Variance estimation under monotone non-response for a panel survey
Section 3. Calibration and complex parameters

In most surveys, a calibration step is used to obtain adjusted weights which enable to improve the accuracy of total estimates. Such calibrated estimators are considered in Section 3.1. Also, more complex parameters than totals are frequently of interest, and a linearization step can be used for variance estimation. This is the purpose of Section 3.2. The estimation of complex parameters with calibrated weights is treated in Section 3.3. In each case, explicit formulas for variance estimation and simplified variance estimation are derived, and the bias of the simplified variance estimator is discussed.

3.1  Variance estimation for calibrated total estimators

Assume that a vector x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaa aa@33E3@ of auxiliary variables is available for any unit i s t , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGPbGaeyicI4Saam4CamaaBaaale aacaWG0baabeaakiaacYcaaaa@3715@ and that the vector of totals X MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGybaaaa@32A9@ on the population U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGvbaaaa@32A6@ is known. Then an additional calibration step (Deville and Särndal, 1992) is usually applied to Y ^ t . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaamiDaa qabaGccaGGUaaaaa@349B@ It consists in modifying the weights d t i = π i 1 ( p ^ i 1 t ) 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGKbWaaSbaaSqaaiaadshacaWGPb aabeaakiaai2dacqaHapaCdaqhaaWcbaGaamyAaaqaaiabgkHiTiaa igdaaaGcdaqadaqaaiqadchagaqcamaaDaaaleaacaWGPbaabaGaaG ymaiabgkziUkaadshaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiab gkHiTiaaigdaaaaaaa@434C@ to obtain calibrated weights w t i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG3bWaaSbaaSqaaiaadshacaWGPb aabeaaaaa@34DB@ which enable to match the real total X , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGybGaaiilaaaa@3359@ in the sense that

i s t w t i x i = X . ( 3.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaadaaeqbqabSqaaiaadMgacqGHiiIZca WGZbWaaSbaaWqaaiaadshaaeqaaaWcbeqdcqGHris5aOGaaGjbVlaa dEhadaWgaaWcbaGaamiDaiaadMgaaeqaaOGaamiEamaaBaaaleaaca WGPbaabeaakiaai2dacaWGybGaaGOlaiaaywW7caaMf8UaaGzbVlaa ywW7caaMf8UaaiikaiaaiodacaGGUaGaaGymaiaacMcaaaa@4D00@

The new calibrated weights are chosen to minimize a distance function with the original weights, while satisfying (3.1). This leads to the calibrated estimator

Y ^ w t = i s t w t i y i t . ( 3.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaam4Dai aadshaaeqaaOGaaGypamaaqafabeWcbaGaamyAaiabgIGiolaadoha daWgaaadbaGaamiDaaqabaaaleqaniabggHiLdGccaaMe8Uaam4Dam aaBaaaleaacaWG0bGaamyAaaqabaGccaWG5bWaaSbaaSqaaiaadMga caWG0baabeaakiaai6cacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVl aacIcacaaIZaGaaiOlaiaaikdacaGGPaaaaa@5037@

The estimated residual for the weighted regression of y i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgacaWG0b aabeaaaaa@34DD@ on x i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaa aa@33E3@ is denoted by

e i t = y i t b ^ t x i ( 3.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGLbWaaSbaaSqaaiaadMgacaWG0b aabeaakiaai2dacaWG5bWaaSbaaSqaaiaadMgacaWG0baabeaakiab gkHiTiqadkgagaqcamaaBaaaleaacaWG0baabeaakiaadIhadaWgaa WcbaGaamyAaaqabaGccaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaa cIcacaaIZaGaaiOlaiaaiodacaGGPaaaaa@4933@

with

b ^ t = ( i s t 1 π i p ^ i 1 t x i x i ) 1 i s t 1 π i p ^ i 1 t x i y i t . ( 3.4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGIbGbaKaadaWgaaWcbaGaamiDaa qabaGccaaI9aWaaeWaaeaadaaeqbqabSqaaiaadMgacqGHiiIZcaWG ZbWaaSbaaWqaaiaadshaaeqaaaWcbeqdcqGHris5aOWaaSaaaeaaca aIXaaabaGaeqiWda3aaSbaaSqaaiaadMgaaeqaaOGabmiCayaajaWa a0baaSqaaiaadMgaaeaacaaIXaGaaGPaVlabgkziUkaaykW7caWG0b aaaaaakiaadIhadaWgaaWcbaGaamyAaaqabaGccaWG4bWaa0baaSqa aiaadMgaaeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaG abbiab=rQivcaaaOGaayjkaiaawMcaamaaCaaaleqabaGaeyOeI0Ia aGymaaaakmaaqafabeWcbaGaamyAaiabgIGiolaadohadaWgaaadba GaamiDaaqabaaaleqaniabggHiLdGcdaWcaaqaaiaaigdaaeaacqaH apaCdaWgaaWcbaGaamyAaaqabaGcceWGWbGbaKaadaqhaaWcbaGaam yAaaqaaiaaigdacaaMc8UaeyOKH4QaaGPaVlaadshaaaaaaOGaamiE amaaBaaaleaacaWGPbaabeaakiaadMhadaWgaaWcbaGaamyAaiaads haaeqaaOGaaGOlaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiik aiaaiodacaGGUaGaaGinaiaacMcaaaa@7FB9@

Replacing in (2.11) the variable y i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgacaWG0b aabeaaaaa@34DD@ with e i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGLbWaaSbaaSqaaiaadMgacaWG0b aabeaaaaa@34C9@ yields the estimator of the variance due to the sampling design

