Register-based sampling for household panels 3. Inclusion weights

3.1 Weighting with inclusion expectations

For design-based inference, first and second order inclusion probabilities for households and persons are required. Let M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamytaaaa@37FE@ denote the number of households in the population, N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOtaaaa@37FF@ the number of persons in the population aged 15 years or over and g k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaWGRbaabeaaaaa@3934@ the number of persons aged 15 years or over that belong to the k th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4AamaaCa aaleqabaGaaeiDaiaabIgaaaaaaa@3A2B@ household. With the sample design described in Section 2, households k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@381C@ can be included more than once but a maximum of g k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaWGRbaabeaaaaa@3934@ times. This complicates the derivation of inclusion probabilities since the probability of selecting household k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@381C@ is equal to the selection probability of the union of its household members ( k , j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGRbGaaiilaiaadQgaaiaawIcacaGLPaaaaaa@3B44@ aged 15 years and over. This probability is defined as:

P ( k s ) = P ( j = 1 g k [ ( k , j ) s ] ) = j = 1 g k P ( ( k , j ) s ) j = 1 g k j = j + 1 g k P ( [ ( k , j ) ( k , j ) ] s ) + j = 1 g k j = j + 1 g k j = j + j + 1 g k P ( [ ( k , j ) ( k , j ) ( k , j ) ] s ) ... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqbaeaabmGaaa qaaiaadcfadaqadaqaaiaadUgacqGHiiIZcaWGZbaacaGLOaGaayzk aaGaeyypa0JaamiuamaabmaabaWaambCaeaadaWadaqaamaabmaaba Gaam4AaiaacYcacaWGQbaacaGLOaGaayzkaaGaeyicI4Saam4CaaGa ay5waiaaw2faaaWcbaGaamOAaiabg2da9iaaigdaaeaacaWGNbWaaS baaWqaaiaadUgaaeqaaaqdcqWIQisvaaGccaGLOaGaayzkaaaabaGa eyypa0ZaaabCaeaacaWGqbWaaeWaaeaadaqadaqaaiaadUgacaGGSa GaamOAaaGaayjkaiaawMcaaiabgIGiolaadohaaiaawIcacaGLPaaa aSqaaiaadQgacqGH9aqpcaaIXaaabaGaam4zamaaBaaameaacaWGRb aabeaaa0GaeyyeIuoaaOqaaaqaaiabgkHiTmaaqahabaWaaabCaeaa caWGqbWaaeWaaeaadaWadaqaamaabmaabaGaam4AaiaacYcacaWGQb aacaGLOaGaayzkaaGaeyykIC8aaeWaaeaacaWGRbGaaiilaiqadQga gaqbaaGaayjkaiaawMcaaaGaay5waiaaw2faaiabgIGiolaadohaai aawIcacaGLPaaaaSqaaiqadQgagaqbaiabg2da9iaadQgacqGHRaWk caaIXaaabaGaam4zamaaBaaameaacaWGRbaabeaaa0GaeyyeIuoaaS qaaiaadQgacqGH9aqpcaaIXaaabaGaam4zamaaBaaameaacaWGRbaa beaaa0GaeyyeIuoaaOqaaaqaaiabgUcaRmaaqahabaWaaabCaeaada aeWbqaaiaadcfadaqadaqaamaadmaabaWaaeWaaeaacaWGRbGaaiil aiaadQgaaiaawIcacaGLPaaacqGHPiYXdaqadaqaaiaadUgacaGGSa GabmOAayaafaaacaGLOaGaayzkaaGaeyykIC8aaeWaaeaacaWGRbGa aiilaiqadQgagaGbaaGaayjkaiaawMcaaaGaay5waiaaw2faaiabgI GiolaadohaaiaawIcacaGLPaaaaSqaaiqadQgagaGbaiabg2da9iaa dQgacqGHRaWkceWGQbGbauaacqGHRaWkcaaIXaaabaGaam4zamaaBa aameaacaWGRbaabeaaa0GaeyyeIuoaaSqaaiqadQgagaqbaiabg2da 9iaadQgacqGHRaWkcaaIXaaabaGaam4zamaaBaaameaacaWGRbaabe aaa0GaeyyeIuoaaSqaaiaadQgacqGH9aqpcaaIXaaabaGaam4zamaa BaaameaacaWGRbaabeaaa0GaeyyeIuoakiabgkHiTiaac6cacaGGUa GaaiOlaaaaaaa@B6D5@

This kind of computation can be avoided by using the concept of inclusion expectations instead of inclusion probabilities. Bethlehem (2009), Chapter 2, generalizes the HT estimator to the concept of inclusion expectation for sampling with replacement. Let a k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGRbaabeaaaaa@392E@ denote the number of times that household k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@381C@ is selected in the sample. In the proposed sample design a k [ 0 , 1 , , g k ] . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGRbaabeaakiabgIGiopaadmaabaGaaGimaiaacYcacaaI XaGaaiilaiablAciljaacYcacaWGNbWaaSbaaSqaaiaadUgaaeqaaa GccaGLBbGaayzxaaGaaiOlaaaa@4419@ Let E ( . ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeyramaabm aabaGaaiOlaaGaayjkaiaawMcaaaaa@3A2F@ denote the expectation with respect to the sample design. Now π k = E ( a k ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaadUgaaeqaaOGaeyypa0JaaeyramaabmaabaGaamyyamaa BaaaleaacaWGRbaabeaaaOGaayjkaiaawMcaaaaa@3F72@ denotes the inclusion expectation of sampling unit k . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaiaac6 caaaa@38CE@ Since a k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGRbaabeaaaaa@392E@ can be larger than one, π k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaadUgaaeqaaaaa@3A05@ can also take values larger than one and can therefore no longer be interpreted as an inclusion probability. It can, however, be interpreted as an expectation.

