 3: If mA > nB then for any p > 0, as
TBE( n
(15)B B
- E( A n B )p n0
"']"" z~A B(~ n
z + Zn n
dA ddif m > m B fr some positive integers dA and d thnn for any p >0,
mA A B'
as n
= ( SB/t n
.-4-- 1
e'q:'? T(~~B-T%/( ~~ A %=0(B/~A n
- - TBr
Bnd
Bv. <' • . . .. E(A %ldB )P 0,
TA +TBndl nd
A B(16)
TB TB(n+l)clB ndB
TA +TB TB(n+l)d nd (n+l)d nd
A A B B
If mA > mB, then for any positive integer d and any p > 0, as n 4
n( d p 0 .TA + TBn nd
(17)
E( n+l)d n- )P 0.A B B-n + T(n+l)d T nd
Remark 1. It may be expected that E(T-B /Yn )P 4 0 under general conditions.n n
17nwever we have only been able to prove it in the very special case of success/
failure trials (see Section 4, Theorem 6).A A B B
Renark 2. The independence of (RIR 2 .... and {RR 2 .... is not
necessary for the results in Theorem 1 and Corollaries 1,2 and 3 to hold.
The results also include the cases when either mA = or mB= 1 or both.
A A K KRemark 3. If there are K processes {RI,R 2 F ... ,R_,R 2 ....} with
means mA,..... mK respectively, and if mA > max(mB,...,mK), then the results
TBBin Theorem 1 and Corollaries 1,2 and 3 will still hold when TB and Z are
B B K B C Kreplaced by T + T +. ..+ T and Z + Z +...+ Z respectively, and the
*dA dB
conditions in the corollaries are changed from that of mA > m B to that ofA dB d
mA > max(m B ... m
,. %
Se-ction 3. Estiimation in a INo.no)tonic Bran ci P~rocess
In this section w2 shall study the estimation of somec of the parameters of
the sep:trate populations, specifically the mean responses mn and the
varince ~?,i B. As each separate population follows,, a Galton-Ja7t son
branchingi process, we shall suppress the superscript and sbciti. Let
R, P.1 ... be iid-J positLive integer-valued random variables with ER =m. Let
- - u be a positive integer and
(1T T 0 , 2, u and 11 U.
For: n > 1, define
Z R_ ++Pn n1 +1 n
(19) T 1 =u-~ + . + +,,
0 n
Ntic tha 2 0. , arnd for this reason we shall call this
c-elton--wlatson branching process a monotoni branching pi.c-ss For each
n > 0, let F n be the a-field generated by [ZQ, ...fZ01n For the mean m,
w-, shall1 consider the following two estimators
T n- lu
now (20)
rr~n z_
*n
7t*. C-i~tr Sr Ilko in the literature. See Dion and Keiding
a. thC: ref rocsthere in. Thec estir ,Itor m i a maiLMlikelihood
nr
T 7- K
9
Fact 1. Assume m < . Then with probability one, as n ,
(i) m - m,mn M
(21)
(ii) m -4m.n
In the following, we shall study the L p-consistency of mn and mn .
Definition. e is an L -consistent estimator of e for some p > 1 if as"_ __.- np-n -
(22) EI(en-G) I p 0.
To establish the L -consistency, we first develop a few results which arep
interesting in their own right.
Theorem 2: Assume that ERp < for some p 1. Let a 1max(,'). Then
Z n+1(23) {Iz -( - m)1p , n > 1 is uniformly integrable.
n
Proof. We decompose, for some K,
(4 R-m= (RI- ERI ) + (RnI - ERI(24) Rn-m (RnI[R n<K] ERn [R n<K n [R n>K] n [R n>K]
-X + Y say.n n
Since ERp < , for all c > 0, we can choose K so that
.. . (25) EJY n IP <
"-'.First, let s > max(2,p). Then by the Narcinkiewicz-zygmund inequality (see,
0 ,e.g. Chow and Teicher (1978), p. 356), for some constant B
9'
-,-... . . ......,'........-...-- -. -- .--.-, .-v, -,.-. ,-,.,-.> .-.. ,.. <. ., "... --- . . .
