4. Results

Eric Graf and Yves Tillé

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Simulations were conducted on three sets of real data to compare and assess the different density function estimation methods, f ^ 1 ( x ), MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIXaaabeaakmaabmqabaGaamiEaaGaayjkaiaa wMcaaiaacYcaaaa@3E20@  see (3.1), f ^ 2 ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIYaaabeaakmaabmqabaGaamiEaaGaayjkaiaa wMcaaaaa@3D71@ , see (3.2) and f ^ 3 ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIZaaabeaakmaabmqabaGaamiEaaGaayjkaiaa wMcaaaaa@3D72@ , see (3.4). These methods are required to estimate the variance of certain poverty and inequality indicators.

  1. The first dataset contains equivalent household incomes from the EU-SILC survey conducted by the Swiss Federal Statistical Office in 2009. It includes 17,534 individuals with a non-zero income.
  2. The second dataset also comes from the 2009 EU-SILC survey, but is limited to salaried individuals. It contains salaries from the register of the Central Compensation Office that has been linked with the survey respondents. We therefore have no non-response issues, and there are 7,922 individuals with a non-zero income.
  3. The third test file, named Ilocos, comes with the R package ineq (Zeileis 2012). It contains 632 observations, which are household incomes in Ilocos, one of the 16 regions of the Philippines. The data come from two surveys by the National Statistics Office of the Philippines, in 1997 and in 1998.

The three datasets have a positive skewness coefficient, which is typical of income distributions. Each data set is considered to be one population, and we initially selected 10,000 simple random samples without replacement of various sizes. The values of the various indicators were calculated for each sample, giving us a Monte Carlo estimate of their variance, var sim ( θ ^ ) , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiaabAhaca qGHbGaaeOCamaaBaaaleaacaqGZbGaaeyAaiaab2gaaeqaaOWaaeWa beaacuaH4oqCgaqcaaGaayjkaiaawMcaaiaacYcaaaa@42D7@  for a poverty or inequality indicator θ . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiabeI7aXj aac6caaaa@3B65@  The variance estimator using linearization is denoted var ^ lin ( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaamaaHaaaba GaaeODaiaabggacaqGYbaacaGLcmaadaWgaaWcbaGaaeiBaiaabMga caqGUbaabeaakmaabmqabaGafqiUdeNbaKaaaiaawIcacaGLPaaaaa a@42E3@  and is calculated using the linearization variable z ^ θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadQhaga qcamaaCaaaleqabaGafqiUdeNbaKaaaaaaaa@3BFF@  estimated for each sample:

var ^ lin ( θ ^ ) = N ( N n ) n var ( z ^ S θ ^ ) , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaamaaHaaaba GaaeODaiaabggacaqGYbaacaGLcmaadaWgaaWcbaGaaeiBaiaabMga caqGUbaabeaakmaabmqabaGafqiUdeNbaKaaaiaawIcacaGLPaaacq GH9aqpdaWcaaqaaiaad6eadaqadeqaaiaad6eacqGHsislcaWGUbaa caGLOaGaayzkaaaabaGaamOBaaaacaqG2bGaaeyyaiaabkhadaqade qaaiqadQhagaqcamaaDaaaleaacaWGtbaabaGafqiUdeNbaKaaaaaa kiaawIcacaGLPaaacaaISaaaaa@52F1@

where n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiaad6gaaa a@39F0@  is the size of the sample used for the simulations and

var ( z ^ S θ ^ ) = 1 n 1 k S ( z ^ S , k θ ^ z ¯ S θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiaabAhaca qGHbGaaeOCamaabmqabaGabmOEayaajaWaa0baaSqaaiaadofaaeaa cuaH4oqCgaqcaaaaaOGaayjkaiaawMcaaiabg2da9maalaaabaGaaG ymaaqaaiaad6gacqGHsislcaaIXaaaamaaqafabaWaaeWabeaaceWG 6bGbaKaadaqhaaWcbaGaam4uaiaaiYcacaWGRbaabaGafqiUdeNbaK aaaaGccqGHsislceWG6bGbaebadaqhaaWcbaGaam4uaaqaaiqbeI7a XzaajaaaaaGccaGLOaGaayzkaaaaleaacaWGRbGaeyicI4Saam4uaa qab0GaeyyeIuoaaaa@5704@

where z ¯ S θ ^ = n 1 S z ^ S , k θ ^ , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadQhaga qeamaaDaaaleaacaWGtbaabaGafqiUdeNbaKaaaaGccqGH9aqpcaWG UbWaaWbaaSqabeaacqGHsislcaaIXaaaaOWaaabeaeqaleaacaWGtb aabeqdcqGHris5aOGaaGPaVlqadQhagaqcamaaDaaaleaacaWGtbGa aGilaiaadUgaaeaacuaH4oqCgaqcaaaakiaacYcaaaa@4B4C@  see (2.1).

The quality of the variance estimator using linearization is assessed by comparing the expected Monte Carlo value of the variance estimated using linearization, denoted E sim [ var ^ lin ( θ ^ ) ] , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiaadweada WgaaWcbaGaae4CaiaabMgacaqGTbaabeaakmaadmqabaWaaecaaeaa caqG2bGaaeyyaiaabkhaaiaawkWaamaaBaaaleaacaqGSbGaaeyAai aab6gaaeqaaOWaaeWabeaacuaH4oqCgaqcaaGaayjkaiaawMcaaaGa ay5waiaaw2faaiaacYcaaaa@4958@  with the “true” Monte Carlo variance var sim ( θ ^ ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiaabAhaca qGHbGaaeOCamaaBaaaleaacaqGZbGaaeyAaiaab2gaaeqaaOWaaeWa beaacuaH4oqCgaqcaaGaayjkaiaawMcaaaaa@4227@  in terms of relative bias:

                                               RB [ var ^ lin ( θ ^ ) ] = E sim [ var ^ lin ( θ ^ ) ] var sim ( θ ^ ) var sim ( θ ^ ) .               ( 4.1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiaabkfaca qGcbWaamWabeaadaqiaaqaaiaabAhacaqGHbGaaeOCaaGaayPadaWa aSbaaSqaaiaabYgacaqGPbGaaeOBaaqabaGcdaqadeqaaiqbeI7aXz aajaaacaGLOaGaayzkaaaacaGLBbGaayzxaaGaeyypa0ZaaSaaaeaa caWGfbWaaSbaaSqaaiaabohacaqGPbGaaeyBaaqabaGcdaWadeqaam aaHaaabaGaaeODaiaabggacaqGYbaacaGLcmaadaWgaaWcbaGaaeiB aiaabMgacaqGUbaabeaakmaabmqabaGafqiUdeNbaKaaaiaawIcaca GLPaaaaiaawUfacaGLDbaacqGHsislcaqG2bGaaeyyaiaabkhadaWg aaWcbaGaae4CaiaabMgacaqGTbaabeaakmaabmqabaGafqiUdeNbaK aaaiaawIcacaGLPaaaaeaacaqG2bGaaeyyaiaabkhadaWgaaWcbaGa ae4CaiaabMgacaqGTbaabeaakmaabmqabaGafqiUdeNbaKaaaiaawI cacaGLPaaaaaGaaGOlaiaabccacaqGGaGaaeiiaiaabccacaqGGaGa aeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabccada qadaqaaiaaisdacaGGUaGaaGymaaGaayjkaiaawMcaaaaa@7724@                     

For the second data set (EU-SILC 2009, income of salaried individuals) we also, in a second step, selected 10,000 random samples without replacement under a stratified sampling design, and then calibrated the sampling weights to agree with the eight known sociodemographic marginal totals for the population of 7,922 individuals. The five strata used correspond to the age groups of the salaried individuals (see Table 4.1).

The eight calibration cells were obtained by crossing the three following dichotomous variables (auxiliary calibration variables):

  1. MARIÉ, which indicates whether or not the individual is married;
  2. CHEF, which indicates whether or not the individual’s job is a management position; and
  3. HOMME, which indicates the individual’s sex.

The totals for the population of 7,922 individuals for these calibration cells are shown in Table 4.2.

