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Estimation by Double SamplingAuthor(s): D. R. CoxSource: Biometrika, Vol. 39, No. 3/4 (Dec., 1952), pp. 217-227Published by: Oxford University Press on behalf of Biometrika TrustStable URL: http://www.jstor.org/stable/2334018Accessed: 01-03-2017 14:58 UTC

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VOLUME 39, PARTS 3 AND 4 1)ECEMBER 1952

ESTIMATION BY DOUBLE SAMPLING

BY D. R. COX

Statistical Laboratory, University of Camnbri(ge

1. INTRODUCTION

This paper is about the following problem: to estimate an unknown paramleter 0 with

assigned accuracy using as few observations as possible. The simplest case is to estimate

0 with given standard error. More generally we may requiire the estimate to have variance

a(0), some given function of 0; an important special case is a(O) = a02 corresponding to

a given fractional standard error Va. Another possibility is to estimate by a /100 confidence interval of predetermined form, for example, of given width. Some practical situations in

which these problems are relevant are discussed briefly in ? 6.

In general it is impossible to get an estimate with the re(quired properties by taking

a sample of some fixed size. The number of observations miist depen(l in sonme way on the

observations themselves, i.e. some formn of sequiential sainpling mnust be used. Usually, however, it is extremely difficult to construict a sequential sampling sclhemne leading to an

estimate with the requiired properties, although Anscombe (1949l), 1952) has given a general large sameple theory. Another disadvantage of ordinary sequential procedures is the step- by-step calculation involved; in many applications this precluides the ulse of any but the very simplest sequential methods.

Here the above problems are solved by double sampling. Trhe basic idea is familiar (see,

for example, Quienouille, 1950); a preliminary sample is uise(d to determnine how large the

total sample should be. Stein's (1945) elegant method for constructing a confidence interval

for a normal mean of assigned width and confidence coefficient is a special case where the

exact distribution theory is known. The present douible-sampling methods are different

from those used in industrial inspection, in that in the latter case the second sample, if taken, is of fixed size.

The theory developed below is a large sainple one, but in practice is likely to give reasonable approximations even with quite small samples. For example, we shall construct

an estimate of 0 with bias O(N-2) and variance a(H) [1 + O(N-2)], where N is the preliminary sample size and a(O) the assigned variance.

For any 0 the mean sample size for the double-sampling solution will be greater than the

corresponding mean sample size for the 'best' sequential procedure. However, we shall

show that the difference is likely to be small except when-the sample sizes are very small.

2. ESTIDIATION WITII GIVEN VARIAN(CE: ONE UN.hKNOWN PARAMEITER

2 1. Theory

Suppose that there is one unknown L)arameter 0 amid that we requiire to estimate it with variance equal to a given fuinction of(}. orhe inain application,s are to the estimation of Poisson and binomial means and of a normal mean when the population variance is known. We give

the large sample theory in which the assigned variance is small. lTo establish clearly the relative magnitll(le of tihe various ternis we shall consider a sequence 8A of )roblems letting the paramenter A ten(d to infinity. ANA is the )roblem; estimate 0 with variance a(0)/A. Biometrika 39 15


218 Estimation by double sampling

Assume that in random samples of any fixed size m we can construct an estimate t(m) of

0 such that

(i) t(mI') is an unbiased estimate of 0 with variance v(0)/m;

(ii) the skewness coefficient Yj of t(m) is asymptotically yl(9) mr- and Y2 of t("') is O(m-') as ni tends to infinity;

(iii) asymptotic means and slandard errors can be derived for a(t(01)), v(t(n,)) and combinations

of these functions by expansion in series. These assumptions are sufficiently general for the present applications. There is no

(lifficulty in considering estinmates with non-zero asymptotic skewness or kurtosis.

Now if & were kinown, a saimiple of size Ano(0) = Av(0)/a(6) would give the required accuracy. This suggests the following approximate sampling scheme:

(a) Take a random sample of size NA and let t1 be the estimate of 0 fromn it.

(b) Let no(t,) = v(t&)/a(t). (1)

(r) 'I'ake al seconid r al(tond samiiple of size Max [0, (no(tl) - N) A]. (If (no(tl) - N) A is not an integer take the nearest integer to it.) Let t2 be the estimate of 0 from the second sample.

(d) Put Nt?+(no(tl)- N) t2 if n0(t1)N, (') n0(ti)OY y0) tl if 0O(tY) < N.

In particular, if t("') is the samnple mean, t is the mean of the pooled sample. We shall say that the sampling scheme (a)-(d) is based on the estimate t(I'?).

That the estimate t has the required properties in large samples scarcely needs formal

proof. The proof is, however, outlined here in order to show the method of obtaining closer

approximations to the exact theory. The precise statement to be proved is that t lhas bias

O(A-1) and variance a(f) A-'[ l + O(A-1)].

Repeated application is made of the following property of conditional expectations (.see,

for example, Kolmnogoroff (1950)).

LEI.xM1A. Let N. Y be randomt, variables and let Ef Y I X) denote the conditional expectation

of' Y given X. Then ,[XYJ = E[XE( 1 X)J. (3)

To obtain the expectation of t we have

E(t) = (t - 7r) tNE,(tjm0(t1)) ? Kl[( -- Nm0(t1)) tj} + nE2(t1) (4)

where tiio(tj) = 1/n0(t1) c a(tl)/v(tl), (5)

IT = pro b ( n0(1) <l )A'1) and h, F2denote conditional expectations given that no(t,) N. Now by (iii) E[n0(tl)] - n0(0) and var [n0(t1)] = O(A '1). Thus if A' < no(0) and the distribution of 0o(tl) dies away exponentially in the lower tail nT = O(A--r) for all r. t If, in addition, n0(tj) is well behaved the possibility that no(tl) < N may be neglected in (4). In future, then, we assume that

(iv) N < no(0) and the distribution of rn0(t1) is sach that the event no(tl) < N may be ignored.

We now write (4) E(t) = NE[t1 rn0(t1)] + E[( 1 - Nmo(tj)) t2].

t T his is a mtuch stronger restult than is nee(ded.


D. R. Cox 219

By the lemma the second term is E[ I - ANto(fl)} E(t2 It,)] and E(t2 1 tl) = 0, since the second sample is independent of the first. Thus

E(t) - NE[t1mo(tj)] + 0-NOE[mo(tj)]. (6)

By (iii) E[t1 mo(t)] = 6mo(O) + O(A-1),

and E[mo(tj)] = mo(O) + O(A-1). Thuts E(t) = + O(A-1). (7)

Similarly, var (t) a(6) A-'[I + O(A-1)]. (8)

Equations (7) and (8) show that t has the required properties when the preliminary sample

size is large. Although for many purposes the above solution will be sufficiently accurate, it is worth while taking the analysis a step further in order to obtain closer approximations

to the exact sampling theory.

Consider the sampling procedure defined by steps (a)-(d) with no(tl) replaced by n(tl)

(lefined by n(tl) = no(tl) {I + b(t)/A}, (9)

where b(tl) is a function to be deternmined. Equation (6) now becomes

E(t) = NE[tn mo(tj) { -b(tj)/A}] + 0

- NE[mo(tj) {1 - b(tj)/A}] + O(A-2). (10)

But by (iii) E(t1mO(t1)) = 1MO()+ da2 (OmO(O))V O 2 o()+dO2 NA and there are similar formuilae for the other expectations in (10). After some reduction we get

E(t) = 0 + MIn(0) v(0) A-1 + O(A-2). (11)

ult t' t -/rno(t) v(t) A-' if N < n(ti).l ti. if N > n(t1).

Trhen ( 11) shows that t' has bias O(A-2).