10
XT +1+...+XT
(26) E n a
n
nEE(- --IXT +1- -- .+XT I sIF +)
Zas n n+I
s sn n+1 nn
Therefore
(27) n zn+l n _4 is uniformly integrable.
n
Next, consider the Y's. By the same Marcinkiewicz-Zygmund inequality, we haveYT " +YT p
(28) E n+l
n
B _ E 2 . 2 )p/2E((YT + 1' y TIFn)
(2)ap n +l
np p p
B E( E( +1T +i "" IF) i
p Zap n n+ln
"zp/2-1 Bp E( n E(IYT +IP +...+I llF )), if > 2,P Z ap Tn +
n
< B p c E n p i-p Z -p T n p' + .+YTn 1II
n
*which can be made arbitrarily small. Therefore
4' - . . .
YT +I'1+(29) a n+ > 1 is uniformly integrable.
Ir nCombining (27) and (29), we have the desired result (23).
Corollary 4: If EP < -for some r 1, then
(i) ['In mlP, n > 1} is uniformly integrable.(30)
(ii) {lm -mlP, n > 11 is uniformly integrable.n
Proof. (i) Since In - mjP < lz n( - mP,(Z > 1 and a < 1),
the result follows from Theorem 2.
(i) By (i), {lii n JP, n > 11 is uniformly integrable. We note that
T T +Z Z Z(3) n+l _nn n < _1_n
T T T - Zn n n n-i
Therefore UT n > 1) is uniformly integrable and it follows thatn
T -u p(32) Tj~ ml ,n > 11 is uniformly integrable.
n
Corollar 5 Assume ERP < - for some p l. Then as n
(i) Elm -mIp -* 0, L -consistency,P
(33)
(ii) Elm -mIp -4 0, L -consistency.p
Proof. The result follows from Fact 1 and Corollary 4.
Corollary 6: If = o om n Var R c (0,a-),
then as n
Z -x 2/2(34) Elz( n l - -CO4 ~ll dx,
n
- 79 a TV %;
12
and for any positive odd integer j < p,
(35) E(Z ( Zn +li m))j 4 0.nZ n
Proof. Since as n 4
Z z n+1 M) D 0 2 )
(36) Z ( m N(0,an Zn
(See e.g. Nagaev (1967) or Dion (1974), the results follow from Theorem 2.
Next we shall establish a result analogous to Theorem 2 for T n . We need
the following elementary lemma.
Lemma 1: Let V,Vl,... be iid positive random variables with P[V > 1] = 1
and EV = m > 1. Then for all q 1 1, there is an a with a c (0,1) such
that
(37) sup E(vn>l l+ V n
Proof. By the strong law of large numbers, as n c o
(38) n 4-< 1VI+...+V n m1"n
with probability one. Since (n/(Vl+...+V n ))q < 1, as n
(39) E( n )q i _VI+-..+V mq
by the bounded convergence theorem. Therefore there is an a < 1 and an
integer n0 such that for all n > no
(40) E( n )q *(40 +.. .+V "
Since E(n/V1+...+Vn))q < 1 for n = l,...,no, take
1* Ev n o )q)
(41) a = max(a ,E )q,...,EV1 1 n
and the result follows.
e
13
Theorem 3: Assume that ERP for some p > 1. Let a =max(:-,i-) Then
for any r <p
(42) { , n > 1} is uniformly integrable.
n-i
Proof. Let r be less than p. Choose s such that s > 1 and r < s < p.
Then by Minkowski and Holder inequalities
(43) [lT n1 -u-mT nSJi/S
zn-i
E (E( Zk- 1 J jZk-mZkl sji/sk1 n-i Zi
< kl( 4j jS ZE Zk'-1 IP]i/p
Let q =asp/(p-s). Then by Lemma 1, for some 0 < (x < 1
Zkiq E(E((' '-1'LL2)'IF( Zn~ Z n_2)
n-l n-1n-2
(44) E (k) E(( n-)"IF)
Zk-lq <n-k
By Theorem 2, sup(EI kakiIP)p < C for some finite constant C. Therefore
from (43) and (44), we have
7 14
[T n 1 -u-mT nS] 1/s
k=l - S
andlar 7 Il~slfo o1' ERn- orsome) p>i and a ax ' then for any rp,
(46) (ITn1-a nl -m)Irn 11 is uniformly integrable.
