Asymptotic optimality of double sampling plans employing generalized regression estimators

Journal

of Statistical

Planning

and Inference

26 (1990) 173-183

173

North-Holland

Asymptotic optimality of double sampling plans employing generalized regression estimators Rahul

Mukerjee

Division

of Statistics,

Arijit

Chaudhuri

Computer Received

Science

Northern

Unit, Indian

18 February

Recommended

Abstract:

We consider

in addition,

generalized

observed

only

increasing

AMS

sampling

square

Subject

Institute,

manuscript

IL 60115, U.S.A.

Calcutta

received

a finite population

auxiliary regression

700 035, India

18 August

errors’

in two phases

size measures variate-values

estimators

for the selected

so as to attain

DeKalb,

1989

Rao

sizes and postulating

design-‘mean

University,

Statistical

using known

ascertained

proposed

Illinois

1988; revised

by J.N.K.

the first phase sample

terized

*

and the second for the initial

for the population

subsample.

Referring

super-population

for the estimators

sample.

probabilities,

Properties

sequences

lower bounds

are derived and optimal

therefrom

choosing utilizing,

are investigated

mean of a variable

to infinite

models,

with varying

phase subsample

of interest

of finite

populations

for asymptotic model-based

for

with values with

model-expected

designs are charac-

these bounds.

Classification:

Key words andphrases:

62DOS.

Asymptotic

analysis;

double sampling;

finite population;

optimal

design; regres-

sion estimator.

1. Introduction The literature on two-phase (or double) sampling allowing selection with varying probabilities in both the phases is not yet developed enough to give adequately general optimality results. Rao and Bellhouse (1978) considered selection of a first phase sample with varying probabilities using size measures, W, followed by that of a subsample therefrom. Based on values of an auxiliary variable, x, for the first and of a variable of interest, y, possibly with superimposed errors, for the second phase sample, they showed that for P, the population mean of y, a regression estimator * Now at the Indian

0378-3758/90/$03.50

Statistical

0

Institute,

1990-Elsevier

Calcutta,

India.

Science Publishers

B.V. (North-Holland)

174

R. Mukerjee, A. Chaudhuri / Optimality of double sampling plans

involving some unknown model parameters is optimal in a class of nonhomogeneous linear estimators. Chaudhuri and Adhikary (1983, 1985) obtained parameter-free estimators which are optimal in a narrow class which excludes the regression estimators. Here we consider a regression estimator appropriate for varying probability sampling in both the phases. Though parameter-free, it is design-biased. Therefore, following the current trend in the literature on survey sampling, we investigate and establish some of its asymptotic properties essentially along the line of Robinson and Sarndal (1983). The principal new feature of the present work is that it considers utilization of the x-values ascertained in the first phase sample in choosing the second phase sample for possible improvement in efficiency. With appropriate linear regression model formulations, we consider such wider classes of designs and characterize a subclass which is optimal in the sense of yielding the minimal limiting value for the model-expected design-‘mean square error’ of a parameter-free generalized regression estimator. The setting has been formulated in detail in the next section. It may be remarked that our results are based on a super-population model not identical with that in Rao and Bellhouse (1978) but in special cases with comparable designs, our strategy turns out asymptotically more efficient than theirs. For other works on two-phase sampling one important reference is Robinson (1985) who reported results on approximate optimal allocation, and for asymptotic studies relevant to the present one, we refer to Erdos and Renyi (1959), Hajek (1960), Rosen (1964, 1972), Brewer (1979), Sarndal (1980), Fuller and Isaki (1981), Isaki and Fuller (1982) and Li (1983) among others and Robinson and Sarndal (1983) in particular.

2. Notation

and preliminaries

Consider a sequence of finite populations such that its t-th member U, consists of Nt units labelled i = 1, . . . . N,, t = 1,2, . . . , and N1--+a as t+m. By lim (lim sup, lim inf) we mean limit (limit superior, limit inferior) as I+ 00. Also t(t) means a quantity which tends to zero as t+m. Let w (2 0), x, y denote respectively a size measure, an auxiliary variable and the variable of main interest. Let their respective values for the i-th unit (i = 1, . . . . Nt) of U, (t = 1,2, . ..) be wil (known for each i), Xi, and Yit. The corresponding totals (means) over U, are W,, X,, Y, (p,, xt, yt). Let wNt) and define X,, Y, similarly. Wt=(w,,, ***> From U, an initial sample sit (each possible sit with a fixed number ni, of distinct units) is supposed to be drawn with a probability p&it) which may involve the are supposed to be ascertained. Wit’s for ie U,. For the units in sit, the x-values From slf a subsample szt (say, each possible szf with a given number n2t (< nlr) of distinct units) is to be chosen with a conditional probability p2r(s2t) =p2t(~2t Isir) given that pl,(sl,)>O. These probabilities may involve Wit for iE Ul and also Xi, for 2ESI,. The overall two-phase (double) sample S, = @if, sZt) has the selection proba-

R. Mukerjee, A. Chaudhuri / Optimality of double sampling plans

175

bility p,(s,) =~&sr~)~~~(s~~ Isit). By pt (plr, pzr), we mean the design corresponding to this double (first phase, second phase) sampling plan. We will need the following: EP, (E,,,, Ep,,) =design expectation operator with respect to pt (plr, p2() - of course, Ep,, is a conditional expectation for each fixed slf with p,,(s,,)>O; I,;, (Z2;() = 1 if i~si, (So,), and = 0 otherwise; for i, je U,, ifj: nlit, n,i~,=inclusion probabilities of i and (i, j) according to plr; rrit= l/nr;,; for every slf with ~,~(sr,)>O, and i, jcs,, (i#j): x~&~,), n2;jt(s,,) = conditional inclusion probabilities of i and (i, j) according A,(.

to

1%);

~2&11)

=

l/~2i,@l,).

For simplicity, hereafter we shall write 712i,, T12ij/3 r2if for n2;fCSlt)~ TC2ijt(Slr)9 r2ilblr) respectively. We may emphasize that n21f, rC2ijlmay involve x,, for k E sit. For every sit with p&i,)>0 and every s2f (Csi,) with ~~~~~~~Isr,)>O, let er,(w)=

C

Wifrli19

iSS,,

e2Aw)

=

C Witrlirr2if. iES2,

Similarly define er,(x), e2t(x) and e2,(y) which are all available from survey Consider the following generalized regression estimator for rr,: T,=JY’MY)

-&(e&)

-ZG) -&(e2,(w)

-

w,)l,

data.

(2.1)

where R, = elf(x) -&el,(w) - W,), and Fiji (j = 1,2) are ascertainable quantities in terms of known w and sampled x, y values. Here &, free from y, depends only on xic, Wil, I esI1. This estimator is suggested by consideration of the following linear regression model, say M, for which the model-expectation (-variance) operator is written Em (V,) giving E,(yitIXi,)=PlX;t+P2W;f,

Em(Xir)=&Wit,

(2.2)

with p,, p2, & as unknown parameters (recall that wil for iE U, are known), cf. random permutation model of Rao and Bellhouse (1978). The estimator (2.1) can be further motivated as follows. Had xit been known for each i E U,, one could have used the regression estimator E =K

’ k2,W

-B&2,(4

-x,)

- B22t(e21(w)

-

WA1

as suggested by the model (2.2). In the present set-up, however, the xir’s are known only for ies,, and, therefore, we consider the estimator Tt which can be obtained from Tt replacing X, by R, which is a regression estimator for X, based on sit. The model (2.2) will not remain meaningful and realistic if the data on w are not available, i.e., if w= 1. But we intend to confine to the situation when w is not so allowing selection with varying probabilities in the first phase. Taking x nonstochastic, one may consider an alternative model which is also not unrealistic. But we treat here x and y on a similar footing except that it is less expensive to procure data on x leading to nl,>nzr.

R. Mukerjee, A. Chaudhuri / Optitnality of double sampling plans

176

We will also consider a more detailed model, Yt, X, the following in addition to (2.2): GCvit)%)=oi:,

J%(oi:)=~i:,

cm(YiO Yjf Ixit, xjt)

= O9

Cm(Xit9

say Mi,

so as to postulate

about

K7(%)=fi;“, xjt) =

O

(Vifj,

i, je U,),

C, being the model-covariance operator. In the above, crz may involve Xit and wit, while hz, _f;; may involve Wit. The model Mi is a natural generalization of the one in Robinson and Sarndal (1983) in the present context. Hereafter, q will denote the probability distribution of (( Yt, X,): t = 1,2, . . . } under the appropriate superpopulation model. The main result of this paper has been presented in Section 4. In passing, some properties of T,, as in (2.1), have been discussed briefly in the next section. The results have been derived under suitable assumptions which will often be appropriate in practice (see, e.g., Robinson and Sarndal (1983)).