The parameter of interest is the population total, which is defined as

t y = k = 1 M j = 1 N k y k j k = 1 M y k . ( 3.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDamaaBa aaleaacaWG5baabeaakiabg2da9maaqahabaWaaabCaeaacaWG5bWa aSbaaSqaaiaadUgacaWGQbaabeaaaeaacaWGQbGaeyypa0JaaGymaa qaaiaad6eadaWgaaadbaGaam4AaaqabaaaniabggHiLdaaleaacaWG RbGaeyypa0JaaGymaaqaaiaad2eaa0GaeyyeIuoakiabggMi6oaaqa habaGaamyEamaaBaaaleaacaWGRbaabeaaaeaacaWGRbGaeyypa0Ja aGymaaqaaiaad2eaa0GaeyyeIuoakiaac6cacaaMf8UaaGzbVlaayw W7caaMf8UaaGzbVlaacIcacaaIZaGaaiOlaiaaigdacaGGPaaaaa@5FAF@

The HT estimator for the population total in (3.1) can be defined as

t ^ y = k = 1 M a k y k π k . ( 3.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmiDayaaja WaaSbaaSqaaiaadMhaaeqaaOGaeyypa0ZaaabCaeaadaWcaaqaaiaa dggadaWgaaWcbaGaam4AaaqabaGccaWG5bWaaSbaaSqaaiaadUgaae qaaaGcbaGaeqiWda3aaSbaaSqaaiaadUgaaeqaaaaakiaac6caaSqa aiaadUgacqGH9aqpcaaIXaaabaGaamytaaqdcqGHris5aOGaaGzbVl aaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaG4maiaac6cacaaIYaGa aiykaaaa@535C@

Since E ( a k ) = π k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaaeyramaabm aabaGaamyyamaaBaaaleaacaWGRbaabeaaaOGaayjkaiaawMcaaiab g2da9iabec8aWnaaBaaaleaacaWGRbaabeaakiaacYcaaaa@4022@ it follows that this HT estimator is design unbiased. Let π k k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaadUgaceWGRbGbauaaaeqaaaaa@3B01@ denote the inclusion expectation of units k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@381C@ and k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabm4Aayaafa Gaaiilaaaa@38D8@ i.e., π k k = E ( a k a k ) . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaadUgacaWGRbGaam4jaaqabaGccqGH9aqpcaqGfbWaaeWa aeaacaWGHbWaaSbaaSqaaiaadUgaaeqaaOGaamyyamaaBaaaleaace WGRbGbauaaaeqaaaGccaGLOaGaayzkaaGaaiOlaaaa@43D8@ The variance of the HT estimator is by definition equal to

V ( t ^ y ) = k = 1 M k = 1 M Cov ( a k a k ) y k π k y k π k = k = 1 M k = 1 M [ E ( a k a k ) E ( a k ) E ( a k ) ] y k π k y k π k = k = 1 M k = 1 M ( π k k π k π k ) y k π k y k π k . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqbaeaabmGaaa qaaiaabAfadaqadaqaaiqadshagaqcamaaBaaaleaacaWG5baabeaa aOGaayjkaiaawMcaaaqaaiabg2da9maaqahabaWaaabCaeaacaqGdb Gaae4BaiaabAhadaqadaqaaiaadggadaWgaaWcbaGaam4AaaqabaGc caWGHbWaaSbaaSqaaiqadUgagaqbaaqabaaakiaawIcacaGLPaaada WcaaqaaiaadMhadaWgaaWcbaGaam4AaaqabaaakeaacqaHapaCdaWg aaWcbaGaam4AaaqabaaaaOWaaSaaaeaacaWG5bWaaSbaaSqaaiqadU gagaqbaaqabaaakeaacqaHapaCdaWgaaWcbaGabm4Aayaafaaabeaa aaaabaGabm4AayaafaGaeyypa0JaaGymaaqaaiaad2eaa0GaeyyeIu oaaSqaaiaadUgacqGH9aqpcaaIXaaabaGaamytaaqdcqGHris5aaGc baaabaGaeyypa0ZaaabCaeaadaaeWbqaamaadmaabaGaaeyramaabm aabaGaamyyamaaBaaaleaacaWGRbaabeaakiaadggadaWgaaWcbaGa bm4AayaafaaabeaaaOGaayjkaiaawMcaaiabgkHiTiaabweadaqada qaaiaadggadaWgaaWcbaGaam4AaaqabaaakiaawIcacaGLPaaacaqG fbWaaeWaaeaacaWGHbWaaSbaaSqaaiqadUgagaqbaaqabaaakiaawI cacaGLPaaaaiaawUfacaGLDbaadaWcaaqaaiaadMhadaWgaaWcbaGa am4AaaqabaaakeaacqaHapaCdaWgaaWcbaGaam4AaaqabaaaaOWaaS aaaeaacaWG5bWaaSbaaSqaaiqadUgagaqbaaqabaaakeaacqaHapaC daWgaaWcbaGabm4AayaafaaabeaaaaaabaGabm4AayaafaGaeyypa0 JaaGymaaqaaiaad2eaa0GaeyyeIuoaaSqaaiaadUgacqGH9aqpcaaI XaaabaGaamytaaqdcqGHris5aaGcbaaabaGaeyypa0ZaaabCaeaada aeWbqaamaabmaabaGaeqiWda3aaSbaaSqaaiaadUgaceWGRbGbauaa aeqaaOGaeyOeI0IaeqiWda3aaSbaaSqaaiaadUgaaeqaaOGaeqiWda 3aaSbaaSqaaiqadUgagaqbaaqabaaakiaawIcacaGLPaaadaWcaaqa aiaadMhadaWgaaWcbaGaam4AaaqabaaakeaacqaHapaCdaWgaaWcba Gaam4AaaqabaaaaOWaaSaaaeaacaWG5bWaaSbaaSqaaiqadUgagaqb aaqabaaakeaacqaHapaCdaWgaaWcbaGabm4Aayaafaaabeaaaaaaba Gabm4AayaafaGaeyypa0JaaGymaaqaaiaad2eaa0GaeyyeIuoaaSqa aiaadUgacqGH9aqpcaaIXaaabaGaamytaaqdcqGHris5aOGaaiOlaa aaaaa@A7B2@

Note that in the case of sampling without replacement a k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGRbaabeaaaaa@392E@ is a dummy taking values zero or one indicating whether unit k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@381C@ is selected in the sample. In this case π k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaadUgaaeqaaaaa@3A05@ and π k k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaadUgaceWGRbGbauaaaeqaaaaa@3B01@ are the usual first and second order inclusion probabilities. This illustrates that the standard HT estimator, based on inclusion probabilities, can be extended easily to inclusion expectations. In the case of sample designs where units can be selected more than once, it is more convenient to work with inclusion expectations, since they are derived relatively easily. In the remainder of this subsection, first and second order inclusion expectations for the sample design described in Section 2 are derived.

Core persons are drawn by means of stratified simple random sampling. Since stratification is based on geographical regions, all members of a household k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@381C@ belong to the same stratum h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiAaaaa@3819@ at the moment of drawing core persons. Let N h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa aaleaacaWGObaabeaaaaa@3918@ denote the number of persons in the population of stratum h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiAaaaa@3819@ aged 15 years or over, n h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOBamaaBa aaleaacaWGObaabeaaaaa@3938@ the number of core persons selected in the sample from stratum h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiAaaaa@3819@ and g k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4zamaaBa aaleaacaWGRbaabeaaaaa@3934@ the number of persons aged 15 years or over, belonging to household k . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaiaac6 caaaa@38CE@ Finally, a j k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGQbGaam4Aaaqabaaaaa@3A1D@ denotes an indicator that is equal to one if person j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOAaaaa@381B@ from household k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@381C@ is selected in the sample and zero otherwise. The first order inclusion expectation of the k th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4AamaaCa aaleqabaGaaeiDaiaabIgaaaaaaa@3A2B@ household equals

π k h = E ( a k ) = E ( j = 1 g k a j k ) = j = 1 g k E ( a j k ) = g k n h N h . ( 3.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaadUgacaWGObaabeaakiabg2da9iaabweadaqadaqaaiaa dggadaWgaaWcbaGaam4AaaqabaaakiaawIcacaGLPaaacqGH9aqpca WGfbWaaeWaaeaadaaeWbqaaiaadggadaWgaaWcbaGaamOAaiaadUga aeqaaaqaaiaadQgacqGH9aqpcaaIXaaabaGaam4zamaaBaaameaaca WGRbaabeaaa0GaeyyeIuoaaOGaayjkaiaawMcaaiabg2da9maaqaha baGaamyramaabmaabaGaamyyamaaBaaaleaacaWGQbGaam4Aaaqaba aakiaawIcacaGLPaaaaSqaaiaadQgacqGH9aqpcaaIXaaabaGaam4z amaaBaaameaacaWGRbaabeaaa0GaeyyeIuoakiabg2da9iaadEgada WgaaWcbaGaam4AaaqabaGcdaWcaaqaaiaad6gadaWgaaWcbaGaamiA aaqabaaakeaacaWGobWaaSbaaSqaaiaadIgaaeqaaaaakiaac6caca aMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaIZaGaaiOlaiaa iodacaGGPaaaaa@6E2C@

Second order inclusion expectations for households k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@381C@ and k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabm4Aayaafa aaaa@3828@ for k k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaiabgc Mi5kqadUgagaqbaaaa@3ADF@ belonging to the same stratum h , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiAaiaacY caaaa@38C9@ equal

π k k = E ( a k a k ) = E ( j = 1 g k a j k j = 1 g k a j k ) = j = 1 g k j = 1 g k E ( a j k a j k ) = g k g k n h ( n h 1 ) N h ( N h 1 ) . ( 3.4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaadUgaceWGRbGbauaaaeqaaOGaeyypa0Jaaeyramaabmaa baGaamyyamaaBaaaleaacaWGRbaabeaakiaadggadaWgaaWcbaGabm 4AayaafaaabeaaaOGaayjkaiaawMcaaiabg2da9iaadweadaqadaqa amaaqahabaGaamyyamaaBaaaleaacaWGQbGaam4AaaqabaaabaGaam OAaiabg2da9iaaigdaaeaacaWGNbWaaSbaaWqaaiaadUgaaeqaaaqd cqGHris5aOWaaabCaeaacaWGHbWaaSbaaSqaaiqadQgagaqbaiqadU gagaqbaaqabaaabaGabmOAayaafaGaeyypa0JaaGymaaqaaiaadEga daWgaaadbaGabm4Aayaafaaabeaaa0GaeyyeIuoaaOGaayjkaiaawM caaiabg2da9maaqahabaWaaabCaeaacaWGfbWaaeWaaeaacaWGHbWa aSbaaSqaaiaadQgacaWGRbaabeaakiaadggadaWgaaWcbaGabmOAay aafaGabm4AayaafaaabeaaaOGaayjkaiaawMcaaaWcbaGabmOAayaa faGaeyypa0JaaGymaaqaaiaadEgadaWgaaadbaGabm4Aayaafaaabe aaa0GaeyyeIuoaaSqaaiaadQgacqGH9aqpcaaIXaaabaGaam4zamaa BaaameaacaWGRbaabeaaa0GaeyyeIuoakiabg2da9iaadEgadaWgaa WcbaGaam4AaaqabaGccaWGNbWaaSbaaSqaaiqadUgagaqbaaqabaGc daWcaaqaaiaad6gadaWgaaWcbaGaamiAaaqabaGcdaqadaqaaiaad6 gadaWgaaWcbaGaamiAaaqabaGccqGHsislcaaIXaaacaGLOaGaayzk aaaabaGaamOtamaaBaaaleaacaWGObaabeaakmaabmaabaGaamOtam aaBaaaleaacaWGObaabeaakiabgkHiTiaaigdaaiaawIcacaGLPaaa aaGaaiOlaiaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaG4maiaac6 cacaaI0aGaaiykaaaa@8F93@

The second order inclusion expectation for household k = k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaiabg2 da9iqadUgagaqbaaaa@3A1E@ from the same stratum h , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiAaiaacY caaaa@38C9@ is given by

π k k = E ( a k a k ) = E ( j = 1 g k a j k j = 1 g k a j k ) = E ( j = 1 g k a j k + j = 1 g k j j = 1 g k a j k a j k ) = j = 1 g k E ( a j k ) + j = 1 g k j j = 1 g k E ( a j k a j k ) = g k n h N h + g k ( g k 1 ) n h ( n h 1 ) N h ( N h 1 ) . ( 3.5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaqbaeaabiGaaa qaaiabec8aWnaaBaaaleaacaWGRbGaam4AaaqabaaakeaacqGH9aqp caqGfbWaaeWaaeaacaWGHbWaaSbaaSqaaiaadUgaaeqaaOGaamyyam aaBaaaleaacaWGRbaabeaaaOGaayjkaiaawMcaaiabg2da9iaadwea daqadaqaamaaqahabaGaamyyamaaBaaaleaacaWGQbGaam4Aaaqaba aabaGaamOAaiabg2da9iaaigdaaeaacaWGNbWaaSbaaWqaaiaadUga aeqaaaqdcqGHris5aOWaaabCaeaacaWGHbWaaSbaaSqaaiqadQgaga qbaiaadUgaaeqaaaqaaiqadQgagaqbaiabg2da9iaaigdaaeaacaWG NbWaaSbaaWqaaiaadUgaaeqaaaqdcqGHris5aaGccaGLOaGaayzkaa Gaeyypa0JaamyramaabmaabaWaaabCaeaacaWGHbWaaSbaaSqaaiaa dQgacaWGRbaabeaaaeaacaWGQbGaeyypa0JaaGymaaqaaiaadEgada WgaaadbaGaam4AaaqabaaaniabggHiLdGccqGHRaWkdaaeWbqaamaa qahabaGaamyyamaaBaaaleaacaWGQbGaam4AaaqabaGccaWGHbWaaS baaSqaaiqadQgagaqbaiaadUgaaeqaaaqaaiqadQgagaqbaiabgcMi 5kaadQgacqGH9aqpcaaIXaaabaGaam4zamaaBaaameaacaWGRbaabe aaa0GaeyyeIuoaaSqaaiaadQgacqGH9aqpcaaIXaaabaGaam4zamaa BaaameaacaWGRbaabeaaa0GaeyyeIuoaaOGaayjkaiaawMcaaaqaaa qaaiabg2da9maaqahabaGaamyramaabmaabaGaamyyamaaBaaaleaa caWGQbGaam4AaaqabaaakiaawIcacaGLPaaaaSqaaiaadQgacqGH9a qpcaaIXaaabaGaam4zamaaBaaameaacaWGRbaabeaaa0GaeyyeIuoa kiabgUcaRmaaqahabaWaaabCaeaacaWGfbWaaeWaaeaacaWGHbWaaS baaSqaaiaadQgacaWGRbaabeaakiaadggadaWgaaWcbaGabmOAayaa faGaam4AaaqabaaakiaawIcacaGLPaaaaSqaaiqadQgagaqbaiabgc Mi5kaadQgacqGH9aqpcaaIXaaabaGaam4zamaaBaaameaacaWGRbaa beaaa0GaeyyeIuoaaSqaaiaadQgacqGH9aqpcaaIXaaabaGaam4zam aaBaaameaacaWGRbaabeaaa0GaeyyeIuoakiabg2da9iaadEgadaWg aaWcbaGaam4AaaqabaGcdaWcaaqaaiaad6gadaWgaaWcbaGaamiAaa qabaaakeaacaWGobWaaSbaaSqaaiaadIgaaeqaaaaakiabgUcaRiaa dEgadaWgaaWcbaGaam4AaaqabaGcdaqadaqaaiaadEgadaWgaaWcba Gaam4AaaqabaGccqGHsislcaaIXaaacaGLOaGaayzkaaWaaSaaaeaa caWGUbWaaSbaaSqaaiaadIgaaeqaaOWaaeWaaeaacaWGUbWaaSbaaS qaaiaadIgaaeqaaOGaeyOeI0IaaGymaaGaayjkaiaawMcaaaqaaiaa d6eadaWgaaWcbaGaamiAaaqabaGcdaqadaqaaiaad6eadaWgaaWcba GaamiAaaqabaGccqGHsislcaaIXaaacaGLOaGaayzkaaaaaiaac6ca aaGaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaIZaGaaiOlaiaaiw dacaGGPaaaaa@CE50@

Second order inclusion expectations for households k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@381C@ and k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabm4Aayaafa aaaa@3828@ for k k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaiabgc Mi5kqadUgagaqbaaaa@3ADF@ belonging to two different strata h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiAaaaa@3819@ and h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmiAayaafa aaaa@3825@ equal

π k k = E ( a k a k ) = E ( j = 1 g k a j k j = 1 g k a j k ) = j = 1 g k j = 1 g k E ( a j k a j k ) = g k h g k h n h n h N h N h . ( 3.6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiWda3aaS baaSqaaiaadUgaceWGRbGbauaaaeqaaOGaeyypa0Jaaeyramaabmaa baGaamyyamaaBaaaleaacaWGRbaabeaakiaadggadaWgaaWcbaGabm 4AayaafaaabeaaaOGaayjkaiaawMcaaiabg2da9iaabweadaqadaqa amaaqahabaGaamyyamaaBaaaleaacaWGQbGaam4AaaqabaaabaGaam OAaiabg2da9iaaigdaaeaacaWGNbWaaSbaaWqaaiaadUgaaeqaaaqd cqGHris5aOWaaabCaeaacaWGHbWaaSbaaSqaaiqadQgagaqbaiqadU gagaqbaaqabaaabaGabmOAayaafaGaeyypa0JaaGymaaqaaiaadEga daWgaaadbaGabm4Aayaafaaabeaaa0GaeyyeIuoaaOGaayjkaiaawM caaiabg2da9maaqahabaWaaabCaeaacaWGfbWaaeWaaeaacaWGHbWa aSbaaSqaaiaadQgacaWGRbaabeaakiaadggadaWgaaWcbaGabmOAay aafaGabm4AayaafaaabeaaaOGaayjkaiaawMcaaiabg2da9iaadEga daWgaaWcbaGaam4AaiaadIgaaeqaaOGaam4zamaaBaaaleaaceWGRb GbauaaceWGObGbauaaaeqaaOWaaSaaaeaacaWGUbWaaSbaaSqaaiaa dIgaaeqaaOGaamOBamaaBaaaleaaceWGObGbauaaaeqaaaGcbaGaam OtamaaBaaaleaacaWGObaabeaakiaad6eadaWgaaWcbaGabmiAayaa faaabeaaaaaabaGabmOAayaafaGaeyypa0JaaGymaaqaaiaadEgada WgaaadbaGabm4Aayaafaaabeaaa0GaeyyeIuoaaSqaaiaadQgacqGH 9aqpcaaIXaaabaGaam4zamaaBaaameaacaWGRbaabeaaa0GaeyyeIu oakiaac6cacaaMf8UaaGzbVlaaywW7caaMf8UaaiikaiaaiodacaGG UaGaaGOnaiaacMcaaaa@8B1A@

An alternative proof based on the definition of an expected value, which does not use the rule that the expected value of a sum of mutual dependent variables is equal to the sum over the expected values of these variables is given by van den Brakel (2013).

As time proceeds the household composition of the core persons changes, which affects the inclusion expectations of the households in the sample. If sampling fractions differ between strata, the inclusion expectations (3.3) through (3.6) become more complicated and require information of stratum membership for all persons belonging to the household of the core persons. This complication is avoided by choosing a self-weighted sampling design. In this case each household member of a core persons has the same inclusion probability and the only household specific information required to derive household inclusion expectations is the number of persons aged 15 years and over in the household of the core person.

Since all members of a selected household are included in the sample, it follows that the first order inclusion expectations for persons belonging to household k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@381C@ are equal to the first order inclusion expectation of household k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@381C@ defined in (3.3). The second order inclusion expectations for persons from two different households k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@381C@ and k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabm4Aayaafa Gaaiilaaaa@38D8@ are equal to (3.4) for two households from the same stratum or (3.6) for two households from two different strata. The second order inclusion expectations for persons from the same household are defined by (3.5).

During the review the question was raised whether the inclusion expectations themselves have a variance that should be taken into account in the variance of HT or GREG estimators when they are based on inclusion expectations instead inclusion probabilities. In the finite population each person and each household has a pre-specified inclusion expectation. For the households observed in the sample these expectations can be calculated exactly without uncertainty since all information required to evaluate the true value of these expectations is available. Substituting inclusion probabilities for expectations, therefore does not result in an additional variance component.

3.2 Generalized Weight Share method

The sample design described in Section 2 can be considered as a special case of indirect sampling (Lavallée 2007). Indirect sampling refers to the situation where the population of interest is sampled through the use of a frame that refers to a different population. Lavallée (1995) develops the Generalized Weight Share method to construct weights for these situations and can be used to derive design weights for households and persons in the sample design described in Section 2.

Following the notation of Lavallée (1995) for the case of indirect sampling, there is a population U A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvamaaCa aaleqabaGaamyqaaaaaaa@38F9@ of size N A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOtamaaCa aaleqabaGaamyqaaaaaaa@38F2@ from which a sample s A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4CamaaCa aaleqabaGaamyqaaaaaaa@3917@ of size n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOBaaaa@381F@ is drawn with selection probabilities π i A . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiWda3aa0 baaSqaaiaadMgaaeaacaWGbbaaaOGaaiOlaaaa@3B86@ In addition, there is the target population U B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvamaaCa aaleqabaGaamOqaaaaaaa@38FA@ of size N B . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOtamaaCa aaleqabaGaamOqaaaakiaac6caaaa@39AF@ This population can be divided in M B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamytamaaCa aaleqabaGaamOqaaaaaaa@38F2@ clusters. Each cluster k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@381C@ contains N k B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOtamaaDa aaleaacaWGRbaabaGaamOqaaaaaaa@39E3@ units, such that N B = k = 1 M B N k B . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOtamaaCa aaleqabaGaamOqaaaakiabg2da9maaqadabaGaamOtamaaDaaaleaa caWGRbaabaGaamOqaaaaaeaacaWGRbGaeyypa0JaaGymaaqaaiaad2 eadaahaaadbeqaaiaadkeaaaaaniabggHiLdGccaGGUaaaaa@43E5@ The situation for the sample design described in Section 2 is depicted in Figure 3.1. The clusters are households, U A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvamaaCa aaleqabaGaamyqaaaaaaa@38F9@ is the population of persons aged 15 years and over, and U B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvamaaCa aaleqabaGaamOqaaaaaaa@38FA@ is the population of all persons residing in the Netherlands. Persons in U A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvamaaCa aaleqabaGaamyqaaaaaaa@38F9@ and U B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvamaaCa aaleqabaGaamOqaaaaaaa@38FA@ are depicted as circles, households in U B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvamaaCa aaleqabaGaamOqaaaaaaa@38FA@ are depicted as shaded squares, and the circles within a shaded square visualise persons belonging to the same household. Figure 3.1 shows respectively, a single person household, a two person household containing for example a divorced parent with a child younger than 15, a two person household containing two adults without children, and a four person household containing two parents with two children and one of the children is younger than 15 while the other is 15 years or older. The arrows depict the links between the units of U A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvamaaCa aaleqabaGaamyqaaaaaaa@38F9@ and U B . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvamaaCa aaleqabaGaamOqaaaakiaac6caaaa@39B6@ In the sample design considered in Section 2, each unit in U A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvamaaCa aaleqabaGaamyqaaaaaaa@38F9@ has exactly one unique link with a unit in U B . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvamaaCa aaleqabaGaamOqaaaakiaac6caaaa@39B6@ Clusters in U B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvamaaCa aaleqabaGaamOqaaaaaaa@38FA@ have at least one link with units in U A . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvamaaCa aaleqabaGaamyqaaaakiaac6caaaa@39B5@ Links are identified with an indicator variable

l i j = { 1 if there is a link between  i U A  and  j U B 0 if there is no link between  i U A  and  j U B . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiBamaaBa aaleaacaWGPbGaamOAaaqabaGccqGH9aqpdaGabaqaauaabaqaciaa aeaacaaIXaaabaGaaeyAaiaabAgacaqGGaGaaeiDaiaabIgacaqGLb GaaeOCaiaabwgacaqGGaGaaeyAaiaabohacaqGGaGaaeyyaiaabcca caqGSbGaaeyAaiaab6gacaqGRbGaaeiiaiaabkgacaqGLbGaaeiDai aabEhacaqGLbGaaeyzaiaab6gacaqGGaGaamyAaiabgIGiolaadwfa daahaaWcbeqaaiaadgeaaaGccaqGGaGaaeyyaiaab6gacaqGKbGaae iiaiaaykW7caaMc8UaamOAaiabgIGiolaadwfadaahaaWcbeqaaiaa dkeaaaaakeaacaaIWaaabaGaaeyAaiaabAgacaqGGaGaaeiDaiaabI gacaqGLbGaaeOCaiaabwgacaqGGaGaaeyAaiaabohacaqGGaGaaeOB aiaab+gacaqGGaGaaeiBaiaabMgacaqGUbGaae4AaiaabccacaqGIb GaaeyzaiaabshacaqG3bGaaeyzaiaabwgacaqGUbGaaeiiaiaadMga cqGHiiIZcaWGvbWaaWbaaSqabeaacaWGbbaaaOGaaeiiaiaabggaca qGUbGaaeizaiaabccacaaMc8UaaGPaVlaadQgacqGHiiIZcaWGvbWa aWbaaSqabeaacaWGcbaaaOGaaiOlaaaaaiaawUhaaaaa@8D76@

If a unit i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@381A@ in U A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvamaaCa aaleqabaGaamyqaaaaaaa@38F9@ is selected in the sample, the entire cluster k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@381C@ to which this unit belongs, is included in the sample. The parameter of interest is the population total in U B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvamaaCa aaleqabaGaamOqaaaaaaa@38FA@ and is similar to (3.1) defined as t y = k = 1 M B j = 1 N k B y k j . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDamaaBa aaleaacaWG5baabeaakiabg2da9maaqadabaWaaabmaeaacaWG5bWa aSbaaSqaaiaadUgacaWGQbaabeaaaeaacaWGQbGaeyypa0JaaGymaa qaaiaad6eadaqhaaadbaGaam4AaaqaaiaadkeaaaaaniabggHiLdaa leaacaWGRbGaeyypa0JaaGymaaqaaiaad2eadaahaaadbeqaaiaadk eaaaaaniabggHiLdGccaGGUaaaaa@4BFD@ An estimator for t y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiDamaaBa aaleaacaWG5baabeaaaaa@394F@ is defined as