Proof. In view of
(47) T1 aT -u Tn iu-mTn T -mI a
n zn-1
the result follows from Theorem 3.
Corollary 8: If ERP < -for some p >2 and Var R acO-,te o
any r <p, as n 4c
(4) 1 T n-u m 1rr. IIr -x2/(48 ET 2( nl -mIr*r ---------d~x
n Tn 2t
and for any positive odd integer j < p,
T -u(49) E(T ( T~ -m)) 0.
Proof. Since as n4
T -u D2T n+1 m) -- N(0, a 2,nl Tn
(see e.g. Dion (1974) or Jagers (197.L)); the result follows from Corollary 7.
Remark 4. We conjecture that the r in Theorem 3 and Corollaries 7 and 8
15
can be improved to p, in which case p > I can replace p > 1 in Theorem 3
and Corollary 7; and j < p can be chc nged to j < p in Corollary 8.2
In the iemainder of this section we shall assume that Var R = a whichK;: 2is finite and positive. We shall study the following estimators of a2:
-2 1 n Zk Zn 2I i) a = £ Zk_(--)-n n n k=l -z Z
k=1 ( k-l n-1(50)
, 2 1 Zk 2i (ii) an =- n k ( -ln n k=1 k-1lzk-
The consistency of these estimators is given in the literature and we collect
them into the following fact.
Fact 2.
-2 2(i) Heyde (1974). As n ,a -* a with probability one.n
(51)
(ii) Dion (1975). As n , n a 2 in probability.n
In the following, we shall study the L -consistency of these two estimatorsp
2of a
Theorem 4: Assume that ER2 p < for some p 2 1 and let a = max(lp).
Then
1-a -2 2p(52) {In -a ~2 )I p , n > 1} is uniformly integrable.
Proof. For each k > 1,
(53) E(Zkl(Z m)2 2 IFkl1k-1
":"1 I (z 2 2- Z 1E((Z ZkIFk_) - a2 = 0.
k-1
Since ER2p < , Theorem 2 implies the uniform integrability of.- Zk
{(Zk I z - m)2 )p , k > 1}. By Lemmas 1 and 2 in Chow and Yu (1984),k- 1
r::-
r' - , ,, -'-' -, =, - ', - : - ",{<' - ; , -,,", .'- ". - - - . , .-.-.- , . .. .. - - " .. ' " .. • .... . . .- - ..-
1~ 16
(54) a (Z k k- z M) 2p n >2 j f 11 is uniformly integrable,n k=1 k-i
which is t'.e desired result.
2p 1i1Theorem_5 Assume that ER2 < for some p 1, and let a =max(-,--).
Then
(55) ~ 1 3a 2)1P, n > 11 is uniformly integrable.n
Proof. We decompose
(5) 1-a~ - 2
iZk 2 2 Zn 2 ZZa E~ (Z k-l(Z - n)a) + (m - Zn kl1n k1k-1i- k=l
n z mZ 1-2 "'(Z -z m)( H
kl k k-i Zn~
In view of Theorem 4, it suffices to show the uniform integrability of
ai {Izj - z) ~ k- JP, n > 11 andn Zn-i k=i
nii aI k=1 Zk -"k-i n-1..n~ }
For (i), by Theorem 2 (for some finite constant C) and by Lemma 1 (for some
CX E (0,1)) and (44) we have
mZ -Z n 7(58 E(1 n-i n 2 p -)
nap kl n-i
n k- m n- n p
nZk-i)P < C n ___l /
-ap z - ap Z -
n1 I n- n I n
17
(EC (n-k)/p~ 0,_asn_-ap~ a (,_l/P)P a
yielding (i). For (ii), by the same results used for (i), we have
n (Z -mzkl (Z -mZ ) Zl(59) na El E~ p n
n ap k=l 4,Z n-ik-i n-i
_____ k-1 P(nl 2 Zk )pl 2~- ap z Z~
ki n-i
nz knap 1 n-i -nap1
yielding (ii). And this completes the proof.
