Register-based sampling for household panels 4. Sample size determination

The purpose of the RIS is to publish income distributions for households and persons at different geographical levels. Income distributions for households for region or area r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOCaaaa@3823@  are defined as

P l r = M l r M + r , l = 1 , , L , ( 4.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiuamaaBa aaleaacaWGSbGaamOCaaqabaGccqGH9aqpdaWcaaqaaiaad2eadaWg aaWcbaGaamiBaiaadkhaaeqaaaGcbaGaamytamaaBaaaleaacqGHRa WkcaWGYbaabeaaaaGccaGGSaGaaGzbVlaaywW7caWGSbGaeyypa0Ja aGymaiaacYcacqWIMaYscaGGSaGaamitaiaacYcacaaMf8UaaGzbVl aaywW7caaMf8UaaGzbVlaacIcacaaI0aGaaiOlaiaaigdacaGGPaaa aa@56D0@

where M l r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGSbGaamOCaaqabaaaaa@3A12@ denotes the number of households from region r , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOCaiaacY caaaa@38D3@ belonging to the l th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiBamaaCa aaleqabaGaaeiDaiaabIgaaaaaaa@3A2C@ income category, and M + r = l M l r , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacqGHRaWkcaWGYbaabeaakiabg2da9maaqababaGaamytamaa BaaaleaacaWGSbGaamOCaaqabaaabaGaamiBaaqab0GaeyyeIuoaki aacYcaaaa@417C@ the total number of households in area r . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOCaiaac6 caaaa@38D5@ This income distribution is estimated as

P ^ l r = M ^ l r M + r , l = 1 , , L , ( 4.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmiuayaaja WaaSbaaSqaaiaadYgacaWGYbaabeaakiabg2da9maalaaabaGabmyt ayaajaWaaSbaaSqaaiaadYgacaWGYbaabeaaaOqaaiaad2eadaWgaa WcbaGaey4kaSIaamOCaaqabaaaaOGaaiilaiaaywW7caaMf8UaamiB aiabg2da9iaaigdacaGGSaGaeSOjGSKaaiilaiaadYeacaGGSaGaaG zbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaGinaiaac6cacaaI YaGaaiykaaaa@56F1@

where M ^ l r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmytayaaja WaaSbaaSqaaiaadYgacaWGYbaabeaaaaa@3A22@ denotes an appropriate direct estimator for the total number of households from area r , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOCaiaacY caaaa@38D3@ classified to the l th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiBamaaCa aaleqabaGaaeiDaiaabIgaaaaaaa@3A2C@ income category. For the moment the HT estimator is assumed as an appropriate estimator for M l r , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGSbGaamOCaaqabaGccaGGSaaaaa@3ACC@ i.e.,

M ^ l r = h r k = 1 m h y k h l π k , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmytayaaja WaaSbaaSqaaiaadYgacaWGYbaabeaakiabg2da9maaqafabaWaaabC aeaadaWcaaqaaiaadMhadaWgaaWcbaGaam4AaiaadIgacaWGSbaabe aaaOqaaiabec8aWnaaBaaaleaacaWGRbaabeaaaaaabaGaam4Aaiab g2da9iaaigdaaeaacaWGTbWaaSbaaWqaaiaadIgaaeqaaaqdcqGHri s5aaWcbaGaamiAaiabgIGiolaadkhaaeqaniabggHiLdGccaGGSaaa aa@4F55@

where y k h l = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGRbGaamiAaiaadYgaaeqaaOGaeyypa0JaaGymaaaa@3CEF@ if household k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Aaaaa@381C@ from stratum h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiAaaaa@3819@ is classified to the l th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiBamaaCa aaleqabaGaaeiDaiaabIgaaaaaaa@3A2C@ income class and y k h l = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGRbGaamiAaiaadYgaaeqaaOGaeyypa0JaaGimaaaa@3CEE@ otherwise and m h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyBamaaBa aaleaacaWGObaabeaaaaa@3937@ the total number of households selected in stratum h . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiAaiaac6 caaaa@38CB@ In the RIS L = 10. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamitaiabg2 da9iaaigdacaaIWaGaaiOlaaaa@3B2A@ Income distributions for persons are defined and estimated similarly to (4.1), (4.2), with M l r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGSbGaamOCaaqabaaaaa@3A12@ the number of persons from area r , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamOCaiaacY caaaa@38D3@ belonging to the l th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiBamaaCa aaleqabaGaaeiDaiaabIgaaaaaaa@3A2C@ income category. The HT estimator for M l r MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGSbGaamOCaaqabaaaaa@3A12@ is now defined as