WNe can derive var (t') by an analogouis method; the answer is

var (t') = a() A-' + A-2{2rn0(0) rnf,(0) fY(3) vi(0)

+ [mI(0) V(O)]2 + 2mO(0) m."(0) v2(0)

+ V2(6) n'(6)/2N - a (0) b(0)}I + O(A-3). (13)

We require t' to have as nearly as possible a v tariance a(0)/A. Therefore choose b(0) to make the second termn in (1:3) vanish; i.e. ptit

16(0) = n0o(f6) v (6') {-2?hP,(6) w,O'(0) 71(0) V-i(0)

M, '2(0) + 2ino(0) ma(0) + in"(0)/2N }. (14)

WVe canl suinmmarize the resuilt as follows:

(a) Take a preliminary samiple of size N.\A an(d let t, be the estimate of 0 from it. (b) Let n(t1) no(t1) {l +b(t)IAb- w-here no(1l) and b(tl) are (lefined by (1) and (14). (c) Take a second saimiple of Max [0, (n(t1) - N) A] and let t2 be the estimate of 0 from it. ((d) D)efine t by equation (2) an(d then t' by (12). Then tunder assumptions (i)-(iv) t' has

bias O(A-2) anid variance a(0) A-'[ I + O(A-2)].

Special cases of these forinulae are discussed in the next section.

15-2


220 Eistmation by double anmpling

2 2. Applications

In developing the above forinulae we considered a sequence of sampling schemes in which

as the parameter A tends to infinity the required variance a(0)/A tends to zero and the

preliminary samlple size NX tends to infinity. In any particular application we have one

particular variaince fuinction which must be small and one particular preliminary sample

size whichi must be large. In (liselissing applications we can therefore set A equal to unity without loss of generality.

Example 1. Esftimation of normnal mean 0 u'ith given standard error ai. This is a trivial case when the population variance o-2 is known and the estimator is the sample mean. We have

n0(o) = o-2a-1 and b(f) = 0. Thus the total sample size is constant. The procedure is

equivalent to taking a single sample of size the nearest integer to -2/a. This is the common-

sense solution and is optimum (?4).

Examlple 2. Estimation of normal mean 0 with given fractional standard error a,. The

poptilation variance is ag;ain supposed known. We have

a(f) = a02, no(0) = -2/(aO2) and 6(0) = 8a + -2/(N02).

Thus the total sa-mple size is

n(tl) = 2/(at2) + 8()2/t2 + (4/(Nat4),

where t1 is the ineani of the preliminary sample. The final estimate is t' =t( I -2a), where f is the mean of the combined sample. The preliminary sample size N should be chosen as

large as possible suibject to N r< ff2/(aO2), i.e. it is desirable to know an upper limit to I 0 1. The proceduire fails if 0 is very near to zero becauise the function ni0(0) = (72/(a02) has a singularity at 0 = 0, invalidating the expansions used in ?2 1. In practice if 0 is likely

to be equal or near to zero we should mo(lify a(0) so that a(O) is finite. This would be roughly

equivalent to truincating the procedure (15).

Exami,ple 3. Estimation of a binomi(al mean 0 with given fractional standard error ai. If the procedure is based on the sample proportion defective, we have

z(0)= a902, v(0) = 0(1--0), yl(J) = (_ 1-'2) [0(1 -_)]-i.

The total sample size is

n(fl) =(1tl)l(atl) + 3[tj(1 1-tj)]-1 + (aNtl)-', ( 16)

and the final estimate is t' = t - at( 1- t)-1. We require N < (1 - 0)/(aO). The procedure fails if 0 is very small for the same reason as Example 2. (16) gives the double-sampling solution corresponding to Haldaine's (1945) inverse binomial sampling.

3. ESTIMATION IN THE PRESENCE OF A NUISANCE PARAMETER

3.1. 7Theory

? 2 gave the theory of double-sample estimation when there is one unknownl parameter. A particular applicationi is to the estimation of a normal mean when the population variance is knowI. We now develop the correspondiiig results for use when the population variance is unknown. Suppose that in addition to the unknown parameter 0, which is to be estimated


1). Ri. Cox 221

with variance a(O)/A, there is an unknown nuisance parameter q6. Assume that in samples of any fixed size m, we can find estimates t(t) andf(lnI) of 0 and 5S such that

(i) t(n') andf(rn) are unbiased estimates and have variance 0/m and KOb2/m, where K is asymptotically constant;

(ii) tOm) and f(m) are uncorrelated and.t(m) has zero skewness and kurtosis;

(iii) if m is large, asymptotic means and standard errors can be developed for combinations of t(m), f(m) and a(t(rn)) by expansion in series.

