Replication variance estimation after sample-based calibration
Section 3. Sample-based raking calibration

In the application to the two 2016 FHWAR surveys, we used raking rather than regression estimation for calibration. As for regression estimation, the goal is to create new raking controls for each of the replicates, so that the replicate variance estimator for the primary survey accounts for the variability of the control totals from the secondary survey. The above results for regression estimation do not apply directly, but we can apply the same reasoning as in Deville and Särndal (1992) to show that they continue to hold for raking. Instead of relying on this general result, we will derive it here explicitly to show how to obtain the adjusted control totals for the replicates.

For simplicity, we describe here the case in which we are controlling for the marginal counts in domains defined by the levels of 2 categorical variables, denoted a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaaaa@36CD@ and b , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiaacY caaaa@377E@ having K MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saaaa@36B7@ and L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamitaaaa@36B8@ levels, respectively. In the primary survey, the estimated counts in the domains defined by the intersections of the two variables are N ^ a k b l = s w i δ i ( a k , b l ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOtayaaja WaaSbaaSqaaiaadggadaWgaaadbaGaam4AaaqabaWccaWGIbWaaSba aWqaaiaadYgaaeqaaaWcbeaakiaaysW7caaI9aGaaGjbVpaaqababa GaaGPaVlaadEhadaWgaaWcbaGaamyAaaqabaGccqaH0oazdaWgaaWc baGaamyAaaqabaGccaaMc8UaaGikaiaadggadaWgaaWcbaGaam4Aaa qabaGccaaISaGaaGjbVlaadkgadaWgaaWcbaGaamiBaaqabaGccaaI PaaaleaacaWGZbaabeqdcqGHris5aOGaaiilaaaa@5255@ with δ i ( a k , b l ) = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aaS baaSqaaiaadMgaaeqaaOGaaGPaVlaaiIcacaWGHbWaaSbaaSqaaiaa dUgaaeqaaOGaaGilaiaaysW7caWGIbWaaSbaaSqaaiaadYgaaeqaaO GaaGykaiaaysW7caaI9aGaaGjbVlaaigdaaaa@4699@ if element i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@36D5@ is in the domain defined by the intersection of a k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGRbaabeaaaaa@37E9@ and b l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyamaaBa aaleaacaWGSbaabeaaaaa@37EB@ and 0 otherwise. The marginal estimated counts are defined analogously, N ^ a k = s w i δ i ( a k , ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOtayaaja WaaSbaaSqaaiaadggadaWgaaadbaGaam4AaaqabaaaleqaaOGaaGjb Vlaai2dacaaMe8+aaabeaeaacaaMc8Uaam4DamaaBaaaleaacaWGPb aabeaakiabes7aKnaaBaaaleaacaWGPbaabeaakiaaykW7caaIOaGa amyyamaaBaaaleaacaWGRbaabeaakiaaiYcacaaMe8UaeyyXICTaaG ykaaWcbaGaam4Caaqab0GaeyyeIuoaaaa@4FC7@ and N ^ b l = s w i δ i ( , b l ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmOtayaaja WaaSbaaSqaaiaadkgadaWgaaadbaGaamiBaaqabaaaleqaaOGaaGjb Vlaai2dacaaMe8+aaabeaeaacaaMc8Uaam4DamaaBaaaleaacaWGPb aabeaakiabes7aKnaaBaaaleaacaWGPbaabeaakiaaykW7caaIOaGa eyyXICTaaGilaiaaysW7caWGIbWaaSbaaSqaaiaadYgaaeqaaOGaaG ykaaWcbaGaam4Caaqab0GaeyyeIuoakiaac6caaaa@5087@ We write δ i = ( δ i ( a 1 ) , , δ i ( a K ) , δ i ( b 1 ) , , δ i ( b L ) ) T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGjcVlaahs 7adaWgaaWcbaGaamyAaaqabaGccaaMe8Uaeyypa0JaaGjbVpaabmqa baGaeqiTdq2aaSbaaSqaaiaadMgaaeqaaOGaaGPaVlaaiIcacaWGHb WaaSbaaSqaaiaaigdaaeqaaOGaaGykaiaaiYcacaaMe8UaeSOjGSKa aGilaiaaysW7cqaH0oazdaWgaaWcbaGaamyAaaqabaGccaaMc8UaaG ikaiaadggadaWgaaWcbaGaam4saaqabaGccaaIPaGaaGilaiaaysW7 cqaH0oazdaWgaaWcbaGaamyAaaqabaGccaaMc8UaaGikaiaadkgada WgaaWcbaGaaGymaaqabaGccaaIPaGaaGilaiaaysW7cqWIMaYscaaI SaGaaGjbVlabes7aKnaaBaaaleaacaWGPbaabeaakiaaykW7caaIOa GaamOyamaaBaaaleaacaWGmbaabeaakiaaiMcaaiaawIcacaGLPaaa daahaaWcbeqaaiaadsfaaaaaaa@6C8B@ for the vector of indicators for the marginal domains for element i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyAaiaac6 caaaa@3787@ The estimated marginal counts in the primary survey are N ^ = s w i δ i = ( N ^ a 1 , , N ^ a K , N ^ b 1 , , N ^ b L ) T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOtayaaja GaaGjbVlabg2da9iaaysW7daaeqaqaaiaaykW7caWG3bWaaSbaaSqa aiaadMgaaeqaaOGaaGjcVlaahs7adaWgaaWcbaGaamyAaaqabaaaba Gaam4Caaqab0GaeyyeIuoakiaaysW7caaI9aGaaGjbVlaaiIcaceWG obGbaKaadaWgaaWcbaGaamyyamaaBaaameaacaaIXaaabeaaaSqaba GccaaISaGaaGjbVlablAciljaaiYcacaaMe8UabmOtayaajaWaaSba aSqaaiaadggadaWgaaadbaGaam4saaqabaaaleqaaOGaaiilaiaays W7ceWGobGbaKaadaWgaaWcbaGaamOyamaaBaaameaacaaIXaaabeaa aSqabaGccaaISaGaaGjbVlablAciljaaiYcacaaMe8UabmOtayaaja WaaSbaaSqaaiaadkgadaWgaaadbaGaamitaaqabaaaleqaaOGaaGyk amaaCaaaleqabaGaamivaaaaaaa@652C@ and the corresponding control totals from the secondary survey are N ^ C = s C w C i δ i = ( N ^ C a 1 , , N ^ C a K , N ^ C b 1 , , N ^ C b L ) T . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOtayaaja WaaSbaaSqaaiaadoeaaeqaaOGaaGjbVlaai2dacaaMe8+aaabeaeaa caaMc8Uaam4DamaaBaaaleaacaWGdbGaamyAaaqabaGccaaMi8UaaG PaVlaahs7adaWgaaWcbaGaamyAaaqabaaabaGaam4CamaaBaaameaa caWGdbaabeaaaSqab0GaeyyeIuoakiaaysW7caaI9aGaaGjbVlaaiI caceWGobGbaKaadaWgaaWcbaGaam4qaiaadggadaWgaaadbaGaaGym aaqabaaaleqaaOGaaGilaiaaysW7cqWIMaYscaaISaGaaGjbVlqad6 eagaqcamaaBaaaleaacaWGdbGaamyyamaaBaaameaacaWGlbaabeaa aSqabaGccaaISaGaaGjbVlqad6eagaqcamaaBaaaleaacaWGdbGaam OyamaaBaaameaacaaIXaaabeaaaSqabaGccaaISaGaaGjbVlablAci ljaaiYcacaaMe8UabmOtayaajaWaaSbaaSqaaiaadoeacaWGIbWaaS baaWqaaiaadYeaaeqaaaWcbeaakiaaiMcadaahaaWcbeqaaiaadsfa aaGccaGGUaaaaa@6D20@ Using the classical raking ratio algorithm of Deming and Stephan (1940) until convergence, the raked weights for the primary survey can be written as

w i * = w i exp ( u ^ a k + u ^ b l ) for δ i ( a k , b l ) = 1 = w i exp ( u ^ T δ i ) ( 3.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaa qaaiaadEhadaqhaaWcbaGaamyAaaqaaiaacQcaaaaakeaacaaI9aGa aGjbVlaaysW7caWG3bWaaSbaaSqaaiaadMgaaeqaaOGaciyzaiaacI hacaGGWbGaaGPaVlaaiIcaceWG1bGbaKaadaWgaaWcbaGaamyyamaa BaaameaacaWGRbaabeaaaSqabaGccaaMe8Uaey4kaSIaaGjbVlqadw hagaqcamaaBaaaleaacaWGIbWaaSbaaWqaaiaadYgaaeqaaaWcbeaa kiaaiMcacaaMe8UaaGjbVlaaysW7caaMi8UaaeOzaiaab+gacaqGYb GaaGjbVlaaysW7caaMe8UaeqiTdq2aaSbaaSqaaiaadMgaaeqaaOGa aGPaVlaaiIcacaWGHbWaaSbaaSqaaiaadUgaaeqaaOGaaGilaiaays W7caWGIbWaaSbaaSqaaiaadYgaaeqaaOGaaGykaiaaysW7caaI9aGa aGjbVlaaigdaaeaaaeaacaaI9aGaaGjbVlaaysW7caWG3bWaaSbaaS qaaiaadMgaaeqaaOGaciyzaiaacIhacaGGWbGaaGPaVlaaiIcaceWH 1bGbaKaadaahaaWcbeqaaiaadsfaaaGccaaMc8UaaCiTdmaaBaaale aacaWGPbaabeaakiaaiMcacaaMf8UaaGzbVlaaywW7caaMf8UaaGzb VlaaywW7caaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7caaMf8 UaaGzbVlaaywW7caGGOaGaaG4maiaac6cacaaIXaGaaiykaaaaaaa@9A39@

where u ^ = ( u a 1 , , u a K , u b 1 , , u b L ) T MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCyDayaaja GaaGjbVlabg2da9iaaysW7caaIOaGaamyDamaaBaaaleaacaWGHbWa aSbaaWqaaiaaigdaaeqaaaWcbeaakiaaiYcacaaMe8UaeSOjGSKaaG ilaiaaysW7caWG1bWaaSbaaSqaaiaadggadaWgaaadbaGaam4saaqa baaaleqaaOGaaGilaiaaysW7caWG1bWaaSbaaSqaaiaadkgadaWgaa adbaGaaGymaaqabaaaleqaaOGaaGilaiaaysW7cqWIMaYscaaISaGa aGjbVlaadwhadaWgaaWcbaGaamOyamaaBaaameaacaWGmbaabeaaaS qabaGccaaIPaWaaWbaaSqabeaacaWGubaaaaaa@5764@ is a solution to the system of K + L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiaays W7cqGHRaWkcaaMe8Uaamitaaaa@3B84@ equations