M ^ l r = h r k = 1 m h 1 π k j = 1 N k y k j h l , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGabmytayaaja WaaSbaaSqaaiaadYgacaWGYbaabeaakiabg2da9maaqafabaWaaabC aeaadaWcaaqaaiaaigdaaeaacqaHapaCdaWgaaWcbaGaam4Aaaqaba aaaOWaaabCaeaacaWG5bWaaSbaaSqaaiaadUgacaWGQbGaamiAaiaa dYgaaeqaaaqaaiaadQgacqGH9aqpcaaIXaaabaGaamOtamaaBaaame aacaWGRbaabeaaa0GaeyyeIuoaaSqaaiaadUgacqGH9aqpcaaIXaaa baGaamyBamaaBaaameaacaWGObaabeaaa0GaeyyeIuoaaSqaaiaadI gacqGHiiIZcaWGYbaabeqdcqGHris5aOGaaiilaaaa@57E1@

where y k j h l = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGRbGaamOAaiaadIgacaWGSbaabeaakiabg2da9iaaigda aaa@3DDE@ 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@ and stratum h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiAaaaa@3819@ is classified to the l th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamiBamaaCa aaleqabaGaaeiDaiaabIgaaaaaaa@3A2C@ income class and y k j h l = 0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGRbGaamOAaiaadIgacaWGSbaabeaakiabg2da9iaaicda aaa@3DDD@ otherwise.

For sample size determination, precision specifications for the estimated income distributions are required. For stratified sampling designs, Neyman allocations are often considered to determine minimum sample sizes and optimal allocations to meet precision requirements at aggregated levels (Cochran 1977). Power allocations are useful to find the right balance between precision requirements for aggregates and strata (Bankier 1988). In this application the minimum sample size is based on precision requirements for the individual strata, i.e., neighbourhoods, which is the most detailed publication level.

If precision requirements are specified for the separate classes of the income distributions, then the income class with the largest population variance determines the minimum required sample size, resulting in unnecessarily large sample sizes. As an alternative the square root of the mean over the variances of the estimated income classes of an income distribution is proposed as a precision measure for the estimated income distributions. With this measure the influence of the most imprecise income class on the minimum sample size will be reduced. The square root of the mean over the variances of the estimated income classes of an income distribution is called the average standard error measure and is defined as

s = 1 L l = 1 L V ( P ^ l r ) . ( 4.3 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Caiabg2 da9maakaaabaWaaSaaaeaacaaIXaaabaGaamitaaaadaaeWbqaaiaa dAfadaqadaqaaiqadcfagaqcamaaBaaaleaacaWGSbGaamOCaaqaba aakiaawIcacaGLPaaaaSqaaiaadYgacqGH9aqpcaaIXaaabaGaamit aaqdcqGHris5aaWcbeaakiaac6cacaaMf8UaaGzbVlaaywW7caaMf8 UaaGzbVlaacIcacaaI0aGaaiOlaiaaiodacaGGPaaaaa@5214@

In this section an exact expression for s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Caaaa@3824@ will be derived as well as an approximation that can be used to estimate the minimum required sample size which does not require information about income distributions or variances.