Table 4.1
Strata used in simulations with 2009 EU-SILC data and three sample sizes (income of salaried individuals, N = 7 , 922 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqabeqabmGabiqaceqabeqadeqabqqaaOqaaiaad6eacq GH9aqpcaaI3aGaaiilaiaaiMdacaaIYaGaaGOmaaaa@3E7C@ )
Table summary
This table displays the results of Strata used in simulations with 2009 EU-SILC data and three sample sizes (income of salaried individuals. The information is grouped by Stratum (appearing as row headers), Description, xxx and % (appearing as column headers).
Stratum  h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqabeqabmGabiqaceqabeqadeqabqqaaOqaaiaadIgaaa a@3C46@ Description N h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqabeqabmGabiqaceqabeqadeqabqqaaOqaaiaad6eada WgaaWcbaGaamiAaaqabaaaaa@3D45@ % n h MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqabeqabmGabiqaceqabeqadeqabqqaaOqaaiaad6gada WgaaWcbaGaamiAaaqabaaaaa@3D65@
1 individuals under 25 1,187 15.0 75 112 150
2 26- to 35-year-olds 1,359 17.2 86 129 171
3 36- to 45-year-olds 2,137 27.0 135 202 270
4 46- to 55-year-olds 1,864 23.5 117 177 235
5 individuals over 55 1,375 17.4 87 130 174
  TOTAL 7,922 100.0 500 750 1,000

 

Table 4.2
Calibration margins in simulations with 2009 EU-SILC data (income of salaried individuals, N = 7 , 922 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqabeqabmGabiqaceqabeqadeqabqqaaOqaaiaad6eacq GH9aqpcaaI3aGaaiilaiaaiMdacaaIYaGaaGOmaaaa@3E7C@ )
Table summary
This table displays the results of Calibration margins in simulations with 2009 EU-SILC data (income of salaried individuals. The information is grouped by Margin (appearing as row headers), Marié, Chef, Homme, Population total and % (appearing as column headers).
Margin Marié Chef Homme Population total %
1 0 0 0 1,487 18.8
2 0 0 1 1,208 15.2
3 0 1 0 323 4.1
4 0 1 1 457 5.8
5 1 0 0 1,759 22.2
6 1 0 1 1,278 16.1
7 1 1 0 328 4.1
8 1 1 1 1,082 13.7
      TOTAL 7,922 100.0

For each stratified sample, a calibration (linear method) was performed to make the sums of the weights agree with the eight margins shown above. Point estimates of the indicators and their linearized variable were computed for each sample using the calibrated weights.

Variance was estimated using the method developed by Deville (2000), which consists of linearizing also with respect to the calibration by calculating the residuals e θ ^ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiaadwgada ahaaWcbeqaaiqbeI7aXzaajaaaaaaa@3BDA@  of the regression (weighted by the sampling weights) of the linearized variables of the indicators for the auxiliary calibration variables. The variance of the total of the residuals thus calculated, under a stratified random sampling plan without replacement is therefore an estimator of the variance of the estimated indicator; it is the quantity of interest:

var ^ lin ( θ ^ ) = h = 1 H N h n h ( N h n h ) s e h θ ^ 2               ( 4.2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaamaaHaaaba GaaeODaiaabggacaqGYbaacaGLcmaadaWgaaWcbaGaaeiBaiaabMga caqGUbaabeaakmaabmqabaGafqiUdeNbaKaaaiaawIcacaGLPaaacq GH9aqpdaaeWbqabSqaaiaadIgacqGH9aqpcaaIXaaabaGaamisaaqd cqGHris5aOWaaSaaaeaacaWGobWaaSbaaSqaaiaadIgaaeqaaaGcba GaamOBamaaBaaaleaacaWGObaabeaaaaGcdaqadeqaaiaad6eadaWg aaWcbaGaamiAaaqabaGccqGHsislcaWGUbWaaSbaaSqaaiaadIgaae qaaaGccaGLOaGaayzkaaGaam4CamaaDaaaleaacaWGLbWaa0baaeaa caWGObaabaGafqiUdeNbaKaaaaaabaGaaGOmaaaakiaabccacaqGGa GaaeiiaiaabccacaqGGaGaaeiiaiaabccacaqGGaGaaeiiaiaabcca caqGGaGaaeiiaiaabccadaqadaqaaiaaisdacaGGUaGaaGOmaaGaay jkaiaawMcaaaaa@65F5@

