1. Calibrated estimators of the population covariance
Aleksandras Plikusas and Dalius Pumputis,
Institute of Mathematics and Informatics,
Vilnius, Lithuania.

2. 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.

3. - 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.

4. 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.

5. 1. The nonlinear calibration
The calibrated weights satisfy the calibration equation

6. 2. The linear calibration
The weights are defined by the calibration equation

7. 3. Calibration of the total
The calibration equations are

8. Different loss functions can be used: ♦ ♦ ♦
♦ ♦ ♦

9. Let us introduce some notation:

10. Proposition 1. The weights , which satisfy the calibration equation
and minimize the loss function can be expressed as with

11. 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

12. Proposition 3. The weights , which satisfy the calibration equation and minimize the loss function can be expressed as with

13. Proposition 4. The weights , which satisfy the calibration equation and minimize the loss function can be expressed as with

14. Simulation results
We compare three types of calibrated estimators with known
estimators of population covariance:

15. 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

17.  Case
Bias
MSE cv
4.53E+13
0.0919
4.52E+13
9.48E+13
0.1265
9.50E+13
0.1267
546136
2.93E+13
0.0808
535256
2.92E+13
0.0807
1.55E+14
0.1836
1.54E+14
0.1835
Case 
 Bias
 MSE
 cv
1.03E+14
0.1493
0.1494
8.37E+13
0.1248
8.40E+13
0.1250
1.50E+14
0.1782
1.49E+14
0.1820
1.48E+14

18. Case 
 Bias
MSE 
 cv
1.73E+14
0.2004
1.70E+14
0.1978
1.56E+14
0.1848
0.1846
1.60E+14
0.1855
1.57E+14
0.1847

20. 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

22. Correlation 0,9 Correlation 0,6 Case Bias MSE cv -345948 1.76E+11
1.76E+11
0.2779
1.74E+11
0.2761
3.85E+11
0.5741
3.82E+11
0.5721
5.03E+10
0.1360
4.83E+10
0.1345
5.08E+11
0.7174
5.09E+11
0.7172
Correlation 0,6
Case
Bias
MSE cv
3.86E+11
0.5744
3.85E+11
0.5740
4.87E+11
0.7170
4.85E+11
0.7150
3.22E+11
0.4770
3.19E+11
0.4758
5.14E+11
0.7300
5.15E+11
0.7299

23. Correlation 0,3 Case Bias MSE cv -444330 4.92E+11 0.7213 -443450
4.92E+11
0.7213
4.90E+11
0.7200
5.24E+11
0.7586
5.23E+11
0.7566
4.27E+11
0.6228
4.26E+11
0.6227
5.13E+11
0.7445
0.7443

25. Correlation 0,8 Correlation 0,4 Case Bias MSE cv -120894 1.36E+10
0.0738
1.77E+10
0.0845
1.78E+10
65118
1.05E+10
0.0643
65239
2.76E+10
0.1055
-66528
Correlation 0,4
Case
Bias
MSE cv
2.50E+10
0.1004
2.40E+10
0.0984
-73870
2.83E+10
0.1067
-74058
2.82E+10
0.1066
2.69E+10
0.1042
-65692