3. Unbiasedness

and consistency

For the sake of notational simplicity, from this section drop the subscript t, except in the Appendix.

onwards

we shall often

Definition. (a) T is asymptotically design-unbiased for Y if lim E,,[(T- F)I Y] = 0 with q-probability one, (b) T is consistent for P if lim Prob(j T- PI > 6 1Y) = 0 with q-probability one for every preassigned positive 6. In (b) above, Prob(. 1Y) denotes the probability calculated with respect to the probability distributionp over s on which Tis based. Then one obtains the following theorems. The proofs of these theorems follow essentially along the line of Robinson and Sarndal (1983), but with more tedious algebra, and hence omitted here. Theorem

3.1.

are such that E,(~jjXi, iesl)=pj (j=l,2) and P) = 0 for all s with p(s)>O, i.e., T is model-unbiased

If p’ (j=1,2,3)

E,,,(&) =&, then E,,,(Tfor F, under model A4.

Theorem 3.2. T is consistent as weN as asymptotically

design-unbiased for P under the following assumptions (which hold certainly or with v-probability unity according to situations which are obvious) and model M: (Al)

(i) lim sup C wF/N<

03, (ii) lim sup C xf/N<

03, (iii) lim sup C yf/N<

where C represents summation over i = 1, . . . , N, (A2) lim sup E,(P^f+~~+fi~[ Y)
, 712;;

03,

R. Mukerjee,

A. Chaudhuri

111

/ Optimality of double sampling plans

(A6) lim SUP U2=0, where 112=SLlpxSUp,, (A7) lim inf Nnr-’ mini rcrj>O.

maxi+jEs,

11;

1712ijr2ir2j-

At this stage it would be helpful to indicate the extent to which the above assumptions as well as those made in the next section in the context of Theorem 4.1 hold in a simple case. To that effect, suppose (a) there exist positive constants Cr, C2, Cs, C,, Cs, free from t, such that C, I xi I C2 Vi, 0
Azxiwis/[

(

iL2xf)(

2,

wf)]

5f?2

Cc113

vs29

with q-probability 1, where e2 is free from t. The condition (d) eliminates any difficulty arising out of possible in the estimation of p,,p2 on the basis of s2. One can take

multicolinearity

(3. la)

(3.lb)

where L= Cies2xfv fLy= Cieszx.y. and so on. The expressions in (3.la, b) are well-defined by (a), (d) above. By (c), the optimal design in Theorem 4.1 below is given by zli=nl/n Vi, and 7r2i=rz2/~l ViEsl wherep,(s,)>O, so that we can consider simple random sampling without replacement at both the phases. Then I

nri=nr/N

Vi,

7C2ixT12/nl Vies,,

,,

Xlij=nl(nl7T2;jzt12(n2-

l)/{N(Nl)/{nr(nr

l)}

Vi#j, - 1))

Vi#jEs,.

(3.2)

We shall verify the assumptions under the conditions (a)-(d) with the ~j (j = 1,2,3) and the sampling design specified by (3.la, b) and (3.2), respectively. It is easily seen that under the model M, the Bj (j= 1,2,3), as in (3.la, b) satisfy the assumption in Theorem 3.1. As for the assumptions in Theorem 3.2, (Al) follows from (a), (A2) follows from (a), (d) and (3.la, b), and (A3)-(A7) are immediate consequences of (b) and (3.2). Similarly, it can be verified that the additional assumptions (A8)-(AlO) made in the context of Theorem 4.1 also hold.

4. Optimal designs We will postulate the model M, now and use the expectation operator E = E,,,Ep and while using it remember that Em operates in two steps, keeping x fixed and

178

R. Mukerjee,

A. Chaudhuri

/ Optimality of double sampling plans

then over variation over x so that we may write E, = EmXE,,,lX.Also EP = EP,E,,,. Further, we will remember and make use of commutativity of these expectation operators

whenever

Theorem

4.1.

valid.

Under model M,,

n,E,,,E,(T-

p)2r V, +<(t), where r(t)-0

as

t+Oo, and

provided,

in addition to (Al)(i),

(A3)-(A7),

the following

assumptions hold:

with q-probability one, for some constants H,, H2, free from t; (A9) there exist constants k,, k2, free from t, such that h,?Ik,
fi2sk2
Vi;

n,ny’a,>O, where a, is as in (AS). The above lower bound is attained provided nlj=nlfi/C~,f, a&‘for iE.sl when pl(s,)>O. ~2i=n2~ifj~1/CiGs, (AlO)

liminf

(i= 1, . . . . N) and

The proof of Theorem 4.1, which requires considerable algebraic manipulations and occasional use of Robinson and Sarndal’s (1983) work is sketched in the Appendix. In most practical situations, the optimal design leading to the attainment of the lower bound in the theorem will not involve any unknown model-parameter (e.g., when c$axflw,92, ffa wig’ with g,, g2, g3 known, say gl = g2 = g3 = 2) and hence will be feasible in terms of actual applications. Also observe that the optimal design is allowed to involve the x-values ascertained from the first phase sample, since, as noted earlier, (Si may involve xi. If one restricts to designs where pz(s21sI) is free from xi’s and hence so are Z2i’S then, under the assumptions made, it follows 4.1 that n,E,E,(TP)2 1 V2 + r(t), where

V2=N-2

ENE h;hj-n2 i#j=l

This

lower

n2hJ-‘/CiEs It is worth

along the line of proof

of Theorem

f i=l

bound is attained , h,f,-’ for ies, noting that

provided rrli=nl~f;:/C~,fi when pl(s,)>O.

(i= 1, . . . . N) and 7c2;=

Vz- V, = ~~~ (hihj-E,,(oioj))/N210, i#j=

1

since JEm,(oioj)j i h,hj Vi#j,

by the Cauchy-Schwarz

inequality.

This indicates

R. Mukerjee, A. Chaudhuri / Optimality of double sampling plans

the advantage

of allowing

gain in efficiency,

p2(sz1s1) to involve

vindicating

the efficacy

179

xi’s for i~s~ in deriving

of the set-up

Remark 4.1. Our strategy, namely the estimator sampling design as in Theorem 4.1, is not strictly

considered

additional

in this paper.

T as in (2.1) together with the comparable with that in Section

4 of Rao and Bellhouse (1978) as the underlying models are different. Our model is based on regression of y on x, w and of x on w while the model in Rao and Bellhouse (1978) is based on bivariate exchangeability of the pair (y/w,x/w). It may, however, be of some interest to compare the two strategies in a special case. To that effect, consider model M, with ai2,=hiz,=h2,

fif=f=,

Vi, t,

(4.1)

where h, f (> 0) are constants free from t. Then the optimal design in Theorem 4.1 is given by one with nlj=n,/N(i= 1, . . . . N), 7r2;= n2/n1 (ins,) and from Theorem 4.1, it is easy to see that for such a design, n,E,E,(TF)2 = V, + r(t), where t(t)-+0 as t-m and ~l=(N-n2)N-‘h2+n2n;1~;f2(N-nl)N-1.

The strategy estimator

in Section

4 of Rao

and

(4.2)

Bellhouse

(1978) involves

the use of the

together with a design for which nli=n,Wi/(N~) (i= 1, . . . . N), Ic2i=n2/n1 (iEsl) (here /3 is a constant involving the model parameters in the Rao and Bellhouse (1978) model). For such a strategy, under model Ml with (4.1), it is not hard to see that

+f2n2Np1

(n+n;l)~(/3-/3)=;~,

wi’

[ + w+;’

;!I w;‘-8:],

and denoting the above by I/*, it follows from (4.2) and the Cauchy-Schwarz inequality that V*? Vi, with strict inequality unless /3 = pi and the Wi’S (i = 1, . . . , N) are all equal. Thus under model Ml with (4.1), our strategy appears to be asymptotically superior to the one in Rao and Bellhouse (1978). In fact, it can be seen that this remains true even if the Rao and Bellhouse estimator is used with our optimal design in the present context, namely one with 7cri= n,/N (i= 1, . . . . N), 7r2;= n,/nr (ies]). It must, however, be clarified that this superiority of our strategy may not hold in general, in particular, it will be inferior to the Rao and Bellhouse (1978) strategy when compared under their model.

180

R. Mukerjee, A. Chaudhuri / Optimality of double sampling plans

Remark 4.2. In order to propose an estimator for the mean square error of T, we make the following additional assumptions: (i) there exist constants pi, &, p3 such that ni(/?, -f13)2~H: (< 03), and n2{(P1-&)2+(P2-P2)2}~H~ (O, where H:, H2* are positive constants free from t, (ii) lim Sup n, maxizj 11< 00, and lim sup n2a2 < m, where a2 is as in 17llijr,ir,j-

assumption (A6). Let V’=QE,(Tand let

P)2. Define

P=n2Np2

J,

@2i-

+

+

Zii=yi_p^rXi-~22Wi,

C C i#jes,

lkfi4r2i+

C @I;iES,

~2i=yi-(~2+/Ji/Js)~i

(~EsZ)

lldir,ir2i

(n2ijr2ir2j_l)ZliZljrlirljn2ij

CC

(nlijr,irlj-

1)Z2iZ2j711~j’7T~j’

i+jesz

1 .

Then we propose P as an estimator of V and under the assumptions (Al), (A3), (A4), (A5), (A7), (AlO), together with the assumptions (i), (ii) above, it can be shown that E,(P) - I/+0 as Woo. The details follow essentially along the line of proof of Theorem 4.1. The actual derivation is rather lengthy and hence omitted here to save space.

Appendix Proof of Theorem

4.1. It will be convenient

to write

(A.11 where Cjr= n$[2NF ‘bj,, and

bit= C ZdZmrz;t- l)rl;r(Y;r-Plx;t-P2W;t), bzt= C (Zlirrlit-l)(yit-plXif-/j2Wil), b3t =

C Umr,;t

bqt=

c

&t=

-(PI

-Su>

bet=

-(PI

-P^dP3

bt

=PI

b,,=h(P3

with

C representing

-

~)((PI

ZdZ2ifr2itC

lPd(P1

-~lt)xit+(P2-IS2t)~it),

Vctr,;t-

l)C%t-P3~ith

-B3J

C Uutrct43t)

- P^~r)xit+ (P2 - B22t)~;th

c

Vd+lif-

lb%

l)(xit-P3Wit), c

summation

(zlir’lir-

lb’%,

over i (i = 1, . . . . N,) throughout

the Appendix.

R. Mukerjee,

A. Chaudhuri

181

/ Optimality of double sampling plans

Now

i,C,,Cr2it-&P,,(& =wY2Ep,,Emx say, and by (A7), (A9), (AlO), lim sup L,,<

say, and by (A7), ulf= C

Uiitriit-

(A9),

llxit9

l12d1

l)=Lzt,

(A-3)

lim sup L2t~ k, lim sup [nt,/(N,

U2t=

%E&:t)

C hz(rlit-

= ~2tNt-2K,&((P,

-i&tht

+ U32 -~2tN2t12

+ t&)+0

as t+ co,

by (Al)(i), (A3), (A4), (AS), (A9), and proceeding

and making

and Sarndal

u,*,= C

ZIitrlitUzitr2it

4’

ZlifrliAz2itr2it-

C

use of (Al)(i),

-

(1983). Similarly

along

as t-co,

+ uG2))+0

and Sarndal

’ c Ulitrl;t - l)(xit -

P3

that (A.6) follows essentially through a conditional i ESPY} are held conditionally fixed at the initial step. IXit; based on (Al)(i), (A3), (A4), (A8) shows that

along

the line of Robinson

K&,(c~J

= n2K2P?

L7,~&k21imsup

argument A similar

as t-m,

and Sarndal

c @lit - 1).1? = LTt

say, and by (A7), (A9), limsup

wit)12 as Wm.

c (r,,-l)~~rH,k,/(N,m~nn,i,)-O

E,,,E,,(c~~)~H,H~N~~~E~,,(v~~)+O

(A.3

(1983). By (A3), (A8), (A9),

Note

proceeding

of

(A4)-(A9),

5 H2Ep,,4&Y

=H2NF2

along the line of proof defining

l)Wit,

the lines of Robinson EmEp,(&

(A.4)

llxit,

E,,,E,,(c&) I H2Ep,E,,,,(Nte2(u~2 again

&)I

-P^d2+(P2-~2t)21(dt+

IH~E,~,E,,(N~~~(u~~

1 in Robinson

mini Irlir)] < m. Defining

C V,itrlit-lN;t9

~~2tN,-2Wp,MP,

Theorem

(A.2)

=Llt9

00, since

E,Ep,(S> = MV2Epl,E,,tx[ C Vlitrl;t =n2,Ni2

1

lP?itd

[nrt/(Ntm:n.tit)]<-.

(A.6) in which argument

(A-7)

(1983). Next (A.8)

182

R. Mukerjee, A. Chaudhuri / Optimality of double sampling plans

Finally, E,E,,(c~~)IP:H,N;2E,,~(u~~)-t0

as 1-03,

(A.9)

by (AL)(i), (A3), (A4), (A@, and proceeding as in Robinson and Sarndal Let 1={1,2,..., 8}, J= { 1,2,7}, Jc =I- J. Then by (A.2)-(A.9), limsup Hence

EJZP,(cj:)<~

ifjEJ,

=0 ifjE,P.

for j E J, j’~ Jc, I~!$&,(c~~c~~~)~5 [E,E,,(cj:)E,Ep,(c~~)]‘“~O

Also, it is straightforward (A. l)-(A.9),

to see that for j,j’~J,

n,,-Q&,(T, where,

as usual,

- &)” = Lr, + as t+m.

c(t)+0

et =Nr2

[

C r&Z

L2t

+ -%t

as t+m. j#j’,

(A. IO)

+ T(t),

Let LI”,=L,,-Q,,

+ $yz

E,,$?,,(c~~c~~~)=O. Hence by

where

lQtlrk,[ (N,m~nn,,)l+~~~ln,l,r,,r,j~-I]-O

Cies,,

with equality

- 6)’ = L,* + it follows

n2;1=n2t,

&

r2idit+

6

‘(

L2t

+ L7t

+

(A.12)

5(t).

from the Cauchy-Schwarz

C

iES,,

inequality

that

rcPit)2Y

if and only if 7t2;t

Hence

as t-+03.

by (A.lO), GGJ&,(T,

Since

(A.ll)

.

(~r;jrrri~rrjt - l)Emx(oitoj,)]

L%,(~~Ed~,%11’2, by W), WV, (A%

Since IEm.x(oitajt)I 5

Hence

(1983).

= n2triiPiJ

by (A.2), L;c, +

C

iES,,

(A.3), L2t

(A.8),

rlitoif

(A.ll),

(A.13)

(i~$t).

one obtains

after some simplification,

+ L7t

rK2

z:z

Ernx(~;Pjt)

-

n2t

C

&I

+

n2,/S2

C

@la

-

l).AS]

.

(A. 14)

[

Since by the Cauchy-Schwarz if and only if 711it=n,,fit/Cfir (A. 12)-(A. 14).

inequality, C rljfxf 2 n, (i= l,..., N,), Theorem

‘( C fir)2,

4.1 now

with equality follows from

183

R. Mukerjee, A. Chaudhuri / Optimality of double sampling plans

Acknowledgement The authors are thankful constructive suggestions.

to the associate

editor

and a referee

for their highly

References Brewer,

K.R.W.

(1979).

A class of robust

sampling

designs

for large-scale

surveys.

J. Amer. Statist.

Assoc. 74, 911-915. Chaudhuri,

A. and A.K.

Adhikary

(1983). On optimality

of double

sampling

probabilities. J. Statist. Plann. Inference 8, 257-265. Chaudhuri, A. and A.K. Adhikary (1985). Some results on admissibility

strategies

in double

with varying

J. Statist.

sampling.

Plann. Inference 12, 199-202. Erdas,

P. and A. Renyi (1959). On the central

limit theorem

for samples

Publ.

from a finite population.

Math. Inst. Hung. Acad. Sci. 4, 49-61. Fuller,

W.A.

and C.T. Isaki (1981). Survey

design under

superpopulation

models.

In: D. Krewski,

Platek and J.N.K. Rao, Eds., Current Topics in Survey Sampling, pp. 199-226. Hajek, J. (1960). Limiting distributions in simple random sampling from a finite population.

R.

Publ.

Math. Inst. Hung. Acad. Sri. 5, 361-374. Isaki,

C.T.

and W.A.

Fuller

(1982).

Survey

designs

under

the regression

superpopulation

model.

J.

Amer. Statist. Assoc. 77, 89-96. Li, Ker-Chau

(1983).

Consistent,

asymptotically

efficient

strategies.

Technical

Report,

Purdue

Uni-

versity. Rao, J.N.K. random Robinson,

and D.R. Bellhouse permutation P.M.

(1978). Optimal

models.

estimator

of a finite population

mean under generalized

J. Statist. Plann. Inference 2, 125-141.

(1985). Approximate

optimal

allocation

in repeated

sampling

from a finite population.

J. Statist. Plann. Inference 11, 135-148. Robinson,

P.M. and C.E. SLrndal

in probability

sampling.

Rosen,

B. (1964). Limit theorems

Rostn,

B. (1972). Asymptotic

ment, Slrndal,

(1983). Asymptotic

properties

of the generalized

regression

estimator

Sankhyd Ser. B 45, 240-248. for sampling

theory

from

for successive

finite populations.

sampling

with varying

Ark. Mat. 5, 383-424. probabilities

without

replace-

I, II. Ann. Math. Staatist. 43, 373-397, 748-776. C.E. (1980). On n-inverse

67, 639-650.

weighting

versus BLU weighting

in probability

sampling.

Biometrika