The following sampling procedure can be justified by an argument exactly analogous to that in ? 9 1.

(a) Take a preliminary sample of size NA and let tl, fi be the estimates from it.

(b) Put n(t1,fJ) = a(t1)fj71 {I + b(t1,f1)/A}, (17)

where b(tI,f1) = 2a" (t1) + a(tl) +2fa(t (18) a(tl) 2Na(tl) N'(8

(c) Take a second sample of Max [0, (n(tI,f) - N) A] and let t2 be the estimate of 0 from it.

(d) Ptt t t=t1 if N > n(t1, f1) Nt1 + (n(t1fJ) - N) t2 if N < n(t1,f1), (19)

n(t1,f1)

aiufl t' = 1 if N > n(tl,f) | (20) t-a'(t)/A if N < n(tl,f1). J

Flren assuming as before that

(iv) the possibility that N > n(tJ,f1) can be ignored, we can shiow that t' has bias O(A-2) and varianIce a(O) A-[ (I + O(A--2)}.

3-2. Applications

As in ? 2 2 we set A = 1 in applications.

Exanple 4. Estimnation of a normal msbean with given standard error a*. Base the method on

the sample mean. Then 0 is the unknown population variance C2 and is estimated by the ordinary estimate of variance, i.e. we takef1 = s2. Then K = 2, a(e) = a and

n(t1,f) & + (21)

an(d the final estimate is the pooled sample mean t, which is easily shown to he exactly unbiased. Tlhe mean sample size is o-2(1 + 2/N)/a. Thus the effect. of not knowing (r is to inierease the sample size by the raitio (1 + 2/N). Examiple .5. Estimation of dili erence between two normnal means with given standardl error a

Let 0 be the (lifference between the means of two nornmal populations and o02 be twice the separate population variances, assumed equal. rhen (21) of Elxample 4 can be applied if we takef1 to be the sum of the sample estimates of variance. First take a preliminary sample of size N from each population and let fi be the sum of the ordinary sample estimates of varianice. Put

n(tl,fl) a (1+N)' (22)



and take a second sample of Max [0, n(t1,f) - N] from each population. The difference between the pooled sample means is the required estimate. If the population variances are not equal, the procedure is still valid if the factor (1 + N-1) in (22) is replaced by

{1 + 2(1 + r2) (1 + r)-2N-1},

where r = f (l)/f (2) iS the ratio of the sample variances in the preliminary samples. However, when the population variances are different, the expected sample size is reduced by dividing the second sample between the two populations in a way depending on r2. This possibility is not analysed here.

More complicated problems with several unknown parameters can be dealt with in

a similar way. Detailed formulae will not be given here. Examples of problems requiring the more general analysis are:

(i) To estimate the difference between, or the ratio of, two binomial proportions 01, 02, with a variance some function of 0, and 02.

(ii) To estimate the difference (01 - 02) between two normal means with a standard error a given percentage of 01.

(iii) To estimate the difference between two normal means with unequal population variances with optimum subdivision of the observations between the two populations.

4. EXPECTED SAMPLE SIZE

The aim of the double-sampling schemes is to produce estimates with assigned accuracy

from as few observations as possible. In this section we show that, except where the

preliminary sample size is small, the best double-sampling procedure has an expected sample size only slightly greater than that for the best sequential procedure.

We consider for simplicity the single-parameter problem of ? 2. From (9) the expected

sample size is E[An(tl)] = Ano(O) + b(0) no(0) + v(0) n"(0)/(2N) + o( l). (23)

To the first approximation the expected sample size is Ano(0) = Av(0)/a(0). It follows immediately that the scheme should be based on the estimate t(m) with minimum variance,

and that if instead an estimate of efficiency E is used, the mean sample size is increased in the ratio E-1.

Now Ano(0) is the fixed sample size that gives an estimate with the assigned variance. It is reasonable to expect, at least when t('tl) is a sufficient estimate for 0, that no sequential

scheme can require fewer than Ano(0) observations on the average. This can be proved formally as follows. Wolfowitz (1947) has generalized the Cramenr-Rao inequality by showing that for any sequential estimation procedure giving an ulnbiased estimate 0* of the parameter 0 of the distribution f(x, 0),

var (0*) > (EnE( lo-f)2}X (24)

where En is the expected sample size. Now for noriial, Poisson and binomnial distributions, the variance of the sample mean is v(0)/rn where

v(0) = E(3l?f).


D. R. Cox 223

Thus considering estimates for which var (0*) < a(0) A-1 we have for the above distributions

En > Av(O)/a(0) = Ano0Q). (25)

Equiation (25) is satisfied asymptotically for all distributions.

To summarize, the mean sample size in the best double-sampling scheme is (23), and in the best sequential scheme it satisfies the inequality (25).

Equation (24) depends on the assumption that the possibility that N > n(tl) may be ignored. If this assumption is not true the mean sample will be increased and it is

likely that the resulting estimate t' will be more accurate than is required (i.e. that (bias of t')2 +var (I') < a() A-'). N should be chosen as large as possible subject to the

cofl(lition that prob (N > n(tl)) should be small.

5. CONFIDENCE INTERVALS

So far we have considered point estimation problems in which the variance of the estimate

is requLired to be of given formi. We now consider estimation by confidence intervals. There are two cases. First, we may want to give a confidence interval for 0 at the end of one

of the sampling proceduires already considered. Secondly, the problem may be stated in terms of confidence intervals, i.e. we may want to estimate 0 by a /3 0? confidence interval of predetermined form.

We shall deal mainly with the first sort of problem. Suppose, in fact, that we have obtained aIi estiimiate t' after a sampling procedure designed to give a variance a(O)/A. If we

want a ( [ 00 - 2a) % confidence interval for 0, we let gqt be the soluition of

f e-it -_ = ex

ani(1 then (lehine f-, O+ by the e(luations

6L+g~ 9af(6L) = (26)

e _ g +ai(0+) = t'j

(- , 9, ) is then the required confidence interval. If an explicit solution is impossible, the e quations (26) are solved by successive approximation; the method is due to Bartlett (1937).

Example 6. Suppose that t' is the estimate obtained after the procedure of Example 2 fol

estimating a normal mean with given fractional standar(d error i. Then a((O) = "Q2 an(d

e(IuationIs (26) give for, say, the 95 % confidence interval

= t'(1 + 1-96ai)-1, f? = t'(1 -I 96aAl)-.

Trhe formulae (26) assuine that t' is normally distribuited. A refinement of the method

(lepends on evaluating the skewness yl, and kurtosis 72 of t' and making a correction for these based on the Cornish-Fisher expansion (Kendall, 1947). We shall not give general formulae here but will illustrate the method by a simple example.

Example 7. Confidence interval of given breadth for a normal mean, variance unknown.

Consider the construction of a confidence interval after the procedure of Example 4 for estimating a normal mean 0 with given standard error at. If t' is the final estimate, in this

case the sample mean, the (1 00 - 20c) % confidence interval is, from (26), (t'- g.af, t' + g.ai).


224 Estimation by double sarnplinq

Now it can be shown that for the distribution of t', y, is zero an(l 72 is 6N-1. Thus according to the Cornish-Fisher inversion of the Edgeworth expansion, the normal multiplier should

be replaced by gYa + (93- 3gx)/(4N) = g*, say. If we use the normal multiplier, the width of the confidence interval is 2gi,ai, and if we use the corrected multiplier, the width is 2g* ai. To solve Stein's problem of arranging that the confidence interval is of width A, we take '= A/(2ga) or A/(2g*). The corresponiding sample-size functions are fromn (21)

- I N) -(27) 42 N F +ra

or in the second case I2 + *2\ g 2 'i1 + 2N (28) 4s N 1 A2 L1 2Nj

In Stein's exact solution the corresponding sample size is 4,s? 2A-2t2 At' where t2, - is the

two-sided 2ax %0 point of the t distribution with N - 1 degrees of freedom.

