varofsamplevar

Comparison with without replacement samples here we

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Unformatted text preview: eplacement samples given in 1]. Let V arwo S 2 denote the variance of variance of without-replacement samples of size n from A. The following is a simpli ed (improved) version from 1]. ; V arwo S 2 = c1 where 4 + c3 2 2 (N ; n ( ; 1) c1 = n (N ; 1) (N);Nn(; N ; nN ; 1) n ; 3) N ; 2) ( N (N ; n) N 2 n ; 3 n ; 3 N 2 + 6 N ; 3 c3 = ; n (n ; 1) (N ; 1)2 (N ; 2) (N ; 3) 1292 (8) (9) Section on Survey Research Methods – JSM 2008 We note ; ; n nn;;31) 2 2 ;( = V ar S 2 ; ; ; ; as expected. The di erence of V arwo S 2 and V ar S 2 is of order 1=N , that is, jV arwo S 2 ; V ar; S 2 j is O(N ;1 ). ; In most practical situations where n = cN for some c;> 0 and 0 < ; < 1, jV arwo S 2 ; V ar S 2 j is O(n; 1 ). p For example, if n = N , then the di erence of V arwo S 2 and V ar S 2 is O(n;2 ). As we did for V ar(S 2 ), we ; represent V arwo S 2 in terms of the moments 02 and 04 about zero by substitution of (7) into (8). lim V arwo S 2 N !1 ; V arwo S 2 = c1 0 4 1 =n + c2 0 3 4 + c3 02 2 + c4 20 2 + c5 4 (10) where c1 and c3 are as before (9) and (N ; n ( ; 1) c2 = ;4 n (N ; 1) (N);Nn(; N ; nN ; 1) n 3) N ; 2) ( N 2 (N ; n) (2 Nn ; 3 N ; 3 n + 3) c4 = 4 n (n ; 1) (N ; 1)2 (N ; 2) (N ; 3) 2 c5 = ;2 N (N ; n) (2 Nn2 ; 3 N ; 3 n + 3) n (n ; 1) (N ; 1) (N ; 2) (N ; 3) Here again, each ci converges to the corresponding coe cient in (7). 4 n ; 3) lim c2 = ; n lim c = 4(2n ; 1) N !1 4 n( lim c = ; n2n ; 3 N !1 5 (n ; 1) N !1 4. Acknowledgment The views expressed in this paper are those of the authors and do not necessarily re ect the p...
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