l = 1 L N ^ a k b l exp ( u a k + u b l ) = N ^ C a k ( k = 1, , K ) k = 1 K N ^ a k b l exp ( u a k + u b l ) = N ^ C b l ( l = 1, , L ) . ( 3.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpu0de9LqFHe9Lq pepeea0xd9q8as0=LqLs=Jirpepeea0=as0Fb9pgea0lrP0xe9Fve9 Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaa qaamaaqahabaGaaGPaVlqad6eagaqcamaaBaaaleaacaWGHbWaaSba aWqaaiaadUgaaeqaaSGaamOyamaaBaaameaacaWGSbaabeaaaSqaba aabaGaamiBaiaai2dacaaIXaaabaGaamitaaqdcqGHris5aOGaaGPa VlGacwgacaGG4bGaaiiCaiaaykW7caaIOaGaamyDamaaBaaaleaaca WGHbWaaSbaaWqaaiaadUgaaeqaaaWcbeaakiaaysW7cqGHRaWkcaaM e8UaamyDamaaBaaaleaacaWGIbWaaSbaaWqaaiaadYgaaeqaaaWcbe aakiaaiMcaaeaacqGH9aqpcaaMe8UaaGjbVlaaykW7ceWGobGbaKaa daWgaaWcbaGaam4qaiaadggadaWgaaadbaGaam4AaaqabaaaleqaaO GaaGzbVlaaiIcacaWGRbGaaGjbVlaai2dacaaMe8UaaGymaiaaiYca caaMe8UaeSOjGSKaaGilaiaaysW7caWGlbGaaGykaaqaamaaqahaba GaaGPaVlqad6eagaqcamaaBaaaleaacaWGHbWaaSbaaWqaaiaadUga aeqaaSGaamOyamaaBaaameaacaWGSbaabeaaaSqabaGcciGGLbGaai iEaiaacchacaaMc8oaleaacaWGRbGaaGypaiaaigdaaeaacaWGlbaa niabggHiLdGccaaIOaGaamyDamaaBaaaleaacaWGHbWaaSbaaWqaai aadUgaaeqaaaWcbeaakiaaysW7cqGHRaWkcaaMe8UaamyDamaaBaaa leaacaWGIbWaaSbaaWqaaiaadYgaaeqaaaWcbeaakiaaiMcaaeaaca aI9aGaaGjbVlaaykW7caaMe8UabmOtayaajaWaaSbaaSqaaiaadoea caWGIbWaaSbaaWqaaiaadYgaaeqaaaWcbeaakiaaywW7caaIOaGaam iBaiaaysW7caaI9aGaaGjbVlaaigdacaaISaGaaGjbVlablAciljaa iYcacaaMe8UaamitaiaaiMcacaaIUaGaaGzbVlaaywW7caaMf8UaaG zbVlaaywW7caGGOaGaaG4maiaac6cacaaIYaGaaiykaaaaaaa@AFAF@

The solution to these equations is not unique, so one of the unknowns can be set to 0 and an equation removed. This does not affect the values of exp ( u a k + u b l ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaciyzaiaacI hacaGGWbGaaGPaVlaaiIcacaWG1bWaaSbaaSqaaiaadggadaWgaaad baGaam4AaaqabaaaleqaaOGaaGjbVlabgUcaRiaaysW7caWG1bWaaS baaSqaaiaadkgadaWgaaadbaGaamiBaaqabaaaleqaaOGaaGykaiaa cYcaaaa@46DC@ and we will set v b L = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamODamaaBa aaleaacaWGIbWaaSbaaWqaaiaadYeaaeqaaaWcbeaakiaaysW7caaI 9aGaaGjbVlaaicdaaaa@3DA3@ and remove the last equation in what follows.

There is no explicit expression for the solution to (3.2), but it can be approximated by using a linearization argument. Under the usual survey asymptotic framework that ensures design consistency of Horvitz-Thompson estimators, the u ^ a k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyDayaaja WaaSbaaSqaaiaadggadaWgaaadbaGaam4Aaaqabaaaleqaaaaa@392B@ and u ^ b l MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmyDayaaja WaaSbaaSqaaiaadkgadaWgaaadbaGaamiBaaqabaaaleqaaaaa@392D@ converge to 0 as the sample sizes of the two surveys increase, so that expansion around 0 is valid. Doing so for the equations in (3.2), we approximate the reduced set of K + L 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiaays W7cqGHRaWkcaaMe8UaamitaiaaysW7cqGHsislcaaMe8UaaGymaaaa @4046@ equations by

l = 1 L N ^ a k b l ( 1 + u a k + u b l + o p ( 1 ) ) = N ^ C a k ( k = 1, , K ) k = 1 K N ^ a k b l ( 1 + u a k + u b l + o p ( 1 ) ) = N ^ C b l ( l = 1, , L 1 ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpu0de9LqFHe9Lq pepeea0xd9q8as0=LqLs=Jirpepeea0=as0Fb9pgea0lrP0xe9Fve9 Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaa qaamaaqahabaGaaGPaVlqad6eagaqcamaaBaaaleaacaWGHbWaaSba aWqaaiaadUgaaeqaaSGaamOyamaaBaaameaacaWGSbaabeaaaSqaba GccaaIOaGaaGymaiaaysW7cqGHRaWkcaaMe8UaamyDamaaBaaaleaa caWGHbWaaSbaaWqaaiaadUgaaeqaaaWcbeaakiaaysW7cqGHRaWkca aMe8UaamyDamaaBaaaleaacaWGIbWaaSbaaWqaaiaadYgaaeqaaaWc beaakiaaysW7cqGHRaWkcaaMe8Uaam4BamaaBaaaleaacaWGWbaabe aakiaaiIcacaaIXaGaaGykaiaaiMcaaSqaaiaadYgacaaMc8Uaeyyp a0JaaGPaVlaaigdaaeaacaWGmbaaniabggHiLdaakeaacqGH9aqpca aMe8UaaGPaVlaaysW7ceWGobGbaKaadaWgaaWcbaGaam4qaiaadgga daWgaaadbaGaam4AaaqabaaaleqaaOGaaGzbVlaaiIcacaWGRbGaaG jbVlabg2da9iaaysW7caaIXaGaaGilaiaaysW7cqWIMaYscaaISaGa aGjbVlaadUeacaaIPaaabaWaaabCaeaacaaMc8UabmOtayaajaWaaS baaSqaaiaadggadaWgaaadbaGaam4AaaqabaWccaWGIbWaaSbaaWqa aiaadYgaaeqaaaWcbeaakiaaiIcacaaIXaGaaGjbVlabgUcaRiaays W7caWG1bWaaSbaaSqaaiaadggadaWgaaadbaGaam4Aaaqabaaaleqa aOGaaGjbVlabgUcaRiaaysW7caWG1bWaaSbaaSqaaiaadkgadaWgaa adbaGaamiBaaqabaaaleqaaOGaaGjbVlabgUcaRiaaysW7caWGVbWa aSbaaSqaaiaadchaaeqaaOGaaGikaiaaigdacaaIPaGaaGykaaWcba Gaam4AaiaaykW7cqGH9aqpcaaMc8UaaGymaaqaaiaadUeaa0Gaeyye IuoaaOqaaiabg2da9iaaysW7caaMe8UaaGPaVlqad6eagaqcamaaBa aaleaacaWGdbGaamOyamaaBaaameaacaWGSbaabeaaaSqabaGccaaM f8UaaGikaiaadYgacaaMe8Uaeyypa0JaaGjbVlaaigdacaaISaGaaG jbVlablAciljaaiYcacaaMe8UaamitaiaaysW7cqGHsislcaaMe8Ua aGymaiaaiMcacaaISaaaaaaa@C02E@

which can be rewritten as

N ^ a k u a k + l = 1 L N ^ a k b l u b l = ( N ^ C a k N ^ a k ) ( 1 + o p ( 1 ) ) ( k = 1, , K ) N ^ b l u b l + k = 1 K N ^ a k b l u a k = ( N ^ C b l N ^ b l ) ( 1 + o p ( 1 ) ) ( l = 1, , L 1 ) . ( 3.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpu0de9LqFHe9Lq pepeea0xd9q8as0=LqLs=Jirpepeea0=as0Fb9pgea0lrP0xe9Fve9 Fve9qapdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaa qaaiqad6eagaqcamaaBaaaleaacaWGHbWaaSbaaWqaaiaadUgaaeqa aaWcbeaakiaadwhadaWgaaWcbaGaamyyamaaBaaameaacaWGRbaabe aaaSqabaGccaaMe8Uaey4kaSIaaGjbVpaaqahabaGaaGPaVlqad6ea gaqcamaaBaaaleaacaWGHbWaaSbaaWqaaiaadUgaaeqaaSGaamOyam aaBaaameaacaWGSbaabeaaaSqabaGccaWG1bWaaSbaaSqaaiaadkga daWgaaadbaGaamiBaaqabaaaleqaaaqaaiaadYgacaaI9aGaaGymaa qaaiaadYeaa0GaeyyeIuoaaOqaaiabg2da9iaaysW7caaMc8UaaGjb VlaaiIcaceWGobGbaKaadaWgaaWcbaGaam4qaiaadggadaWgaaadba Gaam4AaaqabaaaleqaaOGaaGjbVlabgkHiTiaaysW7ceWGobGbaKaa daWgaaWcbaGaamyyamaaBaaameaacaWGRbaabeaaaSqabaGccaaIPa GaaGjbVlaaiIcacaaIXaGaaGjbVlabgUcaRiaaysW7caWGVbWaaSba aSqaaiaadchaaeqaaOGaaGikaiaaigdacaaIPaGaaGykaiaaywW7ca aIOaGaam4AaiaaysW7caaI9aGaaGjbVlaaigdacaaISaGaaGjbVlab lAciljaaiYcacaaMe8Uaam4saiaaiMcaaeaaceWGobGbaKaadaWgaa WcbaGaamOyamaaBaaameaacaWGSbaabeaaaSqabaGccaWG1bWaaSba aSqaaiaadkgadaWgaaadbaGaamiBaaqabaaaleqaaOGaaGjbVlabgU caRiaaysW7daaeWbqaaiaaykW7ceWGobGbaKaadaWgaaWcbaGaamyy amaaBaaameaacaWGRbaabeaaliaadkgadaWgaaadbaGaamiBaaqaba aaleqaaOGaamyDamaaBaaaleaacaWGHbWaaSbaaWqaaiaadUgaaeqa aaWcbeaaaeaacaWGRbGaaGPaVlabg2da9iaaykW7caaIXaaabaGaam 4saaqdcqGHris5aaGcbaGaeyypa0JaaGjbVlaaykW7caaMe8UaaGik aiqad6eagaqcamaaBaaaleaacaWGdbGaamOyamaaBaaameaacaWGSb aabeaaaSqabaGccaaMe8UaeyOeI0IaaGjbVlqad6eagaqcamaaBaaa leaacaWGIbWaaSbaaWqaaiaadYgaaeqaaaWcbeaakiaaiMcacaaMe8 UaaGikaiaaigdacaaMe8Uaey4kaSIaaGjbVlaad+gadaWgaaWcbaGa amiCaaqabaGccaaIOaGaaGymaiaaiMcacaaIPaGaaGzbVlaaiIcaca WGSbGaaGjbVlaai2dacaaMe8UaaGymaiaaiYcacaaMe8UaeSOjGSKa aGilaiaaysW7caWGmbGaaGjbVlabgkHiTiaaigdacaaIPaGaaiOlai aaywW7caaMf8UaaGzbVlaaywW7caaMf8UaaiikaiaaiodacaGGUaGa aG4maiaacMcaaaaaaa@D885@

Ignoring the smaller order remainders, the solution to this system of linear equations can be written in the form u ^ = J ^ 1 ( N ^ C N ^ ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCyDayaaja GaaGjbVlabg2da9iaaysW7ceWHkbGbaKaadaahaaWcbeqaaiabgkHi TiaaigdaaaGccaaMc8UaaGikaiqah6eagaqcamaaBaaaleaacaWGdb aabeaakiaaysW7cqGHsislcaaMe8UabCOtayaajaGaaGykaiaacYca aaa@484A@ where J ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOsayaaja aaaa@36CA@ is a symmetric ( K + L 1 ) × ( K + L 1 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGikaiaadU eacaaMe8Uaey4kaSIaaGjbVlaadYeacaaMe8UaeyOeI0IaaGjbVlaa igdacaaIPaGaaGjbVlabgEna0kaaysW7caaIOaGaam4saiaaysW7cq GHRaWkcaaMe8UaamitaiaaysW7cqGHsislcaaMe8UaaGymaiaaiMca aaa@52A0@ matrix containing estimated domain counts readily obtained from the left-hand side of (3.3). Note that the resulting u ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCyDayaaja aaaa@36F5@ is not a linear estimator, because in the linearization we conditioned on N ^ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOtayaaja GaaiOlaaaa@3780@ Finally, plugging u ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCyDayaaja aaaa@36F5@ into the expression (3.1) and linearizing again, we obtain

w i * w i ( 1+ ( N ^ C N ^ ) T J ^ 1 δ i )(3.4) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaDa aaleaacaWGPbaabaGaaiOkaaaakiaaysW7cqGHijYUcaaMe8Uaam4D amaaBaaaleaacaWGPbaabeaakiaaykW7daqadaqaaiaaigdacaaMe8 Uaey4kaSIaaGjbVlaaiIcaceWHobGbaKaadaWgaaWcbaGaam4qaaqa baGccaaMe8UaeyOeI0IaaGjbVlqah6eagaqcaiaaiMcadaahaaWcbe qaaiaadsfaaaGcceWHkbGbaKaadaahaaWcbeqaaiabgkHiTiaaigda aaGccaaMc8UaaCiTdmaaBaaaleaacaWGPbaabeaaaOGaayjkaiaawM caaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8UaaiikaiaaiodacaGG UaGaaGinaiaacMcaaaa@62B0@

and the estimator after raking is

t ^ y,rak = s w i * y i s w i y i + ( N ^ C N ^ ) T J ^ 1 δ s T W s Y s .(3.5) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiDayaaja WaaSbaaSqaaiaadMhacaaISaGaaGPaVlaabkhacaqGHbGaae4Aaaqa baGccaaMe8UaaGypaiaaysW7daaeqbqaaiaaykW7caWG3bWaa0baaS qaaiaadMgaaeaacaGGQaaaaOGaamyEamaaBaaaleaacaWGPbaabeaa aeaacaWGZbaabeqdcqGHris5aOGaaGjbVlabgIKi7kaaysW7daaeqb qaaiaaykW7caWG3bWaaSbaaSqaaiaadMgaaeqaaOGaamyEamaaBaaa leaacaWGPbaabeaaaeaacaWGZbaabeqdcqGHris5aOGaaGjbVlabgU caRiaaysW7caaIOaGabCOtayaajaWaaSbaaSqaaiaadoeaaeqaaOGa aGjbVlabgkHiTiaaysW7ceWHobGbaKaacaaIPaWaaWbaaSqabeaaca WGubaaaOGabCOsayaajaWaaWbaaSqabeaacqGHsislcaaIXaaaaOGa aGPaVlaahs7adaqhaaWcbaGaam4CaaqaaiaadsfaaaGccaaMc8UaaC 4vamaaBaaaleaacaWGZbaabeaakiaaykW7caWHzbWaaSbaaSqaaiaa dohaaeqaaOGaaGOlaiaaywW7caaMf8UaaGzbVlaaywW7caaMf8Uaai ikaiaaiodacaGGUaGaaGynaiaacMcaaaa@8052@

Note that the size of the control variable δ i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGjcVlaahs 7adaWgaaWcbaGaamyAaaqabaaaaa@39D2@ and associated estimates is now K+L1, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4saiaays W7cqGHRaWkcaaMe8UaamitaiaaysW7cqGHsislcaaMe8UaaGymaiaa cYcaaaa@40F6@ but we maintain the prior notation for simplicity. In (3.5), J ^ 1 δ s T W s Y s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOsayaaja WaaWbaaSqabeaacqGHsislcaaIXaaaaOGaaGPaVlaahs7adaqhaaWc baGaam4CaaqaaiaadsfaaaGccaaMc8UaaC4vamaaBaaaleaacaWGZb aabeaakiaaykW7caWHzbWaaSbaaSqaaiaadohaaeqaaaaa@44A6@ corresponds to β ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOSdyaaja aaaa@3735@ in the regression estimator (2.2). This asymptotic equivalence between the raking estimator and the regression estimator with the same control totals was established by Deville and Särndal (1992). In particular, they provide sufficient conditions under which the asymptotic variance of the equivalent regression estimator can also be used for inference for the raking estimator. Hence, a variance estimator of the form (2.6) can be constructed, with t ^ e ^ = s w i ( y i δ i T J ^ 1 δ s T W s Y s ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiDayaaja WaaSbaaSqaaiqadwgagaqcaaqabaGccaaMe8UaaGypaiaaysW7daae qaqaaiaaykW7caWG3bWaaSbaaSqaaiaadMgaaeqaaOGaaGPaVlaaiI cacaWG5bWaaSbaaSqaaiaadMgaaeqaaOGaaGjbVlabgkHiTiaaysW7 caWH0oWaa0baaSqaaiaadMgaaeaacaWGubaaaOGaaGPaVlqahQeaga qcamaaCaaaleqabaGaeyOeI0IaaGymaaaakiaaykW7caWH0oWaa0ba aSqaaiaadohaaeaacaWGubaaaOGaaGPaVlaahEfadaWgaaWcbaGaam 4CaaqabaGccaaMc8UaaCywamaaBaaaleaacaWGZbaabeaakiaaiMca aSqaaiaadohaaeqaniabggHiLdGccaGGUaaaaa@5FEE@

We now consider the construction of replicate weights for the primary survey that estimate the asymptotic variance of the raking estimator. As before, we construct new replicate control totals N ^ C + a r ( N ^ C (r) N ^ C ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOtayaaja WaaSbaaSqaaiaadoeaaeqaaOGaaGjbVlabgUcaRiaaysW7caWGHbWa aSbaaSqaaiaadkhaaeqaaOGaaGPaVlaaiIcaceWHobGbaKaadaqhaa WcbaGaam4qaaqaaiaaiIcacaWGYbGaaGykaaaakiaaysW7cqGHsisl caaMe8UabCOtayaajaWaaSbaaSqaaiaadoeaaeqaaOGaaGykaaaa@4AF9@ using the replicate estimates N ^ C (r) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCOtayaaja Waa0baaSqaaiaadoeaaeaacaaIOaGaamOCaiaaiMcaaaaaaa@3A1F@ from the secondary survey. Each of the sets of replicate weights w i (r) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaDa aaleaacaWGPbaabaGaaGikaiaadkhacaaIPaaaaaaa@3A5A@ of the primary survey are adjusted by raking to its corresponding set of control totals, to obtain the w i *(r) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4DamaaDa aaleaacaWGPbaabaGaaiOkaiaaiIcacaWGYbGaaGykaaaaaaa@3B08@ and the raked replicate estimates t ^ y,rak (r) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmiDayaaja Waa0baaSqaaiaadMhacaaISaGaaGPaVlaabkhacaqGHbGaae4Aaaqa aiaaiIcacaWGYbGaaGykaaaakiaac6caaaa@403B@ Using the approximation in (3.5) for each replicate, we obtain that the resulting replication variance estimator is consistent for the asymptotic variance of the raking estimator.


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