This paper is concerned with the problem of allocating a fixed number oftrials between K independent populations from the exponential family, in order toestimate a linear combination of the means wthsquared error loss

Internat. J. Math. & Math. Sci.

VOL. 18 NO. (1995) 147-150

147

ASYMPTOTICOPTIMALITY OF^SEQUENTIAL

DESIGNS

FORESTIMATION

KAMEL REKAB

D’paltmcntofAtt)lied.klath’matw’

Floxl(la InsttuWofTechnology Melbourne, FL32901

(Received November 5, 1992)

ABSTRACT. This paper is concerned with the problem of allocating a fixed number oftrials between K independent populations from the exponential family, in order toestimate a linear combination of the means wthsquared error loss. Introducing independent conjugatepriors, ^a batch sequential procedureisproposed andcomparedwth theopnmal.

KEY WORDS AND PHRASES.

Bayes

rusk; Ezponentzalfamhes; Sequential designs;

Umform

ntegrabzhty.

1992AMS SUBJECT CLASSIFICATION CODES. 62L05, 62L10

1. INTRODUCTION.

Given K independent populations

P,... ,PK

indexed by unknown parameters o,...,aK

respectively, it is wished to estimate a linear combination of their respective ^means based on afixed total number of observations M. The problem ^{is one} ofallocating the M observations between the K populationsso as to minimizeexpected squared^error loss. Several special ^cases of thisproblemhave been studied.

Forestimating the difference in two Binomialmeans, Alvo and Cabilio.

(1982)

proposed an allocationprocedure shown to be asymptotically optimal. Rekab

(1989)

^{considered problem of}

estimating theproduct ofmeansof twonormal populations. Robbins, Simon, andStarr

(1967)

considered the problem of estimating the difference of the means of two normal populations with unknownmeansand unknown variances. They proposed ^asampling schemethat has been applied in a variety of sequential estimation problems. Their work was restricted entirely to the ad hoe design. Woodroofe and Hardwick

(1990)

considered the problemof estimating the differencebetweenthemeansof twonormal populationswith ethical costs. Usingaquasi Bayesian approach, theyproposedathreestage procedure shownto beoptimaltosecond order forsquared

errorloss.

In the present article we adopt a fully Bayesian approach ^to the problemof estimating a linear combinationof themeansof

K

populations from the generalexponential family.

A

lower boundon the

Bayes

risk is derived and shown to be achieved asymptotically as M c by a

batch sequentialprocedure.

[48 K. REKAB

’2. THE BAYESIAN ^.I()DEL.

L,t

P1

^{P,. 1),} K inCh,1),nh,1t

lldatiCms

^frm the general,xpCmential ^{fimily inh,xel} tv(1 (/, r’spectiv’ly. That is.Stl)lse that

flr ^I(. It is assmed thr(mghlt that is the natm’al parameter space of the and that _{is len.} It iswellknwn that E,,(Xi) t/,’((*,:) andI,(Xi) ’i,"(ci). Let ^these

trnbeassigned independent conjugate priors, that is suppose that a are independent random wzriables which ^haw’, distributions

dII((ti) exp{r’ipiai

ri/,(ci)}

c(r,,i) d(ti

wtmre

/

c(ri,p,i) exp{ri/ci-

ri(i)}d.i.

LetMbethe fixed total number of observationsand

M,..., MK

^bethe random variables which allocateobservations to

P P-

^with

i ^Mi

^M.

^In

^estimating

^ci’(i)

^withsquared

errorloss, the terminal

Bayes

estimator is

i

^{cii,Mi wherei,Mi istheme}of the posterior distribution

/(i)

^{based on}

Mi

observations. Due to independenceof the priors, the terminal Bayesrisk

r

^{isgiven by}

r ^E _Mi

^ci

^(2.1)

ri

where

Ui,M E{"(i)IXi,,... ,Xi,M,}.

3. LOWER BOUND ON

THE

BAYES RISK.

Webeginthis sectionby derivingalowerboundontheriskincurredby the optimalprocedure.

LEMMA

3.1" Let

r

^{be definedasin}

^(2.1).