t ^ y = k = 1 m j = 1 N k B w k j y k j , ( 3.7 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmiDayaaja WaaSbaaSqaaiaadMhaaeqaaOGaeyypa0ZaaabmaeaadaaeWaqaaiaa dEhadaWgaaWcbaGaam4AaiaadQgaaeqaaOGaamyEamaaBaaaleaaca WGRbGaamOAaaqabaaabaGaamOAaiabg2da9iaaigdaaeaacaWGobWa a0baaWqaaiaadUgaaeaacaWGcbaaaaqdcqGHris5aaWcbaGaam4Aai abg2da9iaaigdaaeaacaWGTbaaniabggHiLdGccaGGSaGaaGzbVlaa ywW7caaMf8UaaGzbVlaaywW7caGGOaGaaG4maiaac6cacaaI3aGaai ykaaaa@5995@

with m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyBaaaa@381E@ the number of unique clusters (households) included in the sample and w k j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaWGRbGaamOAaaqabaaaaa@3A33@ the weight attached to each unit j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOAaaaa@381B@ of cluster k . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaiaac6 caaaa@38CE@ Generally the inverse of the selection probabilities of units ( k , j ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGRbGaaiilaiaadQgaaiaawIcacaGLPaaaaaa@3B44@ observed in the sample are used as weights in the HT estimator. In this situation not all units in the sample have a known inclusion probability. Firstly not all units in U B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvamaaCa aaleqabaGaamOqaaaaaaa@38FA@ have a link to U A . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvamaaCa aaleqabaGaamyqaaaakiaac6caaaa@39B5@ Secondly, as time proceeds household compositions change due to marriages, divorces, departures of children and cohabitation. As a result, as time proceeds, units with a link to U A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvamaaCa aaleqabaGaamyqaaaaaaa@38F9@ enter the clusters in the sample although they are not initially included in the sample drawn from U A . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvamaaCa aaleqabaGaamyqaaaakiaac6caaaa@39B5@ For these units inclusion probabilities are not necessarily known. They affect, however, the inclusion expectations of the clusters included in the sample. Reconstruction of the inclusion probabilities requires information of selection probabilities of all units in the population at the moment that the sample is drawn. In many practical situations this information is not available.

Figure 3.1 of article 14544

Description of Figure 3.1

Figure representing the links between units from the sample frame U A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaW baaSqabeaacaWGbbaaaaaa@3A2D@  and units from the target population U B . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaW baaSqabeaacaWGcbaaaOGaaiOlaaaa@3AEA@  Persons in U A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaW baaSqabeaacaWGbbaaaaaa@3A2D@  and U B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaW baaSqabeaacaWGcbaaaaaa@3A2E@  are depicted as circles, households in U B MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaW baaSqabeaacaWGcbaaaaaa@3A2E@  are depicted as shaded squares, and the circles within a shaded square visualise persons belonging to the same household. Person number 1 from U A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaW baaSqabeaacaWGbbaaaaaa@3A2D@  is linked to person number 1 from U B , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaW baaSqabeaacaWGcbaaaOGaaiilaaaa@3AE8@  who’s the only person in her shaded square (a single person household). Person number 2 from U A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaW baaSqabeaacaWGbbaaaaaa@3A2D@  is linked to person number 2 from U B , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaW baaSqabeaacaWGcbaaaOGaaiilaaaa@3AE8@  who’s with person number 3 in her shaded square (a two person household containing for example a divorced parent with a child younger than 15). People number 3 and 4 from U A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaW baaSqabeaacaWGbbaaaaaa@3A2D@  are linked to people number 4 and 5 from U B , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaW baaSqabeaacaWGcbaaaOGaaiilaaaa@3AE8@  sharing a shaded square (a two person household containing two adults without children). People number 5, 6 and 7 from U A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaW baaSqabeaacaWGbbaaaaaa@3A2D@  are linked to people number 6, 7 and 9 from U B , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGvbWaaW baaSqabeaacaWGcbaaaOGaaiilaaaa@3AE8@  sharing a shaded square (a four person household containing two parents with two children and one of the children is younger than 15 while the other is 15 years or older).

The Generalized Weight Share method can be used to derive non-zero weights for all units in the sample. This method starts by deriving initial weights, which are defined as

w k j * = { δ i A π i A if  ( k , j )  has a link with  i U A 0 otherwise , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4DamaaDa aaleaacaWGRbGaamOAaaqaaiaacQcaaaGccqGH9aqpdaGabaqaauaa baqaciaaaeaadaWcaaqaaiabes7aKnaaDaaaleaacaWGPbaabaGaam yqaaaaaOqaaiabec8aWnaaDaaaleaacaWGPbaabaGaamyqaaaaaaaa keaacaqGPbGaaeOzaiaabccadaqadaqaaiaadUgacaGGSaGaamOAaa GaayjkaiaawMcaaiaabccacaqGObGaaeyyaiaabohacaqGGaGaaeyy aiaabccacaqGSbGaaeyAaiaab6gacaqGRbGaaeiiaiaabEhacaqGPb GaaeiDaiaabIgacaqGGaGaamyAaiabgIGiolaadwfadaahaaWcbeqa aiaadgeaaaaakeaacaaIWaaabaGaae4BaiaabshacaqGObGaaeyzai aabkhacaqG3bGaaeyAaiaabohacaqGLbaaaiaacYcaaiaawUhaaaaa @6774@

with δ i A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiTdq2aa0 baaSqaaiaadMgaaeaacaWGbbaaaaaa@3AB2@ an indicator variable that is equal to one if i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyAaaaa@381A@ is included in the sample s A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4CamaaCa aaleqabaGaamyqaaaaaaa@3917@ and zero otherwise. This expression follows directly from Lavallée (1995), equation (2) in combination with the fact that in this application each unit in U A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvamaaCa aaleqabaGaamyqaaaaaaa@38F9@ has exactly one unique link with a unit in U B , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvamaaCa aaleqabaGaamOqaaaakiaacYcaaaa@39B4@ see Figure 3.1. In a second step a so-called basic weight for each cluster k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@381C@ is derived as the mean of all initial weights within each cluster

w k = j = 1 N k B w k j * j = 1 N k B l k j , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaWGRbaabeaakiabg2da9maalaaabaWaaabmaeaacaWG3bWa a0baaSqaaiaadUgacaWGQbaabaGaaiOkaaaaaeaacaWGQbGaeyypa0 JaaGymaaqaaiaad6eadaqhaaadbaGaam4Aaaqaaiaadkeaaaaaniab ggHiLdaakeaadaaeWaqaaiaadYgadaWgaaWcbaGaam4AaiaadQgaae qaaaqaaiaadQgacqGH9aqpcaaIXaaabaGaamOtamaaDaaameaacaWG RbaabaGaamOqaaaaa0GaeyyeIuoaaaGccaGGSaaaaa@5097@

which follows from Lavallée (1995), equation (7). Finally all persons j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOAaaaa@381B@ that belong to the same household k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@381C@ receive the same weight assigned to their household, i.e., w k j = w k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaWGRbGaamOAaaqabaGccqGH9aqpcaWG3bWaaSbaaSqaaiaa dUgaaeqaaaaa@3D5B@ for all j k . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOAaiabgI GiolaadUgacaGGUaaaaa@3B41@ A proof that the use of the basic weights in (3.7) is an unbiased estimator for the population total is also given by Lavallée (1995).

Let j = 1 N k B l k j = g k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaWaaabmaeaaca WGSbWaaSbaaSqaaiaadUgacaWGQbaabeaakiabg2da9iaadEgadaWg aaWcbaGaam4AaaqabaaabaGaamOAaiabg2da9iaaigdaaeaacaWGob Waa0baaWqaaiaadUgaaeaacaWGcbaaaaqdcqGHris5aaaa@449F@ denote the number of persons in household k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@381C@ aged 15 years and older and a k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyyamaaBa aaleaacaWGRbaabeaaaaa@392E@ the number of core persons in household k , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4AaiaacY caaaa@38CC@ i.e., the number of persons in household k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@381C@ that are included in sample s A . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4CamaaCa aaleqabaGaamyqaaaakiaac6caaaa@39D3@ Since s A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4CamaaCa aaleqabaGaamyqaaaaaaa@3917@ is drawn by means of stratified simple random sampling, it follows that π i A = n h A / N h A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaeqiWda3aa0 baaSqaaiaadMgaaeaacaWGbbaaaOGaeyypa0ZaaSGbaeaacaWGUbWa a0baaSqaaiaadIgaaeaacaWGbbaaaaGcbaGaamOtamaaDaaaleaaca WGObaabaGaamyqaaaaaaaaaa@4180@ with N h A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOtamaaDa aaleaacaWGObaabaGaamyqaaaaaaa@39DF@ the number of persons aged 15 years and older in the population of stratum h , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiAaiaacY caaaa@38C9@ and n h A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOBamaaDa aaleaacaWGObaabaGaamyqaaaaaaa@39FF@ the number of core persons selected in the sample from stratum h . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiAaiaac6 caaaa@38CB@ Then it follows that

w k = a k g k N h A n h A . ( 3.8 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4DamaaBa aaleaacaWGRbaabeaakiabg2da9maalaaabaGaamyyamaaBaaaleaa caWGRbaabeaaaOqaaiaadEgadaWgaaWcbaGaam4AaaqabaaaaOWaaS aaaeaacaWGobWaa0baaSqaaiaadIgaaeaacaWGbbaaaaGcbaGaamOB amaaDaaaleaacaWGObaabaGaamyqaaaaaaGccaGGUaGaaGzbVlaayw W7caaMf8UaaGzbVlaaywW7caGGOaGaaG4maiaac6cacaaI4aGaaiyk aaaa@502D@

Inserting the first order inclusion expectation (3.3) into (3.2) gives the same HT estimator as derived with the Generalized Weight Share method, i.e., inserting (3.8) into (3.7).

The derivation of the inclusion expectations in Subsection 3.1 applies to stratified sampling of households with inclusion expectations proportional to household size and is a special case of the Generalized Weight Share method. An argument to apply a design as outlined in Section 2 is that sampling households proportional to household size is efficient for target variables that are positively correlated with household size.

Lavallée (1995) also provides variance expressions for (3.7) based on the Generalized Weight Share method. This expression is based on the first and second order inclusion probabilities of the sample units drawn from U A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyvamaaCa aaleqabaGaamyqaaaaaaa@38F9@ and a transformation of the target variable. As a result the property that clusters are drawn proportional to their size is not made explicit, nor that the fact they are drawn partially with replacement. In Section 6 it is pointed out that the variance expressions in Lavallée (1995) for this application are equal to the variance expressions based on the inclusion expectations derived in (3.3) through (3.6).

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