Since neighbourhoods are the most detailed areas for which income distributions are published, precision requirements for sample size determination are specified at this level. Since neighbourhoods are used as the stratification variable in the sample design, expressions for s MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4Caaaa@3824@ can be derived under simple random sampling without replacement of core persons within each neighbourhood. It is proved in the appendix that an expression for the average standard error measure s h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGObaabeaaaaa@393D@ in (4.3) for an income distribution is given by

s h = 1 L N h n h n h 1 N h 1 ( N h M h 2 l = 1 L k = 1 M h y k h l g k h l = 1 L ( M l h M h ) 2 ) , ( 4.4 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lq=Je9 vqaqFeFr0xbbG8FaYPYRWFb9fi0FXxbbf9Ff0dfrpm0dXdHqVu0=vr 0=vr0=fdbaqaaeGacaGaaiaabeqaamaabaabaaGcbaGaam4CamaaBa aaleaacaWGObaabeaakiabg2da9maakaaabaWaaSaaaeaacaaIXaaa baGaamitaaaadaWcaaqaaiaad6eadaWgaaWcbaGaamiAaaqabaGccq GHsislcaWGUbWaaSbaaSqaaiaadIgaaeqaaaGcbaGaamOBamaaBaaa leaacaWGObaabeaaaaGcdaWcaaqaaiaaigdaaeaacaWGobWaaSbaaS qaaiaadIgaaeqaaOGaeyOeI0IaaGymaaaadaqadaqaamaalaaabaGa amOtamaaBaaaleaacaWGObaabeaaaOqaaiaad2eadaqhaaWcbaGaam iAaaqaaiaaikdaaaaaaOWaaabCaeaadaaeWbqaamaalaaabaGaamyE amaaBaaaleaacaWGRbGaamiAaiaadYgaaeqaaaGcbaGaam4zamaaBa aaleaacaWGRbGaamiAaaqabaaaaaqaaiaadUgacqGH9aqpcaaIXaaa baGaamytamaaBaaameaacaWGObaabeaaa0GaeyyeIuoaaSqaaiaadY gacqGH9aqpcaaIXaaabaGaamitaaqdcqGHris5aOGaeyOeI0YaaabC aeaadaqadaqaamaalaaabaGaamytamaaBaaaleaacaWGSbGaamiAaa qabaaakeaacaWGnbWaaSbaaSqaaiaadIgaaeqaaaaaaOGaayjkaiaa wMcaaaWcbaGaamiBaiabg2da9iaaigdaaeaacaWGmbaaniabggHiLd GcdaahaaWcbeqaaiaaikdaaaaakiaawIcacaGLPaaaaSqabaGccaGG SaGaaGzbVlaaywW7caaMf8UaaGzbVlaaywW7caGGOaGaaGinaiaac6 cacaaI0aGaaiykaaaa@7BA0@

with M h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS baaSqaaiaadIgaaeqaaaaa@3A4B@ the number of households in stratum h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGObaaaa@394D@ and M l h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS baaSqaaiaadYgacaWGObaabeaaaaa@3B3C@ the number of households in stratum h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGObaaaa@394D@ belonging to the l th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGSbWaaW baaSqabeaacaqG0bGaaeiAaaaaaaa@3B60@ income class. Note that if g k h = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGNbWaaS baaSqaaiaadUgacaWGObaabeaakiabg2da9iaaigdaaaa@3D20@ for all households in the population of stratum h , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGObGaai ilaaaa@39FD@ then it follows that M h = N h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS baaSqaaiaadIgaaeqaaOGaeyypa0JaamOtamaaBaaaleaacaWGObaa beaaaaa@3D47@ and that formula (4.1) simplifies to

V ( P ^ l h ) = N h n h n h 1 N h 1 ( P l h ( 1 P l h ) ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGwbWaae WaaeaaceWGqbGbaKaadaWgaaWcbaGaamiBaiaadIgaaeqaaaGccaGL OaGaayzkaaGaeyypa0ZaaSaaaeaacaWGobWaaSbaaSqaaiaadIgaae qaaOGaeyOeI0IaamOBamaaBaaaleaacaWGObaabeaaaOqaaiaad6ga daWgaaWcbaGaamiAaaqabaaaaOWaaSaaaeaacaaIXaaabaGaamOtam aaBaaaleaacaWGObaabeaakiabgkHiTiaaigdaaaWaaeWaaeaacaWG qbWaaSbaaSqaaiaadYgacaWGObaabeaakmaabmaabaGaaGymaiabgk HiTiaadcfadaWgaaWcbaGaamiBaiaadIgaaeqaaaGccaGLOaGaayzk aaaacaGLOaGaayzkaaGaaiilaaaa@5586@

which can be recognized as the variance of an estimated fraction under simple random sampling without replacement (Cochran 1977, Chapter 3).