Corollary9: If ER 2p < for some p 1 , then
-2 2p(i) {la -a 1, n > 1) is uniformly integrable
(60)
n a 2 1,n > 1} is uniformly integrable.
2 -2 -2Proof. Let a be either or or a .Since a < 1,
ni1-a la2 a 2~ > l0 2- 2~ and the result follows from Theorems 4 and 5.
Coroiiary 10: If ER 2p < for some p 1 , then as n 4
(i) El2
-2
1 0, L -consistency,
* (61)
(ii) Eja2 a2 lP 0, L -consistency.n p
-2 2Proof. By Fact 2, a nand a n converge to a in probability as ni 4 ~
Together with this, Corollary 9 gives the L -consistency.p
1LrI 7e~'
18
Corollary 11: If ER2 < for some 1 < p < 2, then as n-
(i) Efn ' a 2n j o(62) -
(ii) Ejn P -2 2 4P0.
Proof. By Lemma 1 of Chow and Yu (1984) and (53), as n-
(63) Ejnl P (2 2)Pr ~ 0n
By (56), (58), (59) and (63),
E -2 02)J 0.
Corollary 12: If ER2 < for some p 2, then as n
x 2 /(64) Eln 12(a 2 - 2 )P 2
2 2a2p J xj e dx
and for any positive odd integer j p,
(65) E(n( 2 2r j 0n
Consequently as n 4
-2 2 1(66) E(a) a + o(-),
i.e. the bias is of smaller order than n, and
2-2 2a 1
-(67) Va ra Cn) n + oC-).
Proof. 11eyde (1974) has shown that as n co
(68) n (a2 - 02 ) -4 N(0,2a 4,n
By Theorem 5, the results follow.
CI,
, lip
lj! . - 19
S<rt r- . ItOre St.'92 Ploy;ti _'ia-raO'r So- J_n__
"/ t~~~..- 0 Oi: ,.l (__'';,-i<- ci of the Ztlc.; (1969) approa ch to hi nary compalrisons,
->.]1.l c V. E,-:pu'] t-. thc nu:'ho: of tri ais cu treatment i, i =A or F, until the
fi 5 fnib<rL is C'iCL >vel. So the initial generation does not correspond to
z:,, sr" c7 tia !s Lhut cnly to the n,:-er of success runs to be observed in the
1..i f_L c i,.-zatio.i of samp~lingL. on treatnent i. The responses, R have
-c distrilution with finite mean m. = /q i and finite variance
I c-(1-i )/.i whcre qi is the probability of failure. 'The ASN is
.. 1 n(l--c. over the first T+1 u.i trials correspondinyc_ to genera-
t ions throuqhg n. Note that at least nu. trials are run on treatment i,
1,:1 t is5n -o unopir bournd on the actual nun'mer of trials for any
n > 1.
,:o erti l-r tor of p. = 1 - pi-, p. = 1 - 1/mi , reduces to
( V -u )/(T u. which eqals the cumulative number of successes
CU 'id A ky t.: r:un,'r of treatments on tre tment i in this scheme. The
ac:.iA: of i'iferior treatment selections is small if n is large almost
surely arn in thum Lp sense. A'n approximate distribution is available.
InSincr W .(=li, Zn/mi) has a gamma distribution with moment generating
1n
sT. -u.Siof.._. )i(1 +1s/u.)i 1, s > 0, (Harris (1963)) 2u.W. has a chi-+
1
lo.e, distrilrtion with 2u. degrees of freedom. Thus thc ratio," C7. .(o)'j"/Q ( -i) has an alpproxi,,atc- F distribution wit 2uA an! 2u
n-l n-rce., cf frcdom, wlhere the constants are Ci (n) = pi(qi )/(l - qi )
., .by Corollary 2, 1 /In 0((qA/q,) ) with probability
,. ]-o. (A;.>, 05P 0 > '.
Ei :, 6 r5 th,t convergenice obtains as well.
* .p
I -ASo-i % -..i - -) --.. -< - ? .2 ., L ... ' ... -< -i i 'i -) . ) i i - 7 ; -. ; ) ." : .> ;; •< .< i ' "' .< -: < -) i '-'
F 20
Theorem 6: Assume that 1/qA =MA > MB~ 1/qB If P[R1 =X1 (l-q.) ql
x~l2.. i=A,B, then for any p>O0, as n-
BzEi( 4) 0,zn
(ii) T(-)P 0
n
Prcof. without loss of generality, assume that UA =UB =1 and that p is
an integer. Then
-( B P xP(l-q n)x-lq nn x=l
nx-l n n< x(x+l)... .(x+p-1)(l-q) q = l qx=l
E 1 L~ -(- n x-l qnzA x=1 xp Al~ An
n nx-1ln 1 n x-l n
x=l X ~ x=p~l xp
n 1 n x-l n
P+1
S Cnq\n lcg q
< P'-A 1 n , for some constant C.
Hence as n w
7F
(i) E(' )~ = E(ZBP E(~~)-A n A
n n
Next
n1
z<((E(ZoB)P)/p +...+(E(Z )B )p) !/p)p
(1+ (p'cjp) I/p+ ..+(ptq-np l/Ppp
p- + q qn
-1 -(n+l)p
qB - (qBll)
T'E(+1P < E(TB+I )p E 1 p
(ii) -T ( A
znInp
[ 1 n CnqAP log qA
1I P (n+l)p ("'A - n 0 as n .- (qB - )
0 q B -q A
-dr dA -dB
Corollar 13: Assume thatmB qB for some positive
integers d and dBA B1
(i) E(Z A) 0 as n-*nd ndA
(ii) E(TB /'?" ) 0 as n-*wnd A ndA
Remark 5. The delicacy of the above result is noteworthy. It seems plausible
that a similar convergence obtains in non-geometric cases but no proof is
known.
F" mark 6. Certainly other selection procedures exist for the present case,
Bather (1981). Ilowever direct comparisons are difficult because the sample
sizes are random in regenerative sampling but are fixed in Bather's study.
The actual implementation of the trials is only mildly constrained by
i
4' . . .
I I ,:- v : , -.i,.-,-- .-. ..----. -. . - .- --, .- - _-, i. , -< , --" " ",_.
22B/
by the requirement that the generation points TA, T n be reached o,.
both treatments for some fixed value of n. It is interesting that if it isassumed, in addition, that every generation point Ti .... n
be reached on both treatments i = A,B before proceeding to the next, then
the present scheme can be visuclized as an adaptive iteration of the Zelen
scheme. In particular, let ( B=,Zl) PW(UA,UB) denote the number of trials
on the two treatments in a Zelen type experiment which is stopped when uA
failures are observed on treatment A and uB failures are observed on
treatment B. Then the successive sample sizes are generated recursively byAB A B A B
k(,Zk) = PW(Zk I, Zk), k = 2,...,n. The urn scheme
presented in Section 2 can be adapted to the present case in which the total
response is spread over a number of trials. As before, select a ball from
the sampling urn and note its type. Administer the indicated treatment and
observe the response. If it is a success a ball of the corresponding type is
added to the holding urn and the selected ball is returned to the sampling
urn. If a failure is observed then the selected ball is simply transferred
4 to the holding urn. As before, a generation is complete when the sampling
urn becomes empty.
In view of Fact 1 in Section 3 as applied to the present iterative
play-the-winner design, it is plausible to base a test of hypothesis about
the failure rates q, on the number of successes observed over n. sampling
generations, i A,B. In particular, f,-r positive integers nA,nBI a test is
proposed for
nA n nA n(69) H qA > qB vs H : A Bc:
based on the conditional distribution of S= nA uA given S SA + SB ,
inA
23
B B Bwhere S = Z - u . The random variables SA and S are independent and
nB
represent t-.e number of successes on treatments A and B over trials
1,2,...,TA and 1,2,...,T n respectively, if nA = nB 1 and uA uB f
the test is equivalent to that of Zelen (1969).
Since the samples on the two treatments are independent and since the
geometric distribution is preserved under the composition of probability
generating functions in the Galton-Watson branching process (Harris(1963)),
sA'sB and S have negative binomial distributions and the conditional distri-
bution can be presented explic:itly. For r a non-negative integer and
k = 0,1,...,r,
(70) g(klr) P[S = kISA= r]
k+uA-l r-k+uB- r J+UA-1 r-j+uB-1:'.) / E ( U _ )( U _ ) x j
UA-l UB-i j=0 A
where X n(-qA )(-q
nAnA n._
Under qA, = 1 and the distribution with r a nonnegative
integer and x = 0,1,...,r, is
x(71) G(xlr) -P[S A < xIS = r] = E g(kJr).
k=0
The a-level test is proposed:
(72) Reject H in favor of H1 if and only if G(SAIS) > 1-.
In the simplest case, uA = uB 1, the power can be bounded as follows:
Theorem 7: Assume that uA = uB =1, nA and nB are positive integers and
0 < a <. Thenwe
r24
(7 3 ) ( q -B -r n A e -1rA
". B B l A ) -i
< (1-qA )(q + (l-q )(q -
where e = -1 and K = P[G(SAIS) > i-z].
Remark \. The lower bound is exact if a = , and for a small, approximately
(74) q < K < /q 1 ~A
Proof. Since UA = 'B = 1, G(xlr) = (x+l)/(r+l), x = 0,1,...,r.Thus K = P[e(SA+I) > B] Since
(75) P(Reject H0ISB= k] = p[sA+ 1 > ek]
=(I-q nA [ k
and the latter is bounded below by (1-q and above by (-qAk
BK is seen to satisfy the bound stated upon averaging over the values of S
. In view of the interesting outcome of the clinical study by Bartlett et al
(1985), Cornell et al (1986), performance values with qA close to zero are
presented in the Table. Of course, the power is conserved upon truncation of
the favored treatment since the success counts are cumulative. So the null
hypothesis could still be rejected without ever completing a generation on the
better treatment. If it appears that qA= 0 may be true, as in the afore-
mentioned study, then the trial may be concluded at any point after the
specified generation point is reached on treatment B. Of course at least
* uBnB trials shall be run on treatment B with regenerative sampling.
S
B -B
Ix
25
TABLE
Iterative Play-the-Winner
nA ~uA =uB =1, qA .04 and q L
0 A ~B VS B
ASNT\ 25
n KnB (BB ASNB cc--.01 a=cc. 05 c--. 50
LB UB LB UB
1 .80 1.25 .80 .84 .88 .92 .99
2 .64 2.81 .64 .67 .76 .80 .98
3 .512 4.77 .52 .54 .66 .69 .96
5 .328 10.25 .33 .35 .47 .49 .92
8 .168 24.80 .17 .18 .27 .28 .83
26
References
Bartlett, R. H., Roloff, S. W., Cornell, R. G., Andrews, A. F., Dillon, P. W.,Zwischenberger, J. B. (1985), "Extracorporeal Corporeal Circulation inNeonatal Respiratory Failure. A Prospective Randomized Study",Pediatrics, 76, 479-487.
Bather, J. A. (1981), "Randomized Allocation of Treatments in SequentialExperiments", Journal of the Royal Statistical Society, B, 43, 265-292.
Chow, Y. S., Teicher, H. (1978), Probability Theory, Springer-Verlag, New York.
Chow, Y. S., Yu, K. F. (1984), "Some Limit Theorems for a Subcritical BranchingProcess with Immigration", Journal of Applied Probability, 21, 50-57.
Cornell, R. G., Landenberger, B. D., Bartlett, R. H. (1986), "Randomized Play-the-Winner Clinical Trials", Communications in Statistics - Theory andMethod, 15, 159-178.
Dion, J.-P. (1974), "Estimation of the Mean and the Initial Probabilities of a
Branching Process", Journal of Applied Probability, 11, 687-694.
Dion, J.-P. (1975), "Estimation of the Variance of a Branching Process",
Annals of Statistics, 3, 1183-1187.
Dion, J.-P., Keiding, N. (1978), "Statistical Infeerence in BranchingProcesses", Advances in Probability and Related Topics, 5, BranchingProcesses, New York, 105-140.
Harris, T. E. (1963), The Theory of Branching Processes, Springer-VerlagBerlin.
Heyde, C. C. (1974), "On Estimating the Variance of the Offspring Distribu-tion in a Simple Branching Process", Advances in Applied Probability,6, 421-433.
Jagers, P. (1973), "A Limit Theorem for Sums of Random Numbers of IID RandomVariables", Mathematics and Statistics: Essays in Honour of Harold
* Bergstrom, Goteborg, 33-39.
Nagaev, A. V. (1967), "On Estimating the Expected Number of Direct Descendantsof a Particle in a Branching Process", Theory Probability Applications,12, 314-320.
Robbins, H. (1952), "Some Aspects of the Sequential Design of Experiments",
Bulletin of the American Mathematical Society, 58, 527-535.
At
27
Wei, L. J., Durham, S. D. (1978), "The Randomized Play-the-Winner Rule inM. Medical Trials", Journal of the American Statistical Association, 73,840-843.
Zelen, M. (1969), "Play the Winner Rule and the Controlled Clinical Trial",Journal of the American Statistical Association, 64, 131-146.
Department of StatisticsUniversity of South CarolinaColumbia, SC 29208 USA
e.- p•
'U
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V,- °WV1 . 7 1 .7T77 7- .
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11. TITLE ilnciuae Security Clatsaficationl 61102F 2304 A5Regenerative Sampling and Monotonic Branching rocesses
12. PERSONAL AUTHOR(S)Stephen D. Durham and Kai F. Yu
13a. TYPE OF REPORT 13b. TIME COVERED 14. DATE OF REPORT (Yr.. Mo.. Dayj 15. PAGE COUNT
Technical FROM TO _ 1986 May y 2716. SUPPLEMENTARY NOTATION
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FIELD GROUP SUB. GR. Sequential design; play-the-winner, branching process,martingale, estimation, consistency and conditionalhypothesis testing.
19. ABSTRACT (Continue on reverse if necessary and identify by block numberl
A regenerative sampling plan is proposed for the sequential comparison of twopopulations having positive integral response. It is designed to be both an extension andan improvement of the play-the-winner rules for binary trials in the sense that a muchwider variety of responses is allowed, the fraction of inferior selections approaches zero,and the play-the-winner rule is contained as a special case. Almost sure convergence andmoment convergence in the pth order is studied for the fraction of inferior selections andfor a maximum likelihood estimator of the mean response. A conditional test of hypothesisis given for the binary case.
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_________________________________________ Stat.Tech. Rpt. No. 1187. AUTpOR(.) 8. CONTRACT OR GRANT NUMBER(.)
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@1 The view, opinions, and/or findings contained in this report arethose of the author(s) and should not be construed as an officialDepartment of the Army position, policy, or decision, unless so
IS. KEY WORDS (Conifn an ore ,e. od&e ft necessary and Identify by bioc-k ntnber)
Sequential design, play-the-winner, branching process, martingale,
* estimation, consistency and conditional hypothesis testing.
2 20. A fST A CT (Cato lhwe si H IFm- 7 0-d ldeuhlty by block , ber)
regenerative sampling plan is proposed for the sequential comparison of two
populations having positive integral response. It is designed to be both anextension and an improvement of the play-the-winner rules for binary trials in thesense that a much wider variety of responses is allowed, the fraction of inferiorselections approaches zero, and the play-the-winner rule is contained as aspecial case. Almost sure convergence and moment convergence in the pth order isstudied for the fraction of inferior selections and for a maximum likelihoodestimator of the mean response. A tonditional test of hypothesis is given for th
* DO 1473 cOITom or I NOV 6S IS OBSOLETEi Am 73UNCLASSIFIED- - *~~* ~ ~* e. rt,ATION OF, rN?S PAGE W?-e Dot. E,,e'*d)
e r .- .* / .. ... . . . A. * -'. . . . . . . .
p.
~ Ir
'p
r - . . .. - - -- - . - . -
. . .
...........................................