Auatral. J . Statist., 9 (2), 1967, 65-66

ESTIMATORS FOR THE COMPARISON OF SAMPLING DESIGNS1

S . JOHN Australian National University, Canberra

This note records some formulae for estimating from a sampIs drawn according to one sampling design the precision of unbiased linear estimators of the population total in alternative sampling designs. The case in which the two designs coincide is already considered in the literature ; see, for example, Election 12 of Godambe (1965) and papers cited therein.

Following Godambe (1965), we use u to denote a typical element. of the population and U to denote the universe of elements 2c. The letter s will denote a typical sample (collection of elements u) and 8 will denote the set of all samples. Let p(8 ) denote the probability ot 8 in the sampling design by wbich the sample was selected. We consider estimators of the form C p(s,u)a(u) for estimating !C= C m(u).

It will be aasumed that p(s,u)=O, if ub . A necessary and sufEcient condition for the above estimate to be unbiased is that C p(a,u)p(s)=l

UL8 uc l7

88s

for dl UEU. The estimate has then the variance I;C cuua(u)m(ut), u,u'cU

where c u u p = I; { ~ ( s , u ) - 1 } ( ~ ( s , u ' ) -l}p(8). It is desired to obtain

an unbiased estimator of the variance I;C C ~ ~ ~ O ( ~ ) O ( U ' ) of X p*(s,u)m(u),

which we assume is an unbiaeed estimator of 2' under the sampling design assigning probability p*(8) to 8 ;

8SS

u,u'eU U I B

Gu'= C {P*(s,u) -1}(p*(s,u') -l}p*(s). 8ZS

That C c;~*x~(w)/x(u) + XI; ~;u,m(u)m(u')/~(u,u'), U(0d U,U'CB,U ZU'

where ~ ( u ) = I; p(8 ) and x(u,u') =

verified, if we note that it is equal to

Z p(8) , is sucb an estimate is easily 63U t9{uru')

where t(s,u)=l, if UES and zero otherwise. If there exist y(u), UEU, satisfying the condition

1 Manuscript received 24th January, 1967.

86 8. JOHN

as is the case if the sample is drawn without replacement and is of fixed size, and the estimator is the Horvitz and Thompson (1952) estimator, viz. the sum of the m(u)’s in the sample, each divided by its probability of selection, so that y (u ) may be taken equal to the probability of selection of unit u,

zc c:u4u)N4’)= -4 EF 4U*{~(U)/?/(.u) -m(u’)/v(.u’))2y(z)y(u’) U,U’EU u,u EU

[see Godambe (1965)l and an unbiased estimator of it is

To see that this estimator is unbiased, note that it is equal to

To bring multi-stage sampling within the scope of our discussion we now consider the case where the m(u)’s are unknown, but, given s, uncorrelated unbiased estimates s(u)’s of the m(u)’s are available for U E ~ . It is desired to obtain an unbiaaed estimator of

;under a design that assigns probability p*(s) to 8. Since

.( 2 p*(S,U)$(U) UE8

where 02(u)=V($(u) 1 s ) , and

&,~2(u) /x (u) + U,U’E8,U #U’ ~;~~~(u).^(u’)/x(u,u‘)]

i t follows that c bu2(u)/x(u) + cc &(u).^(u’)/n(u,u’) + z {pe(s,4 - ~ : v / w ) ~ “ u ) ,

UEE U,U’M,U #UP

UZI

where ;2(u) is an unbiased estimator of 02(u), is an estimator of the desired kind. If y(u), U E U , exist such that 2 (p*(s,u)-l)y(u)=O,

UE u

References Godambe, V. P. (1985).

of ssmpling.” Rev. Idemu&. &a&&. Inat., 33, 242-258. Horvitz, D. G., and Thompson, D. J. (1952).

replacement from a finite universe.”

‘‘ A review of the contributions towards a unified theory

“ A generdisation of sampling without J . Amer. Statzkt. Aeeoc., 47, 883-886.