where

s e h θ ^ 2 = 1 n h 1 k S h ( e k θ ^ e ¯ θ ^ ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiaadohada qhaaWcbaGaamyzamaaDaaabaGaamiAaaqaaiqbeI7aXzaajaaaaaqa aiaaikdaaaGccqGH9aqpdaWcaaqaaiaaigdaaeaacaWGUbWaaSbaaS qaaiaadIgaaeqaaOGaeyOeI0IaaGymaaaadaaeqbqabSqaaiaadUga cqGHiiIZcaWGtbWaaSbaaeaacaWGObaabeaaaeqaniabggHiLdGcda qadeqaaiaadwgadaqhaaWcbaGaam4AaaqaaiqbeI7aXzaajaaaaOGa eyOeI0IabmyzayaaraWaaWbaaSqabeaacuaH4oqCgaqcaaaaaOGaay jkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaaa@54F3@

The quality of the variance estimator using linearization is assessed analogously to the procedure for simple random sampling, see (4.1).

Tables 4.3, 4.4 and 4.5 show the relative bias of the variance for the three data sets used and described above, using simple random sampling. Table 4.6 shows the relative bias of the variance using stratified random sampling with calibrated weights. The upper portions of the tables give the values for the Gini coefficient and QSR, which do not require estimating the income density function. The estimation of their variance works well. Note that there is a problem involving the underestimation of the variance of the Gini coefficient in the case of stratification with calibration (Table 4.6).

For the first data set, Table 4.3 does not reveal any major differences except that the estimation of income density using f ^ 3 ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIZaaabeaakmaabmqabaGaamiEaaGaayjkaiaa wMcaaaaa@3D72@  gives results that are more conservative. In fact, the relative bias remains of the same order of magnitude, but positive, while it is negative for the other two methods of estimating density. For the second data set, Table 4.4 shows that it is essential to use the logarithm or the nearest neighbour method with minimum bandwidth. The latter, all relative bias falls under 10% when the sample sizes are sufficiently large (see last column in the table). Simulations on the same data with a stratified sampling plan and calibration strengthen and confirm these results (see Table 4.6). For the third data set, Table 4.5 shows the same trends, although the results are less stable as a result of the small sample and population sizes. This is not surprising, since the minimum number of neighbours to consider is fixed at 30. In this case, for the Ilocos data set, simulations with a smaller value of p MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiaadchaaa a@39F2@  fixed at 10 makes no difference ultimately, because the condition h ( p j ) h opt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiaadIgada qadeqaaiaadchadaWgaaWcbaGaamOAaaqabaaakiaawIcacaGLPaaa cqGHLjYScaWGObWaaSbaaSqaaiaab+gacaqGWbGaaeiDaaqabaaaaa@4349@  automatically increases it above 30.

Furthermore, generally speaking, we can see that the greater the use of Gaussian kernel density estimation - f ^ 1 ( x ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIXaaabeaakmaabmqabaGaamiEaaGaayjkaiaa wMcaaaaa@3D70@  - the greater the error. In fact, the relative bias of the variance for the median income of individuals below the ARPT and for the RMPG are almost systematically greater in absolute value that those for the other indicators. For the RMPG, the error may be offset (as in Table 4.3) if there are enough observations, since the density estimation appears in both the numerator and the denominator.

Table 4.3
Relative bias (4.1) of the variance obtained with 10,000 simple random samples without replacement from the 2009 EU-SILC data (equivalent household income, N=17,534 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqabeqabmGabiqaceqabeqadeqabqqaaOqaaiaad6eacq GH9aqpcaaIXaGaaG4naiaacYcacaaI1aGaaG4maiaaisdaaaa@3F66@ )
Table summary
This table displays the results of Relative bias (4.1) of the variance obtained with 10. The information is grouped by Indicator (appearing as row headers), Sample size (sampling rate), calculated using xxx units of measure (appearing as column headers).
Indicator Sample size (sampling rate)
n=500( 2.9% ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqabeqabmGabiqaceqabeqadeqabqqaaOqaaiaad6gacq GH9aqpcaaI1aGaaGimaiaaicdadaqadeqaaiaabkdacaqGUaGaaeyo aiaaysW7caGGLaaacaGLOaGaayzkaaaaaa@4567@ n=750( 4.3% ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqabeqabmGabiqaceqabeqadeqabqqaaOqaaiaad6gacq GH9aqpcaaI3aGaaGynaiaaicdadaqadeqaaiaabsdacaqGUaGaae4m aiaaysW7caGGLaaacaGLOaGaayzkaaaaaa@456A@ n=1,000( 5.7% ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqabeqabmGabiqaceqabeqadeqabqqaaOqaaiaad6gacq GH9aqpcaaIXaGaaiilaiaaicdacaaIWaGaaGimamaabmqabaGaaeyn aiaab6cacaqG3aGaaGjbVlaacwcaaiaawIcacaGLPaaaaaa@46CE@
GINI -0.02 -0.02 -0.02
QSR 0.01 0.00 0.00
  f ^ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIXaaabeaaaaa@3D31@ f ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIYaaabeaaaaa@3D32@ f ^ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIZaaabeaaaaa@3D33@ f ^ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIXaaabeaaaaa@3D31@ f ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIYaaabeaaaaa@3D32@ f ^ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIZaaabeaaaaa@3D33@ f ^ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIXaaabeaaaaa@3D31@ f ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIYaaabeaaaaa@3D32@ f ^ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIZaaabeaaaaa@3D33@
ARPT -0.08 -0.06 0.04 -0.09 -0.07 0.03 -0.09 -0.07 0.04
ARPR -0.05 -0.01 -0.00 -0.09 -0.06 -0.05 -0.08 -0.05 -0.03
RMPG -0.09 -0.07 0.15 -0.10 -0.07 0.12 -0.09 -0.06 0.14
MEDP -0.16 -0.12 0.09 -0.19 -0.13 0.05 -0.18 -0.11 0.07
MED -0.08 -0.06 0.05 -0.08 -0.06 0.04 -0.08 -0.06 0.04

 

Table 4.4
Relative bias (4.1) of the variance obtained with 10,000 simple random samples without replacement from the 2009 EU-SILC data (income of salaried individuals, N=7,922 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqabeqabmGabiqaceqabeqadeqabqqaaOqaaiaad6eacq GH9aqpcaaI3aGaaiilaiaaiMdacaaIYaGaaGOmaaaa@3EAC@ )
Table summary
This table displays the results of Relative bias (4.1) of the variance obtained with 10. The information is grouped by Indicator (appearing as row headers), Sample size (sampling rate), calculated using xxx units of measure (appearing as column headers).
Indicator Sample size (sampling rate)
n=500( 6.3% ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqabeqabmGabiqaceqabeqadeqabqqaaOqaaiaad6gacq GH9aqpcaaI1aGaaGimaiaaicdadaqadeqaaiaabAdacaqGUaGaae4m aiaaysW7caGGLaaacaGLOaGaayzkaaaaaa@4565@ n=750( 9.5% ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqabeqabmGabiqaceqabeqadeqabqqaaOqaaiaad6gacq GH9aqpcaaI3aGaaGynaiaaicdadaqadeqaaiaabMdacaqGUaGaaeyn aiaaysW7caGGLaaacaGLOaGaayzkaaaaaa@4571@ n=1,000( 12.6% ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqabeqabmGabiqaceqabeqadeqabqqaaOqaaiaad6gacq GH9aqpcaaIXaGaaiilaiaaicdacaaIWaGaaGimamaabmqabaGaaeym aiaabkdacaqGUaGaaeOnaiaaysW7caGGLaaacaGLOaGaayzkaaaaaa@477E@
GINI -0.03 -0.03 -0.02
QSR -0.00 0.00 0.00
  f ^ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIXaaabeaaaaa@3D31@ f ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIYaaabeaaaaa@3D32@ f ^ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIZaaabeaaaaa@3D33@ f ^ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIXaaabeaaaaa@3D31@ f ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIYaaabeaaaaa@3D32@ f ^ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIZaaabeaaaaa@3D33@ f ^ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIXaaabeaaaaa@3D31@ f ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIYaaabeaaaaa@3D32@ f ^ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIZaaabeaaaaa@3D33@
ARPT 0.07 0.05 0.13 0.06 0.04 0.10 0.06 0.03 0.08
ARPR -0.05 -0.04 -0.02 -0.05 -0.04 -0.01 -0.06 -0.05 -0.02
RMPG 0.61 0.12 0.15 0.60 0.11 0.08 0.59 0.09 0.05
MEDP 0.73 0.17 0.18 0.72 0.16 0.10 0.72 0.15 0.07
MED 0.07 0.04 0.13 0.06 0.04 0.10 0.05 0.03 0.07

 

Table 4.5
Relative bias (4.1) of the variance obtained with 10,000 simple random samples without replacement from Ilocos data (household income, N=632 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqabeqabmGabiqaceqabeqadeqabqqaaOqaaiaad6eacq GH9aqpcaaI2aGaaG4maiaaikdaaaa@3D38@ )
Table summary
This table displays the results of Relative bias (4.1) of the variance obtained with 10. The information is grouped by Indicator (appearing as row headers), Sample size (sampling rate), calculated using xxx units of measure (appearing as column headers).
Indicator Sample size (sampling rate)
n=50( 7.9% ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqabeqabmGabiqaceqabeqadeqabqqaaOqaaiaad6gacq GH9aqpcaaI1aGaaGimamaabmqabaGaae4naiaab6cacaqG5aGaaGjb VlaacwcaaiaawIcacaGLPaaaaaa@44B2@ n=63( 10.0% ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqabeqabmGabiqaceqabeqadeqabqqaaOqaaiaad6gacq GH9aqpcaaI2aGaaG4mamaabmqabaGaaeymaiaabcdacaqGUaGaaeim aiaaysW7caGGLaaacaGLOaGaayzkaaaaaa@455A@
GINI -0.16 -0.13
QSR 0.00 0.00
  f ^ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIXaaabeaaaaa@3D31@ f ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIYaaabeaaaaa@3D32@ f ^ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIZaaabeaaaaa@3D33@ f ^ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIXaaabeaaaaa@3D31@ f ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIYaaabeaaaaa@3D32@ f ^ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIZaaabeaaaaa@3D33@
ARPT -0.05 -0.06 -0.01 -0.03 -0.03 -0.01
ARPR -0.31 -0.01 -0.12 -0.33 -0.03 -0.18
RMPG 1.55 0.83 0.26 1.54 0.16 0.39
MEDP 1.02 0.28 -0.26 1.05 0.07 -0.11
MED 0.04 0.03 0.08 0.07 0.07 0.09

 

Table 4.6
Relative bias (4.1) of the variance obtained with 10,000 stratified random samples without replacement, with weights calibrated to eight sociodemographic margins, from the 2009 EU-SILC data (income of salaried individuals, N=7,922 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqipu0Je9sqqrpepC0xbbL8F4rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqabeqabmGabiqaceqabeqadeqabqqaaOqaaiaad6eacq GH9aqpcaaI3aGaaiilaiaaiMdacaaIYaGaaGOmaaaa@3EAC@ )
Table summary
This table displays the results of Relative bias (4.1) of the variance obtained with 10. The information is grouped by Indicator (appearing as row headers), Sample size (sampling rate) (appearing as column headers).
Indicator Sample size (sampling rate)
n=500( 6.3% ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqabeqabmGabiqaceqabeqadeqabqqaaOqaaiaad6gacq GH9aqpcaaI1aGaaGimaiaaicdadaqadeqaaiaabAdacaqGUaGaae4m aiaaysW7caGGLaaacaGLOaGaayzkaaaaaa@4565@ n=750( 9.5% ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqabeqabmGabiqaceqabeqadeqabqqaaOqaaiaad6gacq GH9aqpcaaI3aGaaGynaiaaicdadaqadeqaaiaabMdacaqGUaGaaeyn aiaaysW7caGGLaaacaGLOaGaayzkaaaaaa@4571@ n=1,000( 12.6% ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqabeqabmGabiqaceqabeqadeqabqqaaOqaaiaad6gacq GH9aqpcaaIXaGaaiilaiaaicdacaaIWaGaaGimamaabmqabaGaaeym aiaabkdacaqGUaGaaeOnaiaaysW7caGGLaaacaGLOaGaayzkaaaaaa@477E@
GINI -0.21 -0.20 -0.20
QSR -0.06 -0.06 -0.07
  f ^ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIXaaabeaaaaa@3D31@ f ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIYaaabeaaaaa@3D32@ f ^ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIZaaabeaaaaa@3D33@ f ^ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIXaaabeaaaaa@3D31@ f ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIYaaabeaaaaa@3D32@ f ^ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIZaaabeaaaaa@3D33@ f ^ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIXaaabeaaaaa@3D31@ f ^ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIYaaabeaaaaa@3D32@ f ^ 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqk0Jf9crFfpeea0xh9v8qiW7rqasFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaaiqadAgaga qcamaaBaaaleaacaaIZaaabeaaaaa@3D33@
ARPT -0.07 -0.09 -0.01 -0.08 -0.1 -0.04 -0.09 -0.11 -0.06
ARPR -0.10 -0.10 -0.08 -0.07 -0.06 -0.05 -0.06 -0.06 -0.05
RMPG 0.63 0.13 0.13 0.61 0.11 0.08 0.59 0.10 0.04
MEDP 0.71 0.16 0.15 0.68 0.13 0.09 0.66 0.12 0.04
MED -0.07 -0.09 -0.01 -0.08 -0.1 -0.04 -0.08 -0.11 -0.06

In short, we see that the variance can be overestimated ( RB [ var ^ lin ( θ ^ ) ]  >  0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaamaabmaaba GaaeOuaiaabkeadaWadeqaamaaHaaabaGaaeODaiaabggacaqGYbaa caGLcmaadaWgaaWcbaGaaeiBaiaabMgacaqGUbaabeaakmaabmqaba GafqiUdeNbaKaaaiaawIcacaGLPaaaaiaawUfacaGLDbaacaqGGaGa aeOpaiaabccacaaIWaaacaGLOaGaayzkaaaaaa@4ABA@  or underestimated ( RB [ var ^ lin ( θ ^ ) ] < 0 ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaamaabmaaba GaaeOuaiaabkeadaWadeqaamaaHaaabaGaaeODaiaabggacaqGYbaa caGLcmaadaWgaaWcbaGaaeiBaiaabMgacaqGUbaabeaakmaabmqaba GafqiUdeNbaKaaaiaawIcacaGLPaaaaiaawUfacaGLDbaacqGH8aap caaIWaaacaGLOaGaayzkaaaaaa@49B7@  depending on the indicator and the data set. The use of the logarithm ( f ^ 2 ( x ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaamaabmqaba GabmOzayaajaWaaSbaaSqaaiaaikdaaeqaaOWaaeWabeaacaWG4baa caGLOaGaayzkaaaacaGLOaGaayzkaaaaaa@3EFB@  provides significant improvement. The nearest neighbour method ( f ^ 3 ( x ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiFu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9LqFf0x e9q8qqvqFr0dXdbrVc=b0P0xb9peuD0xXdbvk9qq=xd9qqaq=Jf9sr 0=vr0=vrWZqaaeaabiGaaiaacaqabeaadaqaaqaaaOqaamaabmqaba GabmOzayaajaWaaSbaaSqaaiaaiodaaeqaaOWaaeWabeaacaWG4baa caGLOaGaayzkaaaacaGLOaGaayzkaaaaaa@3EFC@  eliminates all problems if there is enough data (as in Tables 4.3, 4.4 and 4.6). Slight problems arise with this method when the samples are small (as in Table 4.5). Illogical variations and bias that persist in the tables may also be the result of a lack of robustness in the linearized variables for certain samples, as stated in Section 3.3.

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