V ^ t p ( Y ^ w t ) = i , j s t Δ i j π i j 1 p ^ i j 1 t e i t π i e j t π j . ( 3.5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaqhaaWcbaGaamiDaa qaaiaadchaaaGcdaqadaqaaiqadMfagaqcamaaBaaaleaacaWG3bGa amiDaaqabaaakiaawIcacaGLPaaacaaI9aWaaabuaeqaleaacaWGPb GaaGilaiaaysW7caWGQbGaeyicI4Saam4CamaaBaaameaacaWG0baa beaaaSqab0GaeyyeIuoakmaalaaabaGaeuiLdq0aaSbaaSqaaiaadM gacaWGQbaabeaaaOqaaiabec8aWnaaBaaaleaacaWGPbGaamOAaaqa baaaaOWaaSaaaeaacaaIXaaabaGabmiCayaajaWaa0baaSqaaiaadM gacaWGQbaabaGaaGymaiaaykW7cqGHsgIRcaaMc8UaamiDaaaaaaGc daWcaaqaaiaadwgadaWgaaWcbaGaamyAaiaadshaaeqaaaGcbaGaeq iWda3aaSbaaSqaaiaadMgaaeqaaaaakmaalaaabaGaamyzamaaBaaa leaacaWGQbGaamiDaaqabaaakeaacqaHapaCdaWgaaWcbaGaamOAaa qabaaaaOGaaGOlaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiik aiaaiodacaGGUaGaaGynaiaacMcaaaa@6E2A@

Similarly, replacing in (2.12) the variable y i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgacaWG0b aabeaaaaa@34DD@ with e i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGLbWaaSbaaSqaaiaadMgacaWG0b aabeaaaaa@34C9@ yields the estimator of the variance due to the non-response

V ^ t nr ( Y ^ w t ) = δ = 1 t i s t p ^ i δ ( 1 p ^ i δ ) p ^ i δ t ( e i t π i p ^ i 1 δ k i δ ( h ^ i δ ) γ ^ t e δ ) 2 ( 3.6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaqhaaWcbaGaamiDaa qaaiaab6gacaqGYbaaaOWaaeWaaeaaceWGzbGbaKaadaWgaaWcbaGa am4DaiaadshaaeqaaaGccaGLOaGaayzkaaGaaGypamaaqahabeWcba GaeqiTdqMaaGypaiaaigdaaeaacaWG0baaniabggHiLdGcdaaeqbqa bSqaaiaadMgacqGHiiIZcaWGZbWaaSbaaWqaaiaadshaaeqaaaWcbe qdcqGHris5aOWaaSaaaeaaceWGWbGbaKaadaqhaaWcbaGaamyAaaqa aiabes7aKbaakmaabmaabaGaaGymaiabgkHiTiqadchagaqcamaaDa aaleaacaWGPbaabaGaeqiTdqgaaaGccaGLOaGaayzkaaaabaGabmiC ayaajaWaa0baaSqaaiaadMgaaeaacqaH0oazcaaMc8UaeyOKH4QaaG PaVlaadshaaaaaaOWaaeWaaeaadaWcaaqaaiaadwgadaWgaaWcbaGa amyAaiaadshaaeqaaaGcbaGaeqiWda3aaSbaaSqaaiaadMgaaeqaaO GabmiCayaajaWaa0baaSqaaiaadMgaaeaacaaIXaGaaGPaVlabgkzi UkaaykW7cqaH0oazaaaaaOGaeyOeI0Iaam4AamaaDaaaleaacaWGPb aabaGaeqiTdqgaaOWaaeWaaeaaceWGObGbaKaadaqhaaWcbaGaamyA aaqaaiabes7aKbaaaOGaayjkaiaawMcaamaaCaaaleqabaWefv3ySL gznfgDOfdaryqr1ngBPrginfgDObYtUvgaiqqacqWFKksLaaGccuaH ZoWzgaqcamaaDaaaleaacaWG0bGaamyzaaqaaiabes7aKbaaaOGaay jkaiaawMcaamaaCaaaleqabaGaaGOmaaaakiaaywW7caaMf8UaaGzb VlaaywW7caGGOaGaaG4maiaac6cacaaI2aGaaiykaaaa@93E4@

γ ^ te δ = { i s t k i δ p ^ i δ ( 1 p ^ i δ ) p ^ i δt h ^ i δ ( h ^ i δ ) } 1 i s t 1 p ^ i δ p ^ i 1t h ^ i δ e it π i .(3.7) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaHZoWzgaqcamaaDaaaleaacaWG0b Gaamyzaaqaaiabes7aKbaakiaai2dadaGadaqaamaaqafabeWcbaGa amyAaiabgIGiolaadohadaWgaaadbaGaamiDaaqabaaaleqaniabgg HiLdGccaaMe8Uaam4AamaaDaaaleaacaWGPbaabaGaeqiTdqgaaOWa aSaaaeaaceWGWbGbaKaadaqhaaWcbaGaamyAaaqaaiabes7aKbaakm aabmaabaGaaGymaiabgkHiTiqadchagaqcamaaDaaaleaacaWGPbaa baGaeqiTdqgaaaGccaGLOaGaayzkaaaabaGabmiCayaajaWaa0baaS qaaiaadMgaaeaacqaH0oazcaaMc8UaeyOKH4QaaGPaVlaadshaaaaa aOGabmiAayaajaWaa0baaSqaaiaadMgaaeaacqaH0oazaaGcdaqada qaaiqadIgagaqcamaaDaaaleaacaWGPbaabaGaeqiTdqgaaaGccaGL OaGaayzkaaWaaWbaaSqabeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0 uy0Hgip5wzaGabbiab=rQivcaaaOGaay5Eaiaaw2haamaaCaaaleqa baGaeyOeI0IaaGymaaaakmaaqafabeWcbaGaamyAaiabgIGiolaado hadaWgaaadbaGaamiDaaqabaaaleqaniabggHiLdGcdaWcaaqaaiaa igdacqGHsislceWGWbGbaKaadaqhaaWcbaGaamyAaaqaaiabes7aKb aaaOqaaiqadchagaqcamaaDaaaleaacaWGPbaabaGaaGymaiaaykW7 cqGHsgIRcaaMc8UaamiDaaaaaaGccaaMe8UabmiAayaajaWaa0baaS qaaiaadMgaaeaacqaH0oazaaGccaaMe8+aaSaaaeaacaWGLbWaaSba aSqaaiaadMgacaWG0baabeaaaOqaaiabec8aWnaaBaaaleaacaWGPb aabeaaaaGccaaIUaGaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaI ZaGaaiOlaiaaiEdacaGGPaaaaa@9E0D@

The global variance estimator for Y ^ w t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaam4Dai aadshaaeqaaaaa@34DB@ is

V ^ t ( Y ^ w t ) = V ^ t p ( Y ^ w t ) + V ^ t nr ( Y ^ w t ) . ( 3.8 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaWgaaWcbaGaamiDaa qabaGcdaqadaqaaiqadMfagaqcamaaBaaaleaacaWG3bGaamiDaaqa baaakiaawIcacaGLPaaacaaI9aGabmOvayaajaWaa0baaSqaaiaads haaeaacaWGWbaaaOWaaeWaaeaaceWGzbGbaKaadaWgaaWcbaGaam4D aiaadshaaeqaaaGccaGLOaGaayzkaaGaey4kaSIabmOvayaajaWaa0 baaSqaaiaadshaaeaacaqGUbGaaeOCaaaakmaabmaabaGabmywayaa jaWaaSbaaSqaaiaadEhacaWG0baabeaaaOGaayjkaiaawMcaaiaai6 cacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaIZaGaaiOl aiaaiIdacaGGPaaaaa@568D@

The simplified estimator of the variance due to non-response is

V ^ t , simp nr ( Y ^ w t ) = i s t 1 p ^ i 1 t ( p ^ i 1 t ) 2 ( e i t π i ) 2 . ( 3.9 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaqhaaWcbaGaamiDai aaiYcacaaMe8Uaae4CaiaabMgacaqGTbGaaeiCaaqaaiaab6gacaqG YbaaaOWaaeWaaeaaceWGzbGbaKaadaWgaaWcbaGaam4Daiaadshaae qaaaGccaGLOaGaayzkaaGaaGypamaaqafabeWcbaGaamyAaiabgIGi olaadohadaWgaaadbaGaamiDaaqabaaaleqaniabggHiLdGcdaWcaa qaaiaaigdacqGHsislceWGWbGbaKaadaqhaaWcbaGaamyAaaqaaiaa igdacaaMc8UaeyOKH4QaaGPaVlaadshaaaaakeaadaqadaqaaiqadc hagaqcamaaDaaaleaacaWGPbaabaGaaGymaiaaykW7cqGHsgIRcaaM c8UaamiDaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaa GcdaqadaqaamaalaaabaGaamyzamaaBaaaleaacaWGPbGaamiDaaqa baaakeaacqaHapaCdaWgaaWcbaGaamyAaaqabaaaaaGccaGLOaGaay zkaaWaaWbaaSqabeaacaaIYaaaaOGaaGzaVlaai6cacaaMf8UaaGzb VlaaywW7caaMf8UaaGzbVlaacIcacaaIZaGaaiOlaiaaiMdacaGGPa aaaa@7403@

Here again, this simplified variance estimator ignores the prediction terms k i δ ( h ^ i δ ) γ ^ t e δ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGRbWaa0baaSqaaiaadMgaaeaacq aH0oazaaGcdaqadaqaaiqadIgagaqcamaaDaaaleaacaWGPbaabaGa eqiTdqgaaaGccaGLOaGaayzkaaWaaWbaaSqabeaatuuDJXwAK1uy0H wmaeHbfv3ySLgzG0uy0Hgip5wzaGabbiab=rQivcaakiqbeo7aNzaa jaWaa0baaSqaaiaadshacaWGLbaabaGaeqiTdqgaaOGaaiOlaaaa@4C9F@ If the underlying calibration model is appropriate, then the explanatory power of h ^ i δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGObGbaKaadaqhaaWcbaGaamyAaa qaaiabes7aKbaaaaa@3589@ for e i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGLbWaaSbaaSqaaiaadMgacaWG0b aabeaaaaa@34C9@ is expected to be small, as well as the bias of the simplified variance estimator. On the other hand, if there remains in e i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGLbWaaSbaaSqaaiaadMgacaWG0b aabeaaaaa@34C9@ some significant part of y i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgacaWG0b aabeaaaaa@34DD@ that may not been explained by x i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaO Gaaiilaaaa@349D@ the bias of the simplified variance estimator may be non-negligible. This may occur in case of domain estimation, when the calibration variables do not include any auxiliary information specific of the domain.

3.2  Variance estimation for complex parameters

We may be interested in estimating more complex parameters than totals. Suppose that the variable of interest y i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgacaWG0b aabeaaaaa@34DD@ is q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGXbGaaGPaVlabgkHiTaaa@353A@ multivariate, and that the parameter of interest is θ ( t ) = f { Y ( t ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH4oqCdaqadaqaaiaadshaaiaawI cacaGLPaaacaaI9aGaamOzamaacmaabaGaamywamaabmaabaGaamiD aaGaayjkaiaawMcaaaGaay5Eaiaaw2haaaaa@3D47@ with f ( ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGMbWaaeWaaeaacqGHflY1aiaawI cacaGLPaaaaaa@368A@ a known function. At time t , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG0bGaaiilaaaa@3375@ substituting Y ^ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaamiDaa qabaaaaa@33DF@ into θ ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH4oqCdaqadaqaaiaadshaaiaawI cacaGLPaaaaaa@3604@ yields the plug-in estimator θ ^ t = f ( Y ^ t ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaH4oqCgaqcamaaBaaaleaacaWG0b aabeaakiaai2dacaWGMbWaaeWaaeaaceWGzbGbaKaadaWgaaWcbaGa amiDaaqabaaakiaawIcacaGLPaaacaGGUaaaaa@3ACB@

The estimated linearized variable of θ ( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH4oqCdaqadaqaaiaadshaaiaawI cacaGLPaaaaaa@3604@ is

u i t = { f ( Y ^ t ) } y i t , ( 3.10 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgacaWG0b aabeaakiaai2dadaGadaqaaiaadAgadaahaaWcbeqaaKqzGfGamai2 gkdiIcaakmaabmaabaGabmywayaajaWaaSbaaSqaaiaadshaaeqaaa GccaGLOaGaayzkaaaacaGL7bGaayzFaaWaaWbaaSqabeaatuuDJXwA K1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGabbiab=rQivcaakiaayg W7caWG5bWaaSbaaSqaaiaadMgacaWG0baabeaakiaaiYcacaaMf8Ua aGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaIZaGaaiOlaiaaigdaca aIWaGaaiykaaaa@5D50@

with f ( Y ^ t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGMbWaaWbaaSqabeaajugybiadaI THYaIOaaGcdaqadaqaaiqadMfagaqcamaaBaaaleaacaWG0baabeaa aOGaayjkaiaawMcaaaaa@3A43@ the q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGXbGaaGPaVlabgkHiTaaa@353A@ vector of first derivatives of f MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGMbaaaa@32B7@ at point Y ^ t . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaamiDaa qabaGccaGGUaaaaa@349B@ Replacing in (2.11) the variable y i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgacaWG0b aabeaaaaa@34DD@ with u i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgacaWG0b aabeaaaaa@34D9@ yields the estimator of the variance due to the sampling design

V ^ t p ( θ ^ t ) = i , j s t Δ i j π i j 1 p ^ i j 1 t u i t π i u j t π j . ( 3.11 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaqhaaWcbaGaamiDaa qaaiaadchaaaGcdaqadaqaaiqbeI7aXzaajaWaaSbaaSqaaiaadsha aeqaaaGccaGLOaGaayzkaaGaaGypamaaqafabeWcbaGaamyAaiaaiY cacaaMe8UaamOAaiabgIGiolaadohadaWgaaadbaGaamiDaaqabaaa leqaniabggHiLdGcdaWcaaqaaiabfs5aenaaBaaaleaacaWGPbGaam OAaaqabaaakeaacqaHapaCdaWgaaWcbaGaamyAaiaadQgaaeqaaaaa kmaalaaabaGaaGymaaqaaiqadchagaqcamaaDaaaleaacaWGPbGaam OAaaqaaiaaigdacaaMc8UaeyOKH4QaaGPaVlaadshaaaaaaOWaaSaa aeaacaWG1bWaaSbaaSqaaiaadMgacaWG0baabeaaaOqaaiabec8aWn aaBaaaleaacaWGPbaabeaaaaGcdaWcaaqaaiaadwhadaWgaaWcbaGa amOAaiaadshaaeqaaaGcbaGaeqiWda3aaSbaaSqaaiaadQgaaeqaaa aakiaai6cacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaI ZaGaaiOlaiaaigdacaaIXaGaaiykaaaa@6EDD@

Similarly, replacing in (2.12) the variable y i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgacaWG0b aabeaaaaa@34DD@ with u i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgacaWG0b aabeaaaaa@34D9@ yields the estimator of the variance due to the non-response

V ^ t nr ( θ ^ t ) = δ = 1 t i s t p ^ i δ ( 1 p ^ i δ ) p ^ i δ t ( u i t π i p ^ i 1 δ k i δ ( h ^ i δ ) γ ^ t θ δ ) 2 ( 3.12 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaqhaaWcbaGaamiDaa qaaiaab6gacaqGYbaaaOWaaeWaaeaacuaH4oqCgaqcamaaBaaaleaa caWG0baabeaaaOGaayjkaiaawMcaaiaai2dadaaeWbqabSqaaiabes 7aKjaai2dacaaIXaaabaGaamiDaaqdcqGHris5aOGaaGPaVpaaqafa beWcbaGaamyAaiabgIGiolaadohadaWgaaadbaGaamiDaaqabaaale qaniabggHiLdGccaaMe8+aaSaaaeaaceWGWbGbaKaadaqhaaWcbaGa amyAaaqaaiabes7aKbaakmaabmaabaGaaGymaiabgkHiTiqadchaga qcamaaDaaaleaacaWGPbaabaGaeqiTdqgaaaGccaGLOaGaayzkaaaa baGabmiCayaajaWaa0baaSqaaiaadMgaaeaacqaH0oazcaaMc8Uaey OKH4QaaGPaVlaadshaaaaaaOWaaeWaaeaadaWcaaqaaiaadwhadaWg aaWcbaGaamyAaiaadshaaeqaaaGcbaGaeqiWda3aaSbaaSqaaiaadM gaaeqaaOGabmiCayaajaWaa0baaSqaaiaadMgaaeaacaaIXaGaaGPa VlabgkziUkaaykW7cqaH0oazaaaaaOGaeyOeI0Iaam4AamaaDaaale aacaWGPbaabaGaeqiTdqgaaOWaaeWaaeaaceWGObGbaKaadaqhaaWc baGaamyAaaqaaiabes7aKbaaaOGaayjkaiaawMcaamaaCaaaleqaba Wefv3ySLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiqqacqWFKksL aaGccaaMb8Uafq4SdCMbaKaadaqhaaWcbaGaamiDaiabeI7aXbqaai abes7aKbaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaakiaa ywW7caaMf8UaaGzbVlaaywW7caGGOaGaaG4maiaac6cacaaIXaGaaG OmaiaacMcaaaa@99F5@

γ ^ tθ δ = { i s t k i δ p ^ i δ ( 1 p ^ i δ ) p ^ i δt h ^ i δ ( h ^ i δ ) } 1 i s t 1 p ^ i δ p ^ i 1t h ^ i δ u it π i .(3.13) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaHZoWzgaqcamaaDaaaleaacaWG0b GaeqiUdehabaGaeqiTdqgaaOGaaGypamaacmaabaWaaabuaeqaleaa caWGPbGaeyicI4Saam4CamaaBaaameaacaWG0baabeaaaSqab0Gaey yeIuoakiaaysW7caWGRbWaa0baaSqaaiaadMgaaeaacqaH0oazaaGc daWcaaqaaiqadchagaqcamaaDaaaleaacaWGPbaabaGaeqiTdqgaaO WaaeWaaeaacaaIXaGaeyOeI0IabmiCayaajaWaa0baaSqaaiaadMga aeaacqaH0oazaaaakiaawIcacaGLPaaaaeaaceWGWbGbaKaadaqhaa WcbaGaamyAaaqaaiabes7aKjaaykW7cqGHsgIRcaaMc8UaamiDaaaa aaGcceWGObGbaKaadaqhaaWcbaGaamyAaaqaaiabes7aKbaakmaabm aabaGabmiAayaajaWaa0baaSqaaiaadMgaaeaacqaH0oazaaaakiaa wIcacaGLPaaadaahaaWcbeqaamrr1ngBPrwtHrhAXaqeguuDJXwAKb stHrhAG8KBLbaceeGae8hPIujaaaGccaGL7bGaayzFaaWaaWbaaSqa beaacqGHsislcaaIXaaaaOWaaabuaeqaleaacaWGPbGaeyicI4Saam 4CamaaBaaameaacaWG0baabeaaaSqab0GaeyyeIuoakiaaysW7daWc aaqaaiaaigdacqGHsislceWGWbGbaKaadaqhaaWcbaGaamyAaaqaai abes7aKbaaaOqaaiqadchagaqcamaaDaaaleaacaWGPbaabaGaaGym aiaaykW7cqGHsgIRcaaMc8UaamiDaaaaaaGccaaMe8UaaGjbVlaays W7ceWGObGbaKaadaqhaaWcbaGaamyAaaqaaiabes7aKbaakiaaysW7 caaMe8UaaGjbVpaalaaabaGaamyDamaaBaaaleaacaWGPbGaamiDaa qabaaakeaacqaHapaCdaWgaaWcbaGaamyAaaqabaaaaOGaaGOlaiaa ywW7caaMf8UaaGzbVlaaywW7caGGOaGaaG4maiaac6cacaaIXaGaaG 4maiaacMcaaaa@A761@

The global variance estimator for θ ^ t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaH4oqCgaqcamaaBaaaleaacaWG0b aabeaaaaa@34B7@ is

V ^ t ( θ ^ t ) = V ^ t p ( θ ^ t ) + V ^ t nr ( θ ^ t ) . ( 3.14 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaWgaaWcbaGaamiDaa qabaGcdaqadaqaaiqbeI7aXzaajaWaaSbaaSqaaiaadshaaeqaaaGc caGLOaGaayzkaaGaaGypaiqadAfagaqcamaaDaaaleaacaWG0baaba GaamiCaaaakmaabmaabaGafqiUdeNbaKaadaWgaaWcbaGaamiDaaqa baaakiaawIcacaGLPaaacqGHRaWkceWGwbGbaKaadaqhaaWcbaGaam iDaaqaaiaab6gacaqGYbaaaOWaaeWaaeaacuaH4oqCgaqcamaaBaaa leaacaWG0baabeaaaOGaayjkaiaawMcaaiaai6cacaaMf8UaaGzbVl aaywW7caaMf8UaaGzbVlaacIcacaaIZaGaaiOlaiaaigdacaaI0aGa aiykaaaa@56D8@

The simplified estimator of the variance due to non-response is

V ^ t , simp nr ( θ ^ t ) = i s t 1 p ^ i 1 t ( p ^ i 1 t ) 2 ( u i t π i ) 2 . ( 3.15 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaqhaaWcbaGaamiDai aaiYcacaaMe8Uaae4CaiaabMgacaqGTbGaaeiCaaqaaiaab6gacaqG YbaaaOWaaeWaaeaacuaH4oqCgaqcamaaBaaaleaacaWG0baabeaaaO GaayjkaiaawMcaaiaai2dadaaeqbqabSqaaiaadMgacqGHiiIZcaWG ZbWaaSbaaWqaaiaadshaaeqaaaWcbeqdcqGHris5aOGaaGjbVpaala aabaGaaGymaiabgkHiTiqadchagaqcamaaDaaaleaacaWGPbaabaGa aGymaiaaykW7cqGHsgIRcaaMc8UaamiDaaaaaOqaamaabmaabaGabm iCayaajaWaa0baaSqaaiaadMgaaeaacaaIXaGaaGPaVlabgkziUkaa ykW7caWG0baaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaa aakiaaysW7daqadaqaamaalaaabaGaamyDamaaBaaaleaacaWGPbGa amiDaaqabaaakeaacqaHapaCdaWgaaWcbaGaamyAaaqabaaaaaGcca GLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaaGzaVlaai6cacaaM f8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaIZaGaaiOlaiaaig dacaaI1aGaaiykaaaa@77C0@

The bias of this simplified variance estimator will depend on the explanatory power for h ^ i δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGObGbaKaadaqhaaWcbaGaamyAaa qaaiabes7aKbaaaaa@3589@ on the linearized variable u i t . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgacaWG0b aabeaakiaac6caaaa@3595@

3.3  Variance estimation for complex parameters under calibration

The calibrated weights w t i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG3bWaaSbaaSqaaiaadshacaWGPb aabeaaaaa@34DB@ may be used to obtain an estimator of the parameter θ ( t ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH4oqCdaqadaqaaiaadshaaiaawI cacaGLPaaacaGGUaaaaa@36B6@ Substituting Y ^ w t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGzbGbaKaadaWgaaWcbaGaam4Dai aadshaaeqaaaaa@34DB@ into θ ( t ) = f { Y ( t ) } MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacqaH4oqCdaqadaqaaiaadshaaiaawI cacaGLPaaacaaMc8UaaGPaVlaai2dacaaMc8UaamOzaiaayIW7daGa daqaaiaadMfadaqadaqaaiaadshaaiaawIcacaGLPaaaaiaawUhaca GL9baaaaa@4379@ yields the calibrated plug-in estimator θ ^ w t = f ( Y ^ w t ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaH4oqCgaqcamaaBaaaleaacaWG3b GaamiDaaqabaGccaaMc8UaaGypaiaaykW7caaMc8UaamOzaiaayIW7 daqadeqaaiqadMfagaqcamaaBaaaleaacaWG3bGaamiDaaqabaaaki aawIcacaGLPaaacaGGUaaaaa@42F6@ To obtain a variance estimator for θ ^ w t , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaH4oqCgaqcamaaBaaaleaacaWG3b GaamiDaaqabaGccaGGSaaaaa@366D@ we first compute the estimated linearized variable u i t = { f ( Y ^ t ) } y i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgacaWG0b aabeaakiaaykW7caaMc8UaaGypaiaaykW7caaMc8+aaiWabeaacaWG MbWaaWbaaSqabeaajugybiadaITHYaIOaaGccaaMi8+aaeWabeaace WGzbGbaKaadaWgaaWcbaGaamiDaaqabaaakiaawIcacaGLPaaaaiaa wUhacaGL9baadaahaaWcbeqaamrr1ngBPrwtHrhAXaqeguuDJXwAKb stHrhAG8KBLbaceeGae8hPIujaaOGaaGzaVlaadMhadaWgaaWcbaGa amyAaiaadshaaeqaaaaa@584D@ and take

e θ i t = u i t b ^ θ t x i ( 3.16 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGLbWaaSbaaSqaaiabeI7aXjaadM gacaWG0baabeaakiaai2dacaWG1bWaaSbaaSqaaiaadMgacaWG0baa beaakiabgkHiTiqadkgagaqcamaaBaaaleaacqaH4oqCcaWG0baabe aakiaadIhadaWgaaWcbaGaamyAaaqabaGccaaMf8UaaGzbVlaaywW7 caaMf8UaaGzbVlaacIcacaaIZaGaaiOlaiaaigdacaaI2aGaaiykaa aa@4D59@

with

b ^ θ t = ( i s t 1 π i p ^ i 1 t x i x i ) 1 i s t 1 π i p ^ i 1 t x i u i t . ( 3.17 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGIbGbaKaadaWgaaWcbaGaeqiUde NaamiDaaqabaGccaaI9aWaaeWaaeaadaaeqbqabSqaaiaadMgacqGH iiIZcaWGZbWaaSbaaWqaaiaadshaaeqaaaWcbeqdcqGHris5aOWaaS aaaeaacaaIXaaabaGaeqiWda3aaSbaaSqaaiaadMgaaeqaaOGabmiC ayaajaWaa0baaSqaaiaadMgaaeaacaaIXaGaaGPaVlabgkziUkaayk W7caWG0baaaaaakiaadIhadaWgaaWcbaGaamyAaaqabaGccaWG4bWa a0baaSqaaiaadMgaaeaatuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy0H gip5wzaGabbiab=rQivcaaaOGaayjkaiaawMcaamaaCaaaleqabaGa eyOeI0IaaGymaaaakmaaqafabeWcbaGaamyAaiabgIGiolaadohada WgaaadbaGaamiDaaqabaaaleqaniabggHiLdGccaaMe8+aaSaaaeaa caaIXaaabaGaeqiWda3aaSbaaSqaaiaadMgaaeqaaOGabmiCayaaja Waa0baaSqaaiaadMgaaeaacaaIXaGaaGPaVlabgkziUkaaykW7caWG 0baaaaaakiaaysW7caWG4bWaaSbaaSqaaiaadMgaaeqaaOGaamyDam aaBaaaleaacaWGPbGaamiDaaqabaGccaaMb8UaaGOlaiaaywW7caaM f8UaaGzbVlaaywW7caaMf8UaaiikaiaaiodacaGGUaGaaGymaiaaiE dacaGGPaaaaa@86CD@

Replacing in (2.11) the variable y i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgacaWG0b aabeaaaaa@34DD@ with e θ i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGLbWaaSbaaSqaaiabeI7aXjaadM gacaWG0baabeaaaaa@367F@ yields the estimator of the variance due to the sampling design

V ^ t p ( θ ^ w t ) = i , j s t Δ i j π i j 1 p ^ i j 1 t e θ i t π i e θ j t π j . ( 3.18 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaqhaaWcbaGaamiDaa qaaiaadchaaaGcdaqadaqaaiqbeI7aXzaajaWaaSbaaSqaaiaadEha caWG0baabeaaaOGaayjkaiaawMcaaiaai2dadaaeqbqabSqaaiaadM gacaaISaGaaGjbVlaadQgacqGHiiIZcaWGZbWaaSbaaWqaaiaadsha aeqaaaWcbeqdcqGHris5aOGaaGjbVpaalaaabaGaeuiLdq0aaSbaaS qaaiaadMgacaWGQbaabeaaaOqaaiabec8aWnaaBaaaleaacaWGPbGa amOAaaqabaaaaOGaaGjbVpaalaaabaGaaGymaaqaaiqadchagaqcam aaDaaaleaacaWGPbGaamOAaaqaaiaaigdacaaMc8UaeyOKH4QaaGPa VlaadshaaaaaaOGaaGjbVpaalaaabaGaamyzamaaBaaaleaacqaH4o qCcaWGPbGaamiDaaqabaaakeaacqaHapaCdaWgaaWcbaGaamyAaaqa baaaaOGaaGjbVpaalaaabaGaamyzamaaBaaaleaacqaH4oqCcaWGQb GaamiDaaqabaaakeaacqaHapaCdaWgaaWcbaGaamOAaaqabaaaaOGa aGOlaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaaiikaiaaiodaca GGUaGaaGymaiaaiIdacaGGPaaaaa@7960@

Similarly, replacing in (2.12) the variable y i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG5bWaaSbaaSqaaiaadMgacaWG0b aabeaaaaa@34DD@ with e θ i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGLbWaaSbaaSqaaiabeI7aXjaadM gacaWG0baabeaaaaa@367F@ yields the estimator of the variance due to the non-response

V ^ t nr ( θ ^ w t ) = δ = 1 t i s t p ^ i δ ( 1 p ^ i δ ) p ^ i δ t ( e θ i t π i p ^ i 1 δ k i δ ( h ^ i δ ) γ ^ t e θ δ ) 2 ( 3.19 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaqhaaWcbaGaamiDaa qaaiaab6gacaqGYbaaaOWaaeWaaeaacuaH4oqCgaqcamaaBaaaleaa caWG3bGaamiDaaqabaaakiaawIcacaGLPaaacaaI9aWaaabCaeqale aacqaH0oazcaaI9aGaaGymaaqaaiaadshaa0GaeyyeIuoakiaaykW7 daaeqbqabSqaaiaadMgacqGHiiIZcaWGZbWaaSbaaWqaaiaadshaae qaaaWcbeqdcqGHris5aOGaaGjbVpaalaaabaGabmiCayaajaWaa0ba aSqaaiaadMgaaeaacqaH0oazaaGcdaqadaqaaiaaigdacqGHsislce WGWbGbaKaadaqhaaWcbaGaamyAaaqaaiabes7aKbaaaOGaayjkaiaa wMcaaaqaaiqadchagaqcamaaDaaaleaacaWGPbaabaGaeqiTdqMaaG PaVlabgkziUkaaykW7caWG0baaaaaakmaabmaabaWaaSaaaeaacaWG LbWaaSbaaSqaaiabeI7aXjaadMgacaWG0baabeaaaOqaaiabec8aWn aaBaaaleaacaWGPbaabeaakiqadchagaqcamaaDaaaleaacaWGPbaa baGaaGymaiaaykW7cqGHsgIRcaaMc8UaeqiTdqgaaaaakiabgkHiTi aadUgadaqhaaWcbaGaamyAaaqaaiabes7aKbaakmaabmaabaGabmiA ayaajaWaa0baaSqaaiaadMgaaeaacqaH0oazaaaakiaawIcacaGLPa aadaahaaWcbeqaamrr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KB LbaceeGae8hPIujaaOGaaGzaVlqbeo7aNzaajaWaa0baaSqaaiaads hacaWGLbGaeqiUdehabaGaeqiTdqgaaaGccaGLOaGaayzkaaWaaWba aSqabeaacaaIYaaaaOGaaGzbVlaaywW7caaMf8UaaGzbVlaacIcaca aIZaGaaiOlaiaaigdacaaI5aGaaiykaaaa@9D88@

γ ^ teθ δ = { i s t k i δ p ^ i δ ( 1 p ^ i δ ) p ^ i δt h ^ i δ ( h ^ i δ ) } 1 i s t 1 p ^ i δ p ^ i 1t h ^ i δ e θit π i .(3.20) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaHZoWzgaqcamaaDaaaleaacaWG0b GaamyzaiabeI7aXbqaaiabes7aKbaakiaai2dadaGadaqaamaaqafa beWcbaGaamyAaiabgIGiolaadohadaWgaaadbaGaamiDaaqabaaale qaniabggHiLdGccaaMe8Uaam4AamaaDaaaleaacaWGPbaabaGaeqiT dqgaaOWaaSaaaeaaceWGWbGbaKaadaqhaaWcbaGaamyAaaqaaiabes 7aKbaakmaabmaabaGaaGymaiabgkHiTiqadchagaqcamaaDaaaleaa caWGPbaabaGaeqiTdqgaaaGccaGLOaGaayzkaaaabaGabmiCayaaja Waa0baaSqaaiaadMgaaeaacqaH0oazcqGHsgIRcaWG0baaaaaakiqa dIgagaqcamaaDaaaleaacaWGPbaabaGaeqiTdqgaaOWaaeWaaeaace WGObGbaKaadaqhaaWcbaGaamyAaaqaaiabes7aKbaaaOGaayjkaiaa wMcaamaaCaaaleqabaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDOb YtUvgaiqqacqWFKksLaaaakiaawUhacaGL9baadaahaaWcbeqaaiab gkHiTiaaigdaaaGcdaaeqbqabSqaaiaadMgacqGHiiIZcaWGZbWaaS baaWqaaiaadshaaeqaaaWcbeqdcqGHris5aOGaaGjbVpaalaaabaGa aGymaiabgkHiTiqadchagaqcamaaDaaaleaacaWGPbaabaGaeqiTdq gaaaGcbaGabmiCayaajaWaa0baaSqaaiaadMgaaeaacaaIXaGaaGPa VlabgkziUkaaykW7caWG0baaaaaakiaaysW7caaMe8UabmiAayaaja Waa0baaSqaaiaadMgaaeaacqaH0oazaaGccaaMe8+aaSaaaeaacaWG LbWaaSbaaSqaaiabeI7aXjaadMgacaWG0baabeaaaOqaaiabec8aWn aaBaaaleaacaWGPbaabeaaaaGccaaIUaGaaGzbVlaaywW7caaMf8Ua aGzbVlaacIcacaaIZaGaaiOlaiaaikdacaaIWaGaaiykaaaa@A232@

The global variance estimator for θ ^ w t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacuaH4oqCgaqcamaaBaaaleaacaWG3b GaamiDaaqabaaaaa@35B3@ is

V ^ t ( θ ^ w t ) = V ^ t p ( θ ^ w t ) + V ^ t nr ( θ ^ w t ) . ( 3.21 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaWgaaWcbaGaamiDaa qabaGcdaqadaqaaiqbeI7aXzaajaWaaSbaaSqaaiaadEhacaWG0baa beaaaOGaayjkaiaawMcaaiaai2daceWGwbGbaKaadaqhaaWcbaGaam iDaaqaaiaadchaaaGcdaqadaqaaiqbeI7aXzaajaWaaSbaaSqaaiaa dEhacaWG0baabeaaaOGaayjkaiaawMcaaiabgUcaRiqadAfagaqcam aaDaaaleaacaWG0baabaGaaeOBaiaabkhaaaGcdaqadaqaaiqbeI7a XzaajaWaaSbaaSqaaiaadEhacaWG0baabeaaaOGaayjkaiaawMcaai aai6cacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaIZaGa aiOlaiaaikdacaaIXaGaaiykaaaa@59CA@

The simplified estimator of the variance due to non-response is

V ^ t , simp nr ( θ ^ w t ) = i s t 1 p ^ i 1 t ( p ^ i 1 t ) 2 ( e θ i t π i ) 2 . ( 3.22 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGwbGbaKaadaqhaaWcbaGaamiDai aaiYcacaaMe8Uaae4CaiaabMgacaqGTbGaaeiCaaqaaiaab6gacaqG YbaaaOWaaeWaaeaacuaH4oqCgaqcamaaBaaaleaacaWG3bGaamiDaa qabaaakiaawIcacaGLPaaacaaI9aWaaabuaeqaleaacaWGPbGaeyic I4Saam4CamaaBaaameaacaWG0baabeaaaSqab0GaeyyeIuoakiaays W7daWcaaqaaiaaigdacqGHsislceWGWbGbaKaadaqhaaWcbaGaamyA aaqaaiaaigdacaaMc8UaeyOKH4QaaGPaVlaadshaaaaakeaadaqada qaaiqadchagaqcamaaDaaaleaacaWGPbaabaGaaGymaiaaykW7cqGH sgIRcaaMc8UaamiDaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaG OmaaaaaaGcdaqadaqaamaalaaabaGaamyzamaaBaaaleaacqaH4oqC caWGPbGaamiDaaqabaaakeaacqaHapaCdaWgaaWcbaGaamyAaaqaba aaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaOGaaGzaVlaa i6cacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaIZaGaai OlaiaaikdacaaIYaGaaiykaaaa@78D3@

Since the variable e θ i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGLbWaaSbaaSqaaiabeI7aXjaadM gacaWG0baabeaaaaa@367F@ is obtained as the residual in the regression of the linearized variable u i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG1bWaaSbaaSqaaiaadMgacaWG0b aabeaaaaa@34D9@ on the calibration variables x i , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWG4bWaaSbaaSqaaiaadMgaaeqaaO Gaaiilaaaa@349D@ the explanatory power for h ^ i δ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaaceWGObGbaKaadaqhaaWcbaGaamyAaa qaaiabes7aKbaaaaa@3589@ on e θ i t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8rrps0l bbf9q8WrFfeuY=Hhbbf9y8WrFr0xc9vqFj0db9qqvqFr0dXdHiVc=b YP0xH8piea0lXxcrpe0db9Wqpepic9qr=xfr=xfr=tmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGLbWaaSbaaSqaaiabeI7aXjaadM gacaWG0baabeaaaaa@367F@ is expected to be small in practice, and the bias of the simplified variance estimator is expected to be small as well.


Date modified: