Register-based sampling for household panels 6. Variance estimation

Parameters of the RIS are estimated as the ratio of two population totals

R ^ = t ^ y t ^ z , ( 6.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGsbGbaK aacqGH9aqpdaWcaaqaaiqadshagaqcamaaBaaaleaacaWG5baabeaa aOqaaiqadshagaqcamaaBaaaleaacaWG6baabeaaaaGccaGGSaGaaG zbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaGOnaiaac6cacaaI XaGaaiykaaaa@4AD3@

where t ^ y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWG0bGbaK aadaWgaaWcbaGaamyEaaqabaaaaa@3A93@ and t ^ z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWG0bGbaK aadaWgaaWcbaGaamOEaaqabaaaaa@3A94@ are GREG estimators defined by (5.1) or (5.2) in the case of person-based or household-based target variables, respectively. The variance of (6.1) under a sample design where core persons are drawn by means of stratified simple random sampling, and all household members of these core persons are included in the sample can be approximated by

V ( R ^ ) = 1 t z 2 h = 1 H N h 2 ( 1 f h ) n h 1 N h 1 k = 1 N h ( e k h g k 1 N h k = 1 N h e k h g k ) 2 , ( 6.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGwbWaae WaaeaaceWGsbGbaKaaaiaawIcacaGLPaaacqGH9aqpdaWcaaqaaiaa igdaaeaacaWG0bWaa0baaSqaaiaadQhaaeaacaaIYaaaaaaakmaaqa habaWaaSaaaeaacaWGobWaa0baaSqaaiaadIgaaeaacaaIYaaaaOWa aeWaaeaacaaIXaGaeyOeI0IaamOzamaaBaaaleaacaWGObaabeaaaO GaayjkaiaawMcaaaqaaiaad6gadaWgaaWcbaGaamiAaaqabaaaaOWa aSaaaeaacaaIXaaabaGaamOtamaaBaaaleaacaWGObaabeaakiabgk HiTiaaigdaaaaaleaacaWGObGaeyypa0JaaGymaaqaaiaadIeaa0Ga eyyeIuoakiaaykW7daaeWbqaamaabmaabaWaaSaaaeaacaWGLbWaaS baaSqaaiaadUgacaWGObaabeaaaOqaaiaadEgadaWgaaWcbaGaam4A aaqabaaaaOGaeyOeI0YaaSaaaeaacaaIXaaabaGaamOtamaaBaaale aacaWGObaabeaaaaGcdaaeWbqaamaalaaabaGaamyzamaaBaaaleaa ceWGRbGbauaacaWGObaabeaaaOqaaiaadEgadaWgaaWcbaGabm4Aay aafaaabeaaaaaabaGabm4AayaafaGaeyypa0JaaGymaaqaaiaad6ea daWgaaadbaGaamiAaaqabaaaniabggHiLdaakiaawIcacaGLPaaaaS qaaiaadUgacqGH9aqpcaaIXaaabaGaamOtamaaBaaameaacaWGObaa beaaa0GaeyyeIuoakmaaCaaaleqabaGaaGOmaaaakiaacYcacaaMf8 UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaI2aGaaiOlaiaaikda caGGPaaaaa@8081@

where f h = n h / N h , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGMbWaaS baaSqaaiaadIgaaeqaaOGaeyypa0ZaaSGbaeaacaWGUbWaaSbaaSqa aiaadIgaaeqaaaGcbaGaamOtamaaBaaaleaacaWGObaabeaaaaGcca GGSaaaaa@4046@ e k h = ( y k h x k h t b y ) R ( z k h x k h t b z ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGLbWaaS baaSqaaiaadUgacaWGObaabeaakiabg2da9maabmaabaGaamyEamaa BaaaleaacaWGRbGaamiAaaqabaGccqGHsislcaWH4bWaa0baaSqaai aadUgacaWGObaabaGaamiDaaaakiaahkgadaWgaaWcbaGaamyEaaqa baaakiaawIcacaGLPaaacqGHsislcaWGsbWaaeWaaeaacaWG6bWaaS baaSqaaiaadUgacaWGObaabeaakiabgkHiTiaahIhadaqhaaWcbaGa am4AaiaadIgaaeaacaWG0baaaOGaaCOyamaaBaaaleaacaWG6baabe aaaOGaayjkaiaawMcaaiaacYcaaaa@5641@ and b y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHIbWaaS baaSqaaiaadMhaaeqaaaaa@3A75@ and b z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHIbWaaS baaSqaaiaadQhaaeqaaaaa@3A76@ are the finite population regression coefficients of the regression of y k h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS baaSqaaiaadUgacaWGObaabeaaaaa@3B67@ and z k h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG6bWaaS baaSqaaiaadUgacaWGObaabeaaaaa@3B68@ respectively on x k h . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bWaaS baaSqaaiaadUgacaWGObaabeaakiaac6caaaa@3C26@ An estimator for the variance specified in (6.2) is given by

V ^ ( R ^ ) = 1 t ^ z 2 h = 1 H ( 1 f h ) n h n h 1 k = 1 n h ( w k e ^ k 1 n h k = 1 n h w k e ^ k h ) 2 , ( 6.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGwbGbaK aadaqadaqaaiqadkfagaqcaaGaayjkaiaawMcaaiabg2da9maalaaa baGaaGymaaqaaiqadshagaqcamaaDaaaleaacaWG6baabaGaaGOmaa aaaaGcdaaeWbqaamaabmaabaGaaGymaiabgkHiTiaadAgadaWgaaWc baGaamiAaaqabaaakiaawIcacaGLPaaadaWcaaqaaiaad6gadaWgaa WcbaGaamiAaaqabaaakeaacaWGUbWaaSbaaSqaaiaadIgaaeqaaOGa eyOeI0IaaGymaaaaaSqaaiaadIgacqGH9aqpcaaIXaaabaGaamisaa qdcqGHris5aOGaaGPaVpaaqahabaWaaeWaaeaacaWG3bWaaSbaaSqa aiaadUgaaeqaaOGabmyzayaajaWaaSbaaSqaaiaadUgaaeqaaOGaey OeI0YaaSaaaeaacaaIXaaabaGaamOBamaaBaaaleaacaWGObaabeaa aaGcdaaeWbqaaiaadEhadaWgaaWcbaGabm4Aayaafaaabeaakiqadw gagaqcamaaBaaaleaaceWGRbGbauaacaWGObaabeaaaeaaceWGRbGb auaacqGH9aqpcaaIXaaabaGaamOBamaaBaaameaacaWGObaabeaaa0 GaeyyeIuoaaOGaayjkaiaawMcaaaWcbaGaam4Aaiabg2da9iaaigda aeaacaWGUbWaaSbaaWqaaiaadIgaaeqaaaqdcqGHris5aOWaaWbaaS qabeaacaaIYaaaaOGaaiilaiaaywW7caaMf8UaaGzbVlaaywW7caaM f8UaaiikaiaaiAdacaGGUaGaaG4maiaacMcaaaa@7CD7@

where e ^ k h = ( y k h x k h t b ^ y ) R ^ ( z k h x k h t b ^ z ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGLbGbaK aadaWgaaWcbaGaam4AaiaadIgaaeqaaOGaeyypa0ZaaeWaaeaacaWG 5bWaaSbaaSqaaiaadUgacaWGObaabeaakiabgkHiTiaahIhadaqhaa WcbaGaam4AaiaadIgaaeaacaWG0baaaOGabCOyayaajaWaaSbaaSqa aiaadMhaaeqaaaGccaGLOaGaayzkaaGaeyOeI0IabmOuayaajaWaae WaaeaacaWG6bWaaSbaaSqaaiaadUgacaWGObaabeaakiabgkHiTiaa hIhadaqhaaWcbaGaam4AaiaadIgaaeaacaWG0baaaOGabCOyayaaja WaaSbaaSqaaiaadQhaaeqaaaGccaGLOaGaayzkaaaaaa@55D1@ and b ^ y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWHIbGbaK aadaWgaaWcbaGaamyEaaqabaaaaa@3A85@ and b ^ z MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWHIbGbaK aadaWgaaWcbaGaamOEaaqabaaaaa@3A86@ are the HT type estimators for b y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHIbWaaS baaSqaaiaadMhaaeqaaaaa@3A75@ and b z . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHIbWaaS baaSqaaiaadQhaaeqaaOGaaiOlaaaa@3B32@ These results follow directly from inserting first and second order inclusion expectations specified in (3.3) through (3.6) in the general approximation for the variance of the ratio of two GREG estimators and its estimator (Särndal et al. 1992, Section 7.13).

The same expressions for the variance can be derived from the variance expressions proposed for the Generalized Weight Share method in the case of indirect sampling. In Lavallée (1995), variance expressions for the HT estimator are based on the sampling design used to select the sample s A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGZbWaa0 baaSqaaaqaaiaadgeaaaaaaa@3A4B@ of n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbaaaa@3953@ units from population U A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaa0 baaSqaaaqaaiaadgeaaaaaaa@3A2D@ with transformed target variables, say z i . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG6bWaaS baaSqaaiaadMgaaeqaaOGaaiOlaaaa@3B35@ In this application each unit in U A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaW baaSqabeaacaWGbbaaaaaa@3A2D@ has exactly one link with a unit in U B . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaW baaSqabeaacaWGcbaaaOGaaiOlaaaa@3AEA@ As a result z i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG6bWaaS baaSqaaiaadMgaaeqaaaaa@3A79@ in Lavallée (1995) is in this case defined as the sum over the target variables of all elements in cluster k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGRbGaai ilaaaa@3A00@ divided by the number of units in cluster k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGRbaaaa@3950@ with a link to population U A , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaW baaSqabeaacaWGbbaaaOGaaiilaaaa@3AE7@ i.e., z i = y k / g k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG6bWaaS baaSqaaiaadMgaaeqaaOGaeyypa0ZaaSGbaeaacaWG5bWaaSbaaSqa aiaadUgaaeqaaaGcbaGaam4zamaaBaaaleaacaWGRbaabeaaaaaaaa@3FCB@ for all i U A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGPbGaey icI4SaamyvamaaCaaaleqabaGaamyqaaaaaaa@3C9F@ that have a link with cluster k U B . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGRbGaey icI4SaamyvamaaCaaaleqabaGaamOqaaaakiaac6caaaa@3D5E@ Inserting the first and second order inclusion probabilities for stratified simple random sampling without replacement and the transformed variables z i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG6bWaaS baaSqaaiaadMgaaeqaaaaa@3A79@ (where the target variable y k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS baaSqaaiaadUgaaeqaaaaa@3A7A@ is preplaced by the residual of the regression on the cluster totals e k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaqacaqaai aadwgadaWgaaWcbaGaam4AaaqabaaakiaawMcaaaaa@3B38@ in the variance formula for a ratio gives (6.2). Result (6.3) follows in a similar way.

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