Decomposition of gender wage inequalities through calibration: Application to the Swiss structure of earnings survey
Section 2. Problem and notation

The question of interest is the estimation of the wage differences between women and men, more specifically, how much of this difference is attributable to discrimination. Assume there is a finite population U MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaaaa@36D5@ of size N MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtaaaa@36CE@ that can be divided into two subpopulations, women and men, denoted by U h , h { F , M } , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvamaaBa aaleaacaWGObaabeaakiaaiYcacaWGObGaeyicI48aaiWaaeaacaWG gbGaaiilaiaad2eaaiaawUhacaGL9baacaGGSaaaaa@404D@ of size N h . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOtamaaBa aaleaacaWGObaabeaakiaac6caaaa@38A3@ Additionally, a random sample S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaaaa@36D3@ is drawn from U , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyvaiaacY caaaa@3785@ which contains both women and men. Sample S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaaaa@36D3@ is selected by means of a sampling design p ( s ) = Pr ( S = s ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaabm aabaGaam4CaaGaayjkaiaawMcaaiaai2daciGGqbGaaiOCamaabmaa baGaam4uaiabg2da9iaadohaaiaawIcacaGLPaaaaaa@4063@ for any s U , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Caiabgk OimlaadwfacaGGSaaaaa@3A79@ where

p ( s ) 0 and s U p ( s ) = 1 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCamaabm aabaGaam4CaaGaayjkaiaawMcaaiabgwMiZkaaicdacaaMe8UaaGjb VlaabggacaqGUbGaaeizaiaaysW7caaMe8+aaabuaeaacaWGWbWaae WaaeaacaWGZbaacaGLOaGaayzkaaGaaGypaiaaigdaaSqaaiaadoha cqGHckcZcaWGvbaabeqdcqGHris5aOGaaGOlaaaa@508B@

Sample S MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaaaa@36D3@ can be split into two subsamples, S h , h { F , M } , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGObaabeaakiaaiYcacaWGObGaeyicI48aaiWaaeaacaWG gbGaaGilaiaad2eaaiaawUhacaGL9baacaGGSaaaaa@4051@ women and men, such that S = S h . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uaiaai2 dacqGHQicYcaWGtbWaaSbaaSqaaiaadIgaaeqaaOGaaiOlaaaa@3BE7@ The variable of interest, denoted by y , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEaiaacY caaaa@37A9@ is in this case the logarithm of the wage. The totals of the variable of interest in the two subpopulations are given by

Y h = k U h y k , h { F , M } , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamywamaaBa aaleaacaWGObaabeaakiaai2dadaaeqbqaaiaadMhadaWgaaWcbaGa am4AaaqabaaabaGaam4AaiabgIGiolaadwfadaWgaaadbaGaamiAaa qabaaaleqaniabggHiLdGccaaISaGaamiAaiabgIGiopaacmaabaGa amOraiaaiYcacaWGnbaacaGL7bGaayzFaaGaaGilaaaa@49D2@

where y k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaBa aaleaacaWGRbaabeaaaaa@3815@ is the logarithm of the wage of the k th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaCa aaleqabaGaaeiDaiaabIgaaaaaaa@38FA@ individual. Since not all units in the subpopulations are observed, the totals can be estimated by

Y ^ h = k S h d k y k , h { F , M } , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaaja WaaSbaaSqaaiaadIgaaeqaaOGaaGypamaaqafabaGaamizamaaBaaa leaacaWGRbaabeaakiaadMhadaWgaaWcbaGaam4AaaqabaaabaGaam 4AaiabgIGiolaadofadaWgaaadbaGaamiAaaqabaaaleqaniabggHi LdGccaaISaGaamiAaiabgIGiopaacmaabaGaamOraiaaiYcacaWGnb aacaGL7bGaayzFaaGaaGilaaaa@4BEF@

where d k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGRbaabeaaaaa@3800@ is a sampling weight assigned to the k th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaCa aaleqabiqaaqUbcaqG0bGaaeiAaaaaaaa@39E7@ unit of the sample. Sampling weights are obtained after several statistical treatments (for example, adjustment for non-response).

The population means of the logarithms of the wages are given by

Y ¯ h = 1 N h k U h y k , h { F , M } , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaara WaaSbaaSqaaiaadIgaaeqaaOGaaGypamaalaaabaGaaGymaaqaaiaa d6eadaWgaaWcbaGaamiAaaqabaaaaOWaaabuaeaacaWG5bWaaSbaaS qaaiaadUgaaeqaaaqaaiaadUgacqGHiiIZcaWGvbWaaSbaaWqaaiaa dIgaaeqaaaWcbeqdcqGHris5aOGaaGilaiaadIgacqGHiiIZdaGada qaaiaadAeacaaISaGaamytaaGaay5Eaiaaw2haaiaaiYcaaaa@4CAB@

and can be estimated by

Y ¯ ^ h = k S h d k y k k S h d k , h { F , M } . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabmywayaary aajaWaaSbaaSqaaiaadIgaaeqaaOGaaGypamaalaaabaWaaabeaeaa caWGKbWaaSbaaSqaaiaadUgaaeqaaOGaamyEamaaBaaaleaacaWGRb aabeaaaeaacaWGRbGaeyicI4Saam4uamaaBaaameaacaWGObaabeaa aSqab0GaeyyeIuoaaOqaamaaqababaGaamizamaaBaaaleaacaWGRb aabeaaaeaacaWGRbGaeyicI4Saam4uamaaBaaameaacaWGObaabeaa aSqab0GaeyyeIuoaaaGccaaISaGaamiAaiabgIGiopaacmaabaGaam OraiaaiYcacaWGnbaacaGL7bGaayzFaaGaaGOlaaaa@5430@

Moreover, assume that for each k th MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AamaaCa aaleqabaGaaeiDaiaabIgaaaaaaa@38FA@ individual in either of the two subsamples, there is a vector of p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiCaaaa@36F0@ auxiliary variables denoted by

x k = ( x k 1 , , x k j , , x k p ) p . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiEamaaBa aaleaacaWGRbaabeaakiaai2dadaqadaqaaiaadIhadaWgaaWcbaGa am4AaiaaigdaaeqaaOGaaGilaiablAciljaaiYcacaWG4bWaaSbaaS qaaiaadUgacaWGQbaabeaakiaaiYcacqWIMaYscaaISaGaamiEamaa BaaaleaacaWGRbGaamiCaaqabaaakiaawIcacaGLPaaadaahaaWcbe qaamXvP5wqSX2qVrwzqf2zLnharyqqYLwySbsvUL2yVrwzG00uaGqb aiaa=jrmaaGccqGHiiIZtuuDJXwAK1uy0HMmaeXbfv3ySLgzG0uy0H giuD3BaGGbaiab+1risnaaCaaaleqabaGaamiCaaaakiaai6caaaa@632A@

This vector is supposed to be known for each unit selected in the sample. The auxiliary variables contain some characteristics of the individual, for instance the age, the education level or the seniority level. They can be quantitative or qualitative variables, thus x k j MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGRbGaamOAaaqabaaaaa@3903@ can be a categorical variable or a quantity. Also assume that the first auxiliary variable is a constant, i.e., x k 1 = 1, MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaBa aaleaacaWGRbGaaGymaaqabaGccaaI9aGaaGymaiaaiYcaaaa@3B11@ for all k U . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4AaiabgI GiolaadwfacaaIUaaaaa@3A01@

The totals of these auxiliary variables at the subpopulation level are given by

X h = k U h x k , h { F , M } . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaCiwamaaBa aaleaacaWGObaabeaakiaai2dadaaeqbqaaiaadIhadaWgaaWcbaGa am4AaaqabaaabaGaam4AaiabgIGiolaadwfadaWgaaadbaGaamiAaa qabaaaleqaniabggHiLdGccaaISaGaamiAaiabgIGiopaacmaabaGa amOraiaaiYcacaWGnbaacaGL7bGaayzFaaGaaGOlaaaa@49D6@

Using the weights d k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamizamaaBa aaleaacaWGRbaabeaaaaa@3800@ defined above, these two totals can be estimated by

X ^ h = k S h d k x k , h { F , M } . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCiwayaaja WaaSbaaSqaaiaadIgaaeqaaOGaaGypamaaqafabaGaamizamaaBaaa leaacaWGRbaabeaakiaahIhadaWgaaWcbaGaam4AaaqabaaabaGaam 4AaiabgIGiolaadofadaWgaaadbaGaamiAaaqabaaaleqaniabggHi LdGccaaISaGaamiAaiabgIGiopaacmaabaGaamOraiaaiYcacaWGnb aacaGL7bGaayzFaaGaaGOlaaaa@4BF7@

Vectors of average values can be analogously estimated. The average values at the subpopulation levels are given by

X ¯ h = 1 N h k U h x k , h { F , M } , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCiwayaara WaaSbaaSqaaiaadIgaaeqaaOGaaGypamaalaaabaGaaGymaaqaaiaa d6eadaWgaaWcbaGaamiAaaqabaaaaOWaaabuaeaacaWH4bWaaSbaaS qaaiaadUgaaeqaaaqaaiaadUgacqGHiiIZcaWGvbWaaSbaaWqaaiaa dIgaaeqaaaWcbeqdcqGHris5aOGaaGilaiaadIgacqGHiiIZdaGada qaaiaadAeacaaISaGaamytaaGaay5Eaiaaw2haaiaaiYcaaaa@4CB1@

and estimated by

X ¯ ^ h = k S h d k x k k S h d k , h { F , M } . ( 2.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrpgpC0xe9LqFf0xc9 qqpeuf0xe9q8qiYRWFGCk9vi=dbvc9G8Wq0db9qqpm0dXdIqpu0=vr 0=vr0=fdbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGabCiwayaary aajaWaaSbaaSqaaiaadIgaaeqaaOGaaGypamaalaaabaWaaabeaeaa caWGKbWaaSbaaSqaaiaadUgaaeqaaOGaaCiEamaaBaaaleaacaWGRb aabeaaaeaacaWGRbGaeyicI4Saam4uamaaBaaameaacaWGObaabeaa aSqab0GaeyyeIuoaaOqaamaaqababaGaamizamaaBaaaleaacaWGRb aabeaaaeaacaWGRbGaeyicI4Saam4uamaaBaaameaacaWGObaabeaa aSqab0GaeyyeIuoaaaGccaaISaGaamiAaiabgIGiopaacmaabaGaam OraiaaiYcacaWGnbaacaGL7bGaayzFaaGaaGOlaiaaywW7caaMf8Ua aGzbVlaaywW7caaMf8UaaiikaiaaikdacaGGUaGaaGymaiaacMcaaa a@5F7E@


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: