AN ASYMPTOTIC OPTIMAL DESIGN

The problem of designing an experiment to estimate the product 
of the means of two normal populations is considered. A Bayesian 
approach is adopted in which the product of the means is estimated by 
its posterior mean. A fully sequential design is proposed and shown to be 
asymptotically optimal.


INTR, ODUCION
Given two independent populations P1 and P2 indexed by unknown parameters/9 and w respectively, it is wished to estimate the product 0w based on a fixed total number of observations N. The question is how to allocate the N observations between the two populations so as to minimize the expected squared error loss.For estimating the difference of two normal means, Srivastava (1970) and Robbins, Simons and Starr (1967) have proposed a class of sequential rules incorporating both a sampling scheme and a stopping rule which are asymptotically optimal.For estimating a linear function of mean vectors, Mukhopadhyay and   Liberman (1989) have proposed various two-stage and sequential procedures.Rekab (1989)   proposed a sequential procedure shown to be asymptotically optimal; however, its application is elusive.We are thus led into seeking an easily expressible procedure which is close to the optimal (in the sense of Bayes) risk.The procedure will be proposed in Section 2. Its form resembles in principle that of Robbins et al. (1967), where the setting is classical rather than Bayesian.
A major difficulty in designing a nonlinear experiment is that the performance of design depends on unknown parameters.To utilize the information fully the experiment must be conducted sequentially.The choice of the next design point is determined by the estimate 1Received: December, 1990.Revised: May, 1991.
of the unknown parameters based on the observations made to date.
Then it follows that ' is a canonical filtering.A sequential procedure will be a sequence of allocation rules {(N, AN)}N>I.
Let #j be the posterior mean of given X1,..., Xj that is "' + E=, x, for j = 1, Similarly let uj be the posterior mean of w given Yx,..., 1 ;that is 8/2 -" Ei=I Y/ j+s for j-1, Then 1 1 e, l& N(p,, n -tr x N(m, mk +'s )" The study proceeds until stage N (fixed).To simplify the notation, nN and mN are denoted by n and m.Now consider the problem of estimating the product 0w with squared error loss.It is well known that the Bayes risk is minimized by taking the posterior means as estimates.Then ()-E m+' + +"m + (+ )(' 4 S) Rekab (1990) derived the following result" E(I 0 + Io I) Td(P) >-N + r + s + o(1/N) (i) m, n + in probability as N --.+oo (ii) rn I01 '* 10-i-+ I i in probability as N N and um--(iii) #2 n m n are uniformly integrable.
In the next section an easily expressible procedure is proposed.
One way of solving the problem is to estimate the unknown parameters 0 and w at each stage.A procedure of this kind is referred to as the fully sequential procedure.To derive the fully sequential procedure 7)*, observe that for all k m: + s 1 } nk+r Since #n and um, are uniformly integrable martingales, one could minimize the above equality by setting n+r lum l" m+, toward the ideal value I,,kl one may define a sequential With a motivation to move nk+r procedure 79" as follows: Start by taking one observation from each population.Then at stage k + 1 choose Y if and X otherwise.As it was mentioned previously, one needs to show that the three conditions listed in Section 1 are verified.
Proof: To show condition (i), note that the proposed sequential procedure can be rewritten as follows: choose Y at stage k + 1 if (3.1) Suppose that nk is bounded.Then the right hand side of (3.1) is bounded.On the other hand, since k cx, the left hand side of (3.1) goes to infinity with probability one.Hence, we have a contradiction.By the same argument it follows that mk c almost surely as k +oo.To show condition (ii) let ' sup(/< ____m' +, < !',' } n,, + r + 1 Condition (ii) follows easily by letting k go to infinity, since un 0 and vm, w by the martingale convergence theorem.Since < +2, nk+r Vm,