Table 1. Estimation of norRmal nean by (100- 2ac) % confide nice intercval of exact and a)proximate solutions

Preliminary sample size 10 Prelirninary sample size 20

0/0 Error of (27) % Error of (28) Jo Error of (27) % Error of (28) 10 3*1 - 2*7 2-5 0.0 5 - 3.4 - 4.5 00 - 11 21 - 9.9 - 6.8 - 3*5 - 1*7 1 - 18-4 - 10-2 - 7*7 - 2*6

2 .-24-6 -13-2 - 10. -3.5

From the exact solution we can compute the percentage error of the approximate

formulae (27) and (28). This has been done in Table 1. Formlula (27) has a fairly small error evein when N is as small as ten, provided that a is not below 24 %. The correction for kurtosis nmakes a substantial improvement. These results suggest that the general approxi- mnate formulae (leveloped in ??2 1, 3-1 will be reasonably acclurate except when N is v-ery small.

If at the end of one of the estimation procedures of ?? 2, 3 we require to test the hypothesis

IIo that 0- 00, we proceed in an analogouis way. If Ho is true t' is (listributed with mean

00 and variance a(00)/A. Hence if we assume t' to be normally distributed, the 2x %0 significance linmits are 00?y.a4(O)A-1. Corrections for skewness and kurtosis can be intro(luced if required.

6. DISCUSSION

There are two nmain situiations in which sequential mnethods are useful. In the first observations only become available at infrequent intervals and must be interpreted as soon as they are obtained. An example is the study of accident rates which may only be obtainable at weekly, monthly, etc., intervals. Double-sampling procedures are useless in problems like thiis. The second type of situation is where the numiber of observations is under the experi- inenter's control, buit observations are expensive, so that the smnaller thie sample size needed the better. t)ouble sampling is often practicable in this sort of problem.


D. R. Cox 225

Examples occur in the routine testing of textiles. For instance, from time to time it is required to compare the mean fibre lengths of two lots A and B of wool. There are three possible purposes for stich a comparison. First, we are very occasionally merely interested in deciding whether or not the two lengths are the same and are not concerned to estimate the magnitude of any difference that may exist. This is a pure problem of significance testing to which, for example, a Wald sequential test could be applied. Secondly, the problemii is sometimes to estimate the difference 0 with given accuracy or given percentage accuracy. This is a pure problem of estimation for which, for example, a double-sampling

procedure could be constructed. The third case is intermediate. Suppose that we want both to test whether 0 is zero and also to estimate the magnitude of any difference that may exist. Further, it sometimes happens that while we want a sensitive test of the null hypotlhesis ilo : (1 - 0, we are content with a comparatively inaccurate estimate when 0 is very different from zero. A Wald sequential test answers this problem provided that the test is supple- mented by a rule for estimating 0 when sampling stops (see Cox (19.5*2) for one such approximate rule). The resulting estimate will be very inaccurate if 0 is very different from

zero. Now we can restate the problem: to construict a scheme for testing Ho with given power near 0 = 0 and for estimating 0 with given accuracy when 0 is different from zero. (For example, if j 6 i > 0 we might require our estimate to have given standard error Va'.) This stuggests trying to solve this sort of problem by using a double-sampling scheme to estimate 0 with variance a(0), where a(0) is a function of the general form shown in Fig. 1, a" an(l the shape of the curve being adjusted to give approximately the required power near 0 = 0.

a(O)'lI

La'

0 ~~~0 ->

Fig. 1. Suggested formn for a(O).

A detailed study of tests of this type will not be given here, but the method will be illustrated by a simple example.

Exainp)le 8. Let ( be the unknown meain of a normal population of unit variance. Suppose that it is required

(i) to test the hypothesis HO: 0 = 0 and to reject H. at the 5 % lev-el with probability 0-975 whenf= ?+l

(ii) to obtain an unbiased estimate of 0 such that when 0 is very different from zero the estinmate has standard error 0.2.

(When I0 >1'0, 25) observations are needed to give the required standard error while abouit 612 observations are needed for a fixed-samiple-size test of the requiired power.)



Suppose that we take a(O) to be the inverted normal frequency curve

a(O) = 0-04- A (29) V(27T)~~~~~~~~(9

where ,u is a constant to be chosen. As shown in ? 5 the estimate t' will be significantly

different from Ho at the 5 % level if I t' I > 1 96a1(0). If this is to be the lower 9712 % point of the distribution of t' when 0 = we must have

1*96ai(0) = 0-5- 1-96ai(i), (30)

and this quadratic equation for ,u gives It = 0*06313. Now that a(8) is fixed we can work out from (12), (14) and (23) the correction for bias a'(8), the sample size function n(O) and

the expected sample size E,(n).

These functionst are given in Table 2, together with a(O) and no(O). The final specification of the sampling scheme is:

(a) take a preliminary sample of 25 and let t, be the sample mean; (b) take a second sample to make the total sample size n(tl); (c) let t be the mean of the pooled sample and t' = t -a'(t); (d) t' is an unbiased estimate of 0 and is tested for significance from 0 = 0 by referring

t'a-i(0) = t'/0 1217 to the normaltables; in particular, if I t' I > 0 2385, t' is significant at 5 %. n(O) is the sample-size function. -a'(a) is the correction for bias. E0(n) is the expected

sample size. ai(0) is the standard error of the estimate. no(0) is the crude first approximation to n(0).

Table 2. Function8 a8sociated with the double sampling scheme

6 n(O) a'(0) Ed(n) ai(O) n,(9) 0.0 73.2 0.000 72.0 0*122 67.5 0.1 72-5 0*002 71.4 0*122 66.9 0)2 70-5 0*005 69.6 0.124 65.3 0*3 67-5 0*007 66.9 0*126 62.8 04 63-7 0*009 63.4 0.129 59.7 0.5 59-6 0011 59*5 0.133 56.3 0 6 55-3 0-013 55-4 0-138 52-7 0-8 47-4 0015 47*7 0.147 46.1 1.0 40*8 0*015 41.1 0.157 404 1-5 30 7 0-012 30 9 0.178 31.4 2-0 26-6 0*007 26*7 0 191 27.3 2*5 25-4 0003 25.4 0*197 25.7

The advantages and disadvantages of the present procedure are as follows.

Advantages. As compared with a fixed sample size test there is an appreciable saving in

observations if 0 is very different from zero. We can give a test of Ho at any required significance level and are not committed to a single level as in the Wald test. Once the prelimninary computation of n(0) and a'(0) has been done, the test requires only one inter-

mediate calculation. There is a definite upper bound (73) to the sample size.

Disadvantages. There is a slight increase in the expected sample size when Ho is true and only a trivial reduction when 0 = j; this compares with the substantial reduction in expected sample size in the Wald test.

t The second-order corrections developed in ?2 are small, so that it is likely that the scheme has the required properties to a high degree of accuracy.


D. R. Cox 227

To sum up, the Wald test and the sort of double-sampling test given here serve quite

different purposes. The first is appropriate when we want to make an irrevocable decision,

such as, for example, in deciding whether to accept or to reject a batch of articles. The

second may be useful when a fairly accurate estimate of the unknown parameter must

always be made after the test is complete.

SUMMARY

Double-sampling methods are developed for estimating an unknown parameter 0 so that

the variance of the estimnate is some function of 0 given in advance. Applications are made

to the estimation of normal and binomial means with given standard error or given fractional

standard error, and to the construction of a new sort of sequential test.

REFERENCES

ANSCOMBE, F. J. (1949). Biometrika, 36, 455. ANSCOMBE, F. J. (1952). Proc. C(amlb. IPhil. Soc. 48, 600.

BARTLETT, M. S. (1937). J.R. Statist. Soc., Suppl. 4, 131. Cox, D. I. (1952). Proc. C(amb. Phil. Soc. 48, 447. HALDANE, J. B. S. (1945). Bionemtrika, 33, 222. KENDALL, M. G. (1947). Advanced T'heory of Statistics, 3rd ed., 1, ?6-3.3. London: Griffin. KOLMOGOROFF, A. N. (1950). Foun(lations of the Theory of Probability. New York: Chelsea Publisling

Co. English translatioin.

QLUENOUILLE, M. H. (1950). Introductory Statistics, ?3.10. Lon(don: Butterworth-Springer. STEIN, C. (1945). Ann. Math. Statist. 16, 243. \VOLFOWITZ, J. (1947). Ann. Math. Statist. 18, 215.


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