1. Estimation of covariance matrix under informative sampling Julia Aru University of Tartu and Statistics Estonia Tartu, June 25-29, 2007

2. Tartu, June 26-29, 20072 Outline Informative sampling Population and sample distribution Multivariate normal distribution and exponential inclusion probabilities Conclusions for normal case Simulation study

3. Tartu, June 26-29, 20073 Informative sampling Probability that an object belongs to the sample depends on the variable we are interested in For example: while studying income we see that people with higher income are not keen to respond Under informative sampling sample distribution of variable(s) of interest differs from that in population

4. Tartu, June 26-29, 20074 Population and sample distribution Vector of study variables Population distribution Sample distribution

5. Tartu, June 26-29, 20075 MVN case (1) Population distribution: multivariate normal with parameters µ and Σ: Inclusion probabilities: Matrix A is symmetrical and such that is positive-definite

6. Tartu, June 26-29, 20076 MVN case (2) Sample distribution is then again normal with parameters

7. Tartu, June 26-29, 20077 Conclusions for MVN case If variables are independent in the population (Σ is diagonal) then independence is preserved only in the case when matrix A is also diagonal Matrix A can be chosen to make variables independent in the sample or dependence structure to be very different from that in the population

8. Tartu, June 26-29, 20078 Simulation study (1) Population is bivariate standard normal with correlation coefficient r : Inclusion probabilities: Repetitions: 1000, population size: 10000, sample size: 1000

9. Tartu, June 26-29, 20079 Simulation study (2) rR -0.8-0.26-0.25 -0.60.020.01 -0.40.160.17 -0.20.260.27 00.33 0.20.400.39 0.40.46 0.60.540.53 0.80.670.68 111

10. Tartu, June 26-29, 200710 Thank you!

11. Tartu, June 26-29, 200711 Exponential family (1) Population distribution belongs to expontial family With canonocal representation And inclusion probabilities have the form

12. Tartu, June 26-29, 200712 Exponential family (2) Then sample distribution belonds to the same family of distributions with canonical parameters