Calibrated estimators of the population covariance Aleksandras Plikusas and Dalius Pumputis, Institute of Mathematics and Informatics, Vilnius, Lithuania.
Let y and z be two study variables defined on the population Consider a finite population : of N elements. Let y and z be two study variables defined on the population U and taking values and respectively.
- are sample design weights. We are interested in the estimation of the covariance Let us consider the standard estimator: - denotes a probability sample set. - are sample design weights. - is a probability of inclusion of the element k into the sample set s.
Let be known auxiliary variables with known covariance We construct three types of estimators of the covariance The weights are defined using three different calibration equations.
1. The nonlinear calibration The calibrated weights satisfy the calibration equation
2. The linear calibration The weights are defined by the calibration equation
3. Calibration of the total The calibration equations are
Different loss functions can be used: ♦ ♦ ♦ ♦ ♦ ♦
Let us introduce some notation:
Proposition 1. The weights , which satisfy the calibration equation and minimize the loss function can be expressed as with
Proposition 2. The weights , which satisfy the calibration equation and minimize the loss function can be expressed as with here is a properly chosen root of the equation
Proposition 3. The weights , which satisfy the calibration equation and minimize the loss function can be expressed as with
Proposition 4. The weights , which satisfy the calibration equation and minimize the loss function can be expressed as with
Simulation results We compare three types of calibrated estimators with known estimators of population covariance:
We have used for simulation distance functions and and three types of calibration equation. Six cases: I type calibration: and II type calibration: and III type calibration: and
Case Bias MSE cv -3490848 4.53E+13 0.0919 -3489778 4.52E+13 -6104197 9.48E+13 0.1265 -6111007 9.50E+13 0.1267 546136 2.93E+13 0.0808 535256 2.92E+13 0.0807 -5602394 1.55E+14 0.1836 -5455553 1.54E+14 0.1835 Case Bias MSE cv -4209064 1.03E+14 0.1493 -4216576 0.1494 -5069762 8.37E+13 0.1248 -5085344 8.40E+13 0.1250 -5874261 1.50E+14 0.1782 -5881074 -5082344 1.49E+14 0.1820 -4934908 1.48E+14
Case Bias MSE cv -4690847 1.73E+14 0.2004 -4854206 1.70E+14 0.1978 -5562585 1.56E+14 0.1848 -5561796 0.1846 -5912708 1.60E+14 0.1855 -5911844 -5662564 1.57E+14 -5515768 0.1847
Correlation 0,8 Correlation 0,4 Case Bias MSE cv -3675 9.17E+08 0.0867 -3675 9.17E+08 0.0867 -3450 9.03E+08 0.0861 3671 1.43E+09 0.1063 3402 4484 7.32E+08 0.0752 4828 7.16E+08 0.0741 -125 1.58E+09 0.1134 306 Correlation 0,4 Case Bias MSE cv 2063 1.57E+09 0.1124 2427 1.53E+09 0.1108 2498 1.60E+09 0.1131 2292 1.59E+09 0.1130 20653 2.64E+09 0.1269 20720 2.58E+09 0.1251 -130 0.1127 341 0.1126
Correlation 0,9 Correlation 0,6 Case Bias MSE cv -345948 1.76E+11 -345948 1.76E+11 0.2779 -344035 1.74E+11 0.2761 -446667 3.85E+11 0.5741 -442287 3.82E+11 0.5721 -176199 5.03E+10 0.1360 -170965 4.83E+10 0.1345 -286681 5.08E+11 0.7174 -283628 5.09E+11 0.7172 Correlation 0,6 Case Bias MSE cv -450255 3.86E+11 0.5744 -448187 3.85E+11 0.5740 -382883 4.87E+11 0.7170 -380149 4.85E+11 0.7150 -436150 3.22E+11 0.4770 -433248 3.19E+11 0.4758 -291121 5.14E+11 0.7300 -288024 5.15E+11 0.7299
Correlation 0,3 Case Bias MSE cv -444330 4.92E+11 0.7213 -443450 -444330 4.92E+11 0.7213 -443450 4.90E+11 0.7200 -402524 5.24E+11 0.7586 -401682 5.23E+11 0.7566 -473459 4.27E+11 0.6228 -471243 4.26E+11 0.6227 -293768 5.13E+11 0.7445 -281019 0.7443
Correlation 0,8 Correlation 0,4 Case Bias MSE cv -120894 1.36E+10 0.0738 -121548 -129208 1.77E+10 0.0845 -129470 1.78E+10 65118 1.05E+10 0.0643 65239 -118643 2.76E+10 0.1055 -66528 Correlation 0,4 Case Bias MSE cv -112093 2.50E+10 0.1004 -111965 -162762 2.40E+10 0.0984 -162495 -73870 2.83E+10 0.1067 -74058 2.82E+10 0.1066 -117655 2.69E+10 0.1042 -65692