Read e-book online Survey Sampling: Theory and Methods PDF

By Federico Rosei, Tudor Johnston

ISBN-10: 1860946402

ISBN-13: 9781860946400

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Y N ) and bsi = 0 for i ∈ / s. Then, the mean square error may be written as M p (t) = E p N Y i (bsi − 1) 2 N = YiY j dij i=1 j =1 with d i j = E p (bsi − 1) (bsj − 1). Let ( p, t) be representative with respect to a given vector X = ( X 1 , . . , X N ) , X i > 0 , i ∈ U . Then, writing Zi = Yi Xi we get M p (t) = Zi Z j ( X i X j d i j ) such that X i X j d i j = 0. i j Define ai j = X i X j d i j . cls 16 dk2429˙ch02 January 27, 2005 11:19 Chaudhuri and Stenger is a non-negative quadratic form in Zi ; i = 1, .

2 Hansen–Hurwitz Strategy For the HA NSEN–HURWITZ estimator t2 , which is unbiased for Y , when based on PPSWR sampling, the variance is well known to be V2 = M = N 1 n = 1 n = 1 n Y i2 −Y2 Pi 1 Yi −Y Pi Pi Pi P j i< j 2 Yi Yj − Pi Pj 2 admitting a well-known non-negative estimator v2 = = 1 n2 (n − 1) 1 n(n − 1) r

1) i∈s with Qi > 0; i = 1, 2, . . , N to be determined; note that asi = / s. 2 Minimizing Eq. 1) subject to the calibration equation bsi xi = x leads to N t˜ = bsi Y i i=1 N =  asi Y i + x − i=1 N   asi xi   i=1 N −1 Qi xi xi  i=1 N Qi xi Y i . 2) PROOF : Consider the Lagrange function N  (bsi − asi ) 2 Qi − 2 · λ  i=1 N  bsi xi − x  i=1 with partial derivative ∂/∂bsi 2(bsi − asi )/Qi − 2λ xi where λ = (λ1 , . . , λk ) is a vector of Lagrange factors. cls 32 dk2429˙ch02 January 27, 2005 11:19 Chaudhuri and Stenger leading to N Qi λ xi + asi xi = x i=1  λ = x − N   asi xi   i=1 N −1 Qi xi xi  .

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Survey Sampling: Theory and Methods by Federico Rosei, Tudor Johnston

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