Register-based sampling for household panels 5. Linear weighting

For household surveys like the RIS, estimates are required for person characteristics as well as household characteristics. Let t y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG0bWaaS baaSqaaiaadMhaaeqaaaaa@3A83@  denote the total of a target variable y . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG5bGaai Olaaaa@3A10@  With linear weighting, an estimator for a person based target variable is defined as

t ^ y = h = 1 H k 1 m h j k w k j y k j h , ( 5.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWG0bGbaK aadaWgaaWcbaGaamyEaaqabaGccqGH9aqpdaaeWbqaamaaqahabaWa aabuaeaacaWG3bWaaSbaaSqaaiaadUgacaWGQbaabeaakiaadMhada WgaaWcbaGaam4AaiaadQgacaWGObaabeaaaeaacaWGQbGaeyicI4Sa am4Aaaqab0GaeyyeIuoaaSqaaiaadUgacqGHiiIZcaaIXaaabaGaam yBamaaBaaameaacaWGObaabeaaa0GaeyyeIuoaaSqaaiaadIgacqGH 9aqpcaaIXaaabaGaamisaaqdcqGHris5aOGaaiilaiaaywW7caaMf8 UaaGzbVlaaywW7caaMf8UaaiikaiaaiwdacaGGUaGaaGymaiaacMca aaa@6163@

with y k j h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS baaSqaaiaadUgacaWGQbGaamiAaaqabaaaaa@3C56@ the value of the target variable for person ( k , j , h ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaadaqadaqaai aadUgacaGGSaGaamOAaiaacYcacaWGObaacaGLOaGaayzkaaaaaa@3E15@ and w k j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG3bWaaS baaSqaaiaadUgacaWGQbaabeaaaaa@3B67@ a weight for person j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGQbaaaa@394F@ belonging to household k . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGRbGaai Olaaaa@3A02@ An estimator for a household based target variable is given by

t ^ y = h = 1 H k = 1 m h w k y k h , ( 5.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWG0bGbaK aadaWgaaWcbaGaamyEaaqabaGccqGH9aqpdaaeWbqaamaaqahabaGa am4DamaaBaaaleaacaWGRbaabeaakiaadMhadaWgaaWcbaGaam4Aai aadIgaaeqaaaqaaiaadUgacqGH9aqpcaaIXaaabaGaamyBamaaBaaa meaacaWGObaabeaaa0GaeyyeIuoaaSqaaiaadIgacqGH9aqpcaaIXa aabaGaamisaaqdcqGHris5aOGaaiilaiaaywW7caaMf8UaaGzbVlaa ywW7caaMf8UaaiikaiaaiwdacaGGUaGaaGOmaiaacMcaaaa@5982@

with y k h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS baaSqaaiaadUgacaWGObaabeaaaaa@3B67@ the value of the target variable for household k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGRbaaaa@3950@ from stratum h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGObaaaa@394D@ and w k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG3bWaaS baaSqaaiaadUgaaeqaaaaa@3A78@ a weight for the corresponding household.

Weights are obtained by means of the GREG estimator to use auxiliary variables which are observed in the sample and for which the population totals are known from other sources (Särndal et al. 1992). Consequently, the weights reflect the (unequal) inclusion expectations of the sampling units and an adjustment such that for auxiliary variables the weighted observations sum to the known population totals. Often categorical variables like gender, age, marital status or region are used as auxiliary variables. Due to the fact that the values of auxiliary variables differ from person to person within the same household, different weights can be derived for people from the same household. To ensure that relationships between household variables and person variables are reflected in estimated totals, it is relevant to apply a weighting method which yields one unique household weight for all its household members. If the weights for persons within a household are the same, then household and person based estimates of the same target variables are consistent with each other (for example the total income estimated from households and that from persons). This can be achieved with so-called integrated weighting methods.

Lemaître and Dufour (1987) apply an integrated weighting method at the persons level and replace the original auxiliary variables defined at the person level by the corresponding household mean. In this way, members of the same household have the same inclusion expectation and share the same auxiliary information, and therefore the resulting regression weights are forced to be the same. Nieuwenbroek (1993) proposes a slightly more general approach by applying the linear weighting method at the household level, where the auxiliary information of person based characteristics is aggregated at the household level. Nieuwenbroek (1993) mentions that the linear weighting method at the household level is equal to the linear weighting method of Lemaître and Dufour (1987) at the person level, if the residual variance of the regression model at the household level is chosen proportional to the number of persons within the household. Steel and Clark (2007) and Estevao and Särndal (2006) further generalize the integrated weighting of person and household surveys. Steel and Clark (2007) address the issue of whether the cosmetic benefits of integrated weighting result in an increased design variance of the GREG estimates. They show that large-sample design variances obtained by linear weighting at the household level is less than or equal to the design variance obtained with linear weighting at the person level. For small samples there can be a small increase in the design variance due to integrated weighting. As a result there is little or no loss in efficiency by applying an integrated weighting method.

In this paper the integrated weighting approach at the household level is applied. Let x k h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bWaaS baaSqaaiaadUgacaWGObaabeaaaaa@3B6A@ denote a q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGXbaaaa@3956@ -vector containing q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGXbaaaa@3956@ auxiliary variables for household k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGRbaaaa@3950@ from stratum h . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGObGaai Olaaaa@39FF@ Person based characteristics are aggregated to household totals. The GREG estimator is derived from a linear regression model that specifies the relation between the target variable and the available auxiliary variables for which population totals are known, and is defined as:

y k h = x k h t β + e k h , with E m ( e k h ) = 0 , V m ( e k h ) = σ k h 2 . ( 5.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS baaSqaaiaadUgacaWGObaabeaakiabg2da9iaahIhadaqhaaWcbaGa am4AaiaadIgaaeaacaWG0baaaOGaaCOSdiabgUcaRiaadwgadaWgaa WcbaGaam4AaiaadIgaaeqaaOGaaiilaiaaywW7caqG3bGaaeyAaiaa bshacaqGObGaaGzbVlaabweadaWgaaWcbaGaamyBaaqabaGcdaqada qaaiaadwgadaWgaaWcbaGaam4AaiaadIgaaeqaaaGccaGLOaGaayzk aaGaeyypa0JaaGimaiaacYcacaaMf8UaaeOvamaaBaaaleaacaWGTb aabeaakmaabmaabaGaamyzamaaBaaaleaacaWGRbGaamiAaaqabaaa kiaawIcacaGLPaaacqGH9aqpcqaHdpWCdaqhaaWcbaGaam4AaiaadI gaaeaacaaIYaaaaOGaaiOlaiaaywW7caaMf8UaaGzbVlaaywW7caaM f8UaaiikaiaaiwdacaGGUaGaaG4maiaacMcaaaa@6FC8@

In (5.3) β MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWHYoaaaa@399E@ denotes a vector containing the q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGXbaaaa@3956@ regression coefficients of the regression of y k h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG5bWaaS baaSqaaiaadUgacaWGObaabeaaaaa@3B67@ on x k h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bWaaS baaSqaaiaadUgacaWGObaabeaaaaa@3B6A@ and e k h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGLbWaaS baaSqaaiaadUgacaWGObaabeaaaaa@3B53@ the residuals and E m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaqGfbWaaS baaSqaaiaad2gaaeqaaaaa@3A46@ and V m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaqGwbWaaS baaSqaaiaad2gaaeqaaaaa@3A57@ denote the expectation and variance with respect to the regression model. In this application, the variance structure is taken proportional to the household size, i.e., σ h k 2 = g k σ 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqaHdpWCda qhaaWcbaGaamiAaiaadUgaaeaacaaIYaaaaOGaeyypa0Jaam4zamaa BaaaleaacaWGRbaabeaakiabeo8aZnaaCaaaleqabaGaaGOmaaaaki aac6caaaa@4373@ Nieuwenbroek (1993) shows that in this case the weighting applied at the household level is equal to the method of Lemaître and Dufour (1987).

Regression weights for the households are finally obtained by

w k = 1 π k ( 1 + ( t x t ^ x π ) t ( k = 1 m x k h x k h t π k g k ) 1 x k h g k ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG3bWaaS baaSqaaiaadUgaaeqaaOGaeyypa0ZaaSaaaeaacaaIXaaabaGaeqiW da3aaSbaaSqaaiaadUgaaeqaaaaakmaabmaabaGaaGymaiabgUcaRm aabmaabaGaaCiDamaaBaaaleaacaWH4baabeaakiabgkHiTiqahsha gaqcamaaBaaaleaacaWH4bGaaCiWdaqabaaakiaawIcacaGLPaaada ahaaWcbeqaaiaadshaaaGcdaqadaqaamaaqahabaWaaSaaaeaacaWH 4bWaaSbaaSqaaiaadUgacaWGObaabeaakiaahIhadaqhaaWcbaGaam 4AaiaadIgaaeaacaWG0baaaaGcbaGaeqiWda3aaSbaaSqaaiaadUga aeqaaOGaam4zamaaBaaaleaacaWGRbaabeaaaaaabaGaam4Aaiabg2 da9iaaigdaaeaacaWGTbaaniabggHiLdaakiaawIcacaGLPaaadaah aaWcbeqaaiabgkHiTiaaigdaaaGcdaWcaaqaaiaahIhadaWgaaWcba Gaam4AaiaadIgaaeqaaaGcbaGaam4zamaaBaaaleaacaWGRbaabeaa aaaakiaawIcacaGLPaaacaGGSaaaaa@6714@

with t x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH0bWaaS baaSqaaiaahIhaaeqaaaaa@3A8A@ a q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGXbaaaa@3956@ vector containing the known population totals of the auxiliary variables x , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH4bGaai ilaaaa@3A11@ t ^ x π MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWH0bGbaK aadaWgaaWcbaGaaCiEaiaahc8aaeqaaaaa@3BE6@ the HT estimator for t x . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWH0bWaaS baaSqaaiaahIhaaeqaaOGaaiOlaaaa@3B46@ The weights calculated at the household level can be used for weighting person based characteristics of the corresponding household members, using formula (5.1) since w k j = w k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG3bWaaS baaSqaaiaadUgacaWGQbaabeaakiabg2da9iaadEhadaWgaaWcbaGa am4Aaaqabaaaaa@3E8F@ for all persons belonging to the same household k . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGRbGaai Olaaaa@3A02@

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