^Then

E(E_I(I

ci

V/O"(ai)) ) M+ E//;lr

PROOF:Theterminal

Bayes

riskcanbewritten asthesumof

K K

(M + ri)-lE(-

^ci

v/Ui,M,)

i=1 i=1

and

(M+ ri)-lE ^((Mi + ^{ri) lcj} /U,M, (M + )I , _v/Ui,Mi

(Mi + ri)(M1+ r)

i=1 i=1 j=i+l

The lemma followsbythemartingaleproperties of

Ui,Mi.

ASYMPTOTIC OPTIMALITY OF SEQUENTIAL DESIGNS 149

4. A BATCH

SEQUENTIAL

PROCEDURE.

In

this section a batch sequential procedure is proposed and shown to be asymptotically optimal. Let

M,,b

denote the number of observations sampled from population

P,

up tostage b.

Thenatstage bthe

Bayes

risk is minimizedby

K K

=! =I

andachieves its minimumwhen thefollowingequality^issatisfiedfor all and j

weproposeabatch sequentialprocedureas Withamotivationtomove

follows. First start withaninitialsampleof

K

observations,oneobservation from eachpopulation.

The remaining M-

K

observationswillbe allocated in

B

batches where each batch consistsof

K-

1 observations.

Suppose

uptobatch bwealready^observed

Ml,b,..., MK,b,

then select one additionalobservationfrom each populationexcluding population

P,

innext batch if

M,, +

^r,

> max{

Thefollowinglemmasareneeded for theproofofthe theorem.

LEMMA

4.1: Using the proposed batch sequentialprocedure,

M,,b

^{cx almost} surely as b-, for all i.

PROOF:The batch sequentialprocedurecanbewrittenasfollows: Sampleoneobservation from each populationexcluding

Pi

^if

(M,,, + ",)I

_S

[c, v/U,,M,.’-F ^[cj v"U.,M,,, >

(M,,b + ",)I ,

for allj

#

^i.

^Suppose

^that

M,,

^isbounded. Then Mj,, ^arebounded for all j

#

^i.

^Hence

^wehave

acontradiction.

LEMMA

4.2:

For

blarge

enough,

^let

k^(’)

sup{k <

^b"

Mi

IA-ri

> max{

ji

for all 1,..., K. Then

i

)<

150 K. REKAB

PROOF.

Ontheotherhand,

M,,b +

^r,

_< ^M,

^{k(,) q-r,q-1}

^M,,k(, +

^r,

Mj,b

+

^r

M,(, +

^r

M,,(, +

^r -t-

, _v/U,,M

<

^O)

+

¹

-,c.

THEOREM

4.1" Let r, be therisk incurredbythe proposed batchsequential procedure.

Suppose

thereexists p> 1 such that

E("(a,))

^p

<

^cxforall 1,...,K.

, ^E(E,(I

^c,

v/"(,)) )

+ o(1/M)

asM---+o.

PROOF: The proof of the theoremwill follow ifweestablish

and

K K

E(-lc, x/U,,M,) E(Ylc, V/"(,))

^{0 as}

M

^-+o

(4.1)

=1 =1

E{

_{,=, ,=,+}

^{((M, + ,)1 , v/U,,,,} (M, + r,)(M ^(M, + ⁺ r,) ^{,) I, v/U,,,)}

⁰

(4.2)

as

M

oo. Let

M,..., MK

^betheMlocation variablesforthe batchsequentiMprocedure. Since

M,

md

<,M,

^eiformly integrable,

(4.1)

^{issatisfied. Using}

Lemma ^(4.2),

M, "(,)

a.s.

Theproofwillbeestablished if

_M+r

u’+’

(c)2U,M,

^isuniforyintegrable. UsingLemma

(4.2),

c

M ⁺ r r

TheprooffollowsbyDoob’s nequMity fornonnegativesubmtingMes.

FENCES

[1] AIvo,

M. ^d Cabilio, P.

(1982).

^{Bayesiestimation of the difference betwn} two pro- portions. CaJ. Statist. I0,139-145.

[2]

b, K.

(1989).

Asymptotic efficiency in quentiM designs forestimation. SequentiM AnMysis 8, 269-280.

[3]

bbins,

H.,

Simons,

G.,

^d

Str,

N.

(1967). ^A

sequentiM

Moe

of the Behren-Fisher problem. A. Math. Stat 38, 1384-1388.

[4] Woodroofe, M.,

Hdwick,

J. (1990).

SequentiM MIocation for estimation problem with ethicMcosts. Ann. Stat 18, 1358-1377.