Minimum sample size requirements based on (4.4) require information about the income distribution and its variance from preceding periods. Since this information is generally not available at the design phase of a panel, it is useful to have an upper bound for the average standard error measure for the income distribution in (4.4). This is comparable to taking the variance for a parameter defined as a proportion, which reaches a maximum when the proportion is 0.5 for calculating the minimum sample size for a survey. It is shown in the appendix that an upper bound for the average standard error measure s h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGZbWaaS baaSqaaiaadIgaaeqaaaaa@3A71@ for an income distribution, specified in (4.4) is given by

s h 1 L N h n h n h 1 N h 1 ( N h M h 2 t = 1 T M t h t 1 L ) , ( 4.5 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGZbWaaS baaSqaaiaadIgaaeqaaOGaeyizIm6aaOaaaeaadaWcaaqaaiaaigda aeaacaWGmbaaamaalaaabaGaamOtamaaBaaaleaacaWGObaabeaaki abgkHiTiaad6gadaWgaaWcbaGaamiAaaqabaaakeaacaWGUbWaaSba aSqaaiaadIgaaeqaaaaakmaalaaabaGaaGymaaqaaiaad6eadaWgaa WcbaGaamiAaaqabaGccqGHsislcaaIXaaaamaabmaabaWaaSaaaeaa caWGobWaaSbaaSqaaiaadIgaaeqaaaGcbaGaamytamaaDaaaleaaca WGObaabaGaaGOmaaaaaaGcdaaeWbqaamaalaaabaGaamytamaaBaaa leaacaWG0bGaamiAaaqabaaakeaacaWG0baaaiabgkHiTmaalaaaba GaaGymaaqaaiaadYeaaaaaleaacaWG0bGaeyypa0JaaGymaaqaaiaa dsfaa0GaeyyeIuoaaOGaayjkaiaawMcaaaWcbeaakiaacYcacaaMf8 UaaGzbVlaaywW7caaMf8UaaGzbVlaacIcacaaI0aGaaiOlaiaaiwda caGGPaaaaa@6816@

with M t h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGnbWaaS baaSqaaiaadshacaWGObaabeaaaaa@3B44@ the number of households of size t MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWG0baaaa@3959@ in stratum h . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGObGaai Olaaaa@39FF@

If g k h = 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGNbWaaS baaSqaaiaadUgacaWGObaabeaakiabg2da9iaaigdaaaa@3D20@ for all households in the population of stratum h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGObaaaa@394D@ and the number of classes of the income distribution L = 2 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGmbGaey ypa0JaaGOmaiaacYcaaaa@3BA3@ then it follows that the approximation for the average standard error measure s h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGZbWaaS baaSqaaiaadIgaaeqaaaaa@3A71@ in (4.5) can be simplified to

s h N h n h n h 1 ( N h 1 ) 1 4 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGZbWaaS baaSqaaiaadIgaaeqaaOGaeyizIm6aaOaaaeaadaWcaaqaaiaad6ea daWgaaWcbaGaamiAaaqabaGccqGHsislcaWGUbWaaSbaaSqaaiaadI gaaeqaaaGcbaGaamOBamaaBaaaleaacaWGObaabeaaaaGcdaWcaaqa aiaaigdaaeaadaqadaqaaiaad6eadaWgaaWcbaGaamiAaaqabaGccq GHsislcaaIXaaacaGLOaGaayzkaaaaamaalaaabaGaaGymaaqaaiaa isdaaaaaleqaaOGaaiilaaaa@4B9E@

which equals the square root of the maximum variance of an estimated fraction at P ^ = 0.5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGqbGbaK aacqGH9aqpcaaIWaGaaiOlaiaaiwdaaaa@3C76@ under simple random sampling. This illustrates that the approximation for the average standard error measure in (4.5) can be interpreted as a generalization of the approximation of the maximum variance of an estimated fraction at P ^ = 0.5 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGqbGbaK aacqGH9aqpcaaIWaGaaiOlaiaaiwdacaGGSaaaaa@3D26@ often used in sample size determination. The average standard error measure has its maximum value in the case of an equal distribution of the households over the income categories, i.e., P ^ l h = 1 / L MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaaceWGqbGbaK aadaWgaaWcbaGaamiBaiaadIgaaeqaaOGaeyypa0ZaaSGbaeaacaaI XaaabaGaamitaaaaaaa@3E01@ for l = 1 , , L . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGSbGaey ypa0JaaGymaiaacYcacqWIMaYscaGGSaGaamitaiaac6caaaa@3F17@ In this situation the approximation for s h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGZbWaaS baaSqaaiaadIgaaeqaaaaa@3A71@ is exact, which follows directly from equation (4.3).

Equating the expression for s h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGZbWaaS baaSqaaiaadIgaaeqaaaaa@3A71@ in (4.5) to a pre-specified maximum value, say Δ h , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacqqHuoarda WgaaWcbaGaamiAaaqabaGccaGGSaaaaa@3B99@ results in the following expression for the minimum sample size of core persons

n h ( N h M h ) 2 t = 1 T M t h t N h L ( N h 1 ) L Δ h 2 + N h M h 2 t = 1 T M t h t 1 L . ( 4.6 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbWaaS baaSqaaiaadIgaaeqaaOGaeyyzIm7aaSaaaeaadaqadaqaamaalaaa baGaamOtamaaBaaaleaacaWGObaabeaaaOqaaiaad2eadaWgaaWcba GaamiAaaqabaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaa aOWaaabCaeaadaWcaaqaaiaad2eadaWgaaWcbaGaamiDaiaadIgaae qaaaGcbaGaamiDaaaacqGHsisldaWcaaqaaiaad6eadaWgaaWcbaGa amiAaaqabaaakeaacaWGmbaaaaWcbaGaamiDaiabg2da9iaaigdaae aacaWGubaaniabggHiLdaakeaadaqadaqaaiaad6eadaWgaaWcbaGa amiAaaqabaGccqGHsislcaaIXaaacaGLOaGaayzkaaGaamitaiabfs 5aenaaDaaaleaacaWGObaabaGaaGOmaaaakiabgUcaRmaalaaabaGa amOtamaaBaaaleaacaWGObaabeaaaOqaaiaad2eadaqhaaWcbaGaam iAaaqaaiaaikdaaaaaaOWaaabCaeaadaWcaaqaaiaad2eadaWgaaWc baGaamiDaiaadIgaaeqaaaGcbaGaamiDaaaacqGHsisldaWcaaqaai aaigdaaeaacaWGmbaaaaWcbaGaamiDaiabg2da9iaaigdaaeaacaWG ubaaniabggHiLdaaaOGaaiOlaiaaywW7caaMf8UaaGzbVlaaywW7ca aMf8UaaiikaiaaisdacaGGUaGaaGOnaiaacMcaaaa@7794@

The information required to estimate the minimum sample size is the total number of persons and the total number of equally sized households for neighbourhoods. No information about the expected income distribution or its variance is required. More precise estimates for the minimum sample size can be obtained with the expression in (4.4), but require sample information from, for example, previous periods about the income distributions.

Expression (4.6) gives the minimum sample size for core persons. Subsequently all household members of each core person are included in the sample. As a result, households can be included in the sample more than once and the sample size in terms of unique households and unique persons is random. To plan a survey and control survey costs, it is necessary to know the expected number of unique households and unique persons if a sample of core persons of size n h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbWaaS baaSqaaiaadIgaaeqaaaaa@3A6C@ is drawn. In the appendix it is proved that the expected number of unique households in a sample of n h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbWaaS baaSqaaiaadIgaaeqaaaaa@3A6C@ core persons, drawn by means of simple random sampling without replacement from a finite population of size N h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobWaaS baaSqaaiaadIgaaeqaaaaa@3A4C@ is given by

