BY NORMAN STARR AND MICHAEL B. WOODROOFE. UNIVERSITY OF MICHIGAk. Communicated by Paul A. Smith, March 12, 1969. Abstract.-We consider ...
REMARKS ON SEQUENTIAL POINT ESTIMATION BY NORMAN STARR AND MICHAEL B. WOODROOFE UNIVERSITY OF
MICHIGAk
Communicated by Paul A. Smith, March 12, 1969
Abstract.-We consider utilizing a sequential experiment for estimating the mean of a normal distribution when the variance of the distribution is unknown. For a suitable choice of the design constants of the experiment and for loss structures of practical interest, it is shown that the difference in the expected loss with optional stopping and the expected loss which would be incurred if a were known is a bounded function of a. 1. Introduction.-H. Robbins' considered the problem of estimating the mean of a normal distribution and suggested a sequential procedure that Starr2 subsequently studied in some detail. Since this procedure is easy to use in practice and applies to a rather frequently encountered statistical problem, results relating to its performance should have considerable practical importance. 2. The Problem.-The results of this section are given in reference 2, and we refer the reader to that paper for proofs. Let x1,x2, . . . be independent N(yU,)2) random variables;n and, for given s > 0, suppose the loss incurred when we estimate j by tn = 1/n E xs on the basis of a sample of size n > 0 is defined by n=1
Ln = A n For fixed n the resulting risk is
vn(cr)
=
I
(A > 0).
+n
E(Ln) = -Ko'n-12 + n,
(1)
where the constant K = 2(a-1)/2 Ar(s + l/2)7r-12 depends on the choice of As only. Define the constant , - K2"$; then, treating n > 0 as a continuous variable, the risk (1) is minimized by taking n to be no, where by definition (0 < a < cx). n= (2)8/(s+2) (2) Replacing this optimal choice of n in (1), the minimum risk is easily seen to be
v(a)
=
= v.()
(3)
+ 1 no.
When a is unknown, a poor choice of n in relation to a in advance of experimentation will magnify (1). Consequently, we consider, as in references 1 and 2, determining a random sample size N by means of the following sequential procedure. Define n
s 2
=
(x-z2 EX, 1/n -1 i21 285
n >
2,
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PROC. N. A. S.
and let
N = least integer n > m for which n > (#8 2)s/(8+2) where the starting sample size m > 2 is a given integer. The risk i when the sample size N is sequentially determined can be shown to be = E(LN)
2
=
5
,n0 (s+2) 12E(N-s/2) + EN.
(4)
As possible measures of the usefulness of this procedure, we define for each a, 0 < of < c, the risk efficiency
P(CT) v(X) and the regret
w~o~) = v(a)
(a).
-
It is known2 that s2 ~~s+ 2+
I1 lim()=
1 + c
+
m
s
a- --O.c
a0
0 depends on A,s. Thus the procedure is asymptotically risk-efficient, provided that the starting sample size m is chosen so that m > S2/ (S + 2) + 1. Moreover, Robbins' computations' suggest that the procedure should be satisfactory for all cr. We shall here support that thesis by proving that the regret suffered in using the sequential procedure in ignorance of a is uniformly bounded, provided that m > s + 1. 3. Results.-We begin by stating two useful lemmas. LEMMA 1. P(N = m) = Oe(m-'+l), where O denotes exact order. LEMMA 2. For fixed 0, 0 < 0 < 1, P(m < N < Ono) = O(-m). COROLLARY 1. P(N < Ono) = O(-m+n ). The lemmas were proved by Starr2 and have been rederived by G. Simons' for a related problem. THEOREM.
As
a
-X,
0(1)
(6)
+ 1.
(7)
if and only if m >
8
MATHEMATICS: STARR AND WOODROOFE
VOL. 63, 1969
287
Proof: For simplicity in what follows, we let c with or without affixes denote a positive constant that does not depend on a and is not necessarily the same from one usage to the next. From (3) and (4)
cw(o) = - no(8+2) /2E(N-1/2- no- /2) + E(N - no). 8
(8)
For fixed t > 0 t
Nt-nO-t =
+1(N-no) +
t (t +1)
2
(N -no)2 nt+2
(9)
where ni is an intermediate value of no,N. Setting t = s/2, we obtain from (8)
a(0) = (S + 2) no (8+2)/2E[nC-(8+4)/2(N
-
) 2]
= Oe(S)E[nli(8+4)/2(N - no)2].
Since it can be verified from (9) that on the event {N = m} (no > m), cno4/(8+4) the necessity of (7) for (6) follows from Lemma 1 by ni
@(a)
>
()
cno4/(4+) (no 2 m). Thus,
W()
=
-
Oe(f' [fe nl1(8+4)/2 (N N ~~~Ono
< Oe(a8){cP(N < Ono) + O(f-)E[
no)dP]
n
= 0(or8+1-m) + c' E [(NonO)2]
Hence, the sufficiency of (7) follows from LEMMA 3. As ao-co 0(1). E O(N-n)2 no
(10)
Proof of the lemma: It suffices to consider no > 10. Integrating by parts, we have
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MATHEMA TICS: STARR AND WOODROOFE
PROC. N. A. S.
E[(N no1 no)] < 1 + 2 XP(N -no < - X Vno)dX + 2fJ1AXP(N-no > X no)dX.
(11)
The first integral in (9) does not exceed the sum of noP(N < 1/2no) and 21/2Vno
< 2 JXP
{8k
{
( no2X 2 V s + 1, noP(N < 1/2no) is bounded by Corollary 1, while no- (8/8+2) El|Sn2I 021 4 < co- (8/8+2) n2208 < CInoo- (8/8+4) o.8 = Cf-8..8...C is bounded in any case. Thus, we have shown that the first integral in (11) is bounded if m > s + 1, and a similar (somewhat simpler) argument will establish the boundedness of the second, proving (10). Remark: Robbins' computations' suggest that 0(1) in (6) is small, and useful estimates of this constant depending on m,o- could perhaps be found by refining the argument given above. -
1 Robbins, H., "Sequential estimation of the mean of a normal population," in Probability and Statistics, ed. Harold Cram6r (Uppsala, Sweden: Almqvist and Wiksell, 1959), pp. 235245. 2 Starr, N., "On the asymptotic efficiency of a sequential procedure for estimating the mean," Ann. Math. Statist., 37, 1173-1185 (1966). 8 Simons, G., "On the cost of not knowing the variance when making a fixed-width confidence interval for the mean," Ann. Math. Statist., 39, 1946-1952 (1968). 4Doob, J. L., Stochastic Processes (New York: John Wiley & Sons, 1953), p. 34. 6 Starr, N., and M. B. Woodroofe, "Remarks on a stopping time," these PROCEEDINGS, 61, 1215 (1968).