D h = t = 1 T M t h ( 1 i = 0 t 1 ( N h n h i ) i = 0 t 1 ( N h i ) ) . ( 4.7 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGebWaaS baaSqaaiaadIgaaeqaaOGaeyypa0ZaaabCaeaacaWGnbWaaSbaaSqa aiaadshacaWGObaabeaakmaabmaabaGaaGymaiabgkHiTmaalaaaba WaaebCaeaadaqadaqaaiaad6eadaWgaaWcbaGaamiAaaqabaGccqGH sislcaWGUbWaaSbaaSqaaiaadIgaaeqaaOGaeyOeI0IaamyAaaGaay jkaiaawMcaaaWcbaGaamyAaiabg2da9iaaicdaaeaacaWG0bGaeyOe I0IaaGymaaqdcqGHpis1aaGcbaWaaebCaeaadaqadaqaaiaad6eada WgaaWcbaGaamiAaaqabaGccqGHsislcaWGPbaacaGLOaGaayzkaaaa leaacaWGPbGaeyypa0JaaGimaaqaaiaadshacqGHsislcaaIXaaani abg+GivdaaaaGccaGLOaGaayzkaaaaleaacaWG0bGaeyypa0JaaGym aaqaaiaadsfaa0GaeyyeIuoakiaac6cacaaMf8UaaGzbVlaaywW7ca aMf8UaaGzbVlaacIcacaaI0aGaaiOlaiaaiEdacaGGPaaaaa@702C@

The expected number of unique persons in a sample of n h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGUbWaaS baaSqaaiaadIgaaeqaaaaa@3A6C@ core persons, drawn by means of simple random sampling without replacement from a finite population of size N h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGobWaaS baaSqaaiaadIgaaeqaaaaa@3A4C@ follows directly from (4.7) and is given by

D h [ p ] = t = 1 T t M t h ( 1 i = 0 t 1 ( N h n h i ) i = 0 t 1 ( N h i ) ) . ( 4.8 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGebWaa0 baaSqaaiaadIgaaeaadaWadaqaaiaadchaaiaawUfacaGLDbaaaaGc cqGH9aqpdaaeWbqaaiaadshacaWGnbWaaSbaaSqaaiaadshacaWGOb aabeaakmaabmaabaGaaGymaiabgkHiTmaalaaabaWaaebCaeaadaqa daqaaiaad6eadaWgaaWcbaGaamiAaaqabaGccqGHsislcaWGUbWaaS baaSqaaiaadIgaaeqaaOGaeyOeI0IaamyAaaGaayjkaiaawMcaaaWc baGaamyAaiabg2da9iaaicdaaeaacaWG0bGaeyOeI0IaaGymaaqdcq GHpis1aaGcbaWaaebCaeaadaqadaqaaiaad6eadaWgaaWcbaGaamiA aaqabaGccqGHsislcaWGPbaacaGLOaGaayzkaaaaleaacaWGPbGaey ypa0JaaGimaaqaaiaadshacqGHsislcaaIXaaaniabg+GivdaaaaGc caGLOaGaayzkaaaaleaacaWG0bGaeyypa0JaaGymaaqaaiaadsfaa0 GaeyyeIuoakiaac6cacaaMf8UaaGzbVlaaywW7caaMf8UaaGzbVlaa cIcacaaI0aGaaiOlaiaaiIdacaGGPaaaaa@740E@

Since the expected numbers of unique households and persons are random variables, it would be useful to have an uncertainty measure for these expected values. Variance expressions for (4.7) and (4.8) are however not straightforward and therefore left for further research.

Sample size calculations are conducted at the level of neighbourhoods. It was finally decided to select core persons with a sampling fraction of 0.16. With this sample size, the maximum value for the average standard error measure s h MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpipeea0xe9Lqpe0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXddrpe0=1qpeea0=yrVue9 Fve9Fve8meaabaqaciaacaGaaeqabaWaaeaaeaaakeaacaWGZbWaaS baaSqaaiaadIgaaeqaaaaa@3A71@ at the level of neighbourhoods amounts to about 0.01 for the estimated household income distributions. With a total population of about 12 million persons, this resulted in a sample size of about 2.1 million core persons and an expected sample size of about 4.6 million unique persons. This sample was drawn in 1994, which was the start of the panel for the Dutch RIS.

Report a problem on this page

Is something not working? Is there information outdated? Can't find what you're looking for?

Please contact us and let us know how we can help you.

Privacy notice

Date modified: