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Estimation of covariance matrix under informative sampling

Estimation of covariance matrix under informative sampling. Julia Aru University of Tartu and Statistics Estonia. Tartu , June 2 5 - 29 , 200 7. Outline. Informative sampling Population and sample distribution Multivariate normal distribution and exponential inclusion probabilities

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Estimation of covariance matrix under informative sampling

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

  2. Outline • Informative sampling • Population and sample distribution • Multivariate normal distribution and exponential inclusion probabilities • Conclusions for normal case • Simulation study Tartu, June 26-29, 2007

  3. 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 Tartu, June 26-29, 2007

  4. Population and sample distribution • Vector of study variables • Population distribution • Sample distribution Tartu, June 26-29, 2007

  5. MVN case (1) • Population distribution: multivariate normal with parameters µ and Σ: • Inclusion probabilities: • Matrix A is symmetrical and such that is positive-definite Tartu, June 26-29, 2007

  6. MVN case (2) • Sample distribution is then again normal with parameters Tartu, June 26-29, 2007

  7. 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 Tartu, June 26-29, 2007

  8. Simulation study (1) • Population is bivariate standard normal with correlation coefficient r : • Inclusion probabilities: • Repetitions: 1000, population size: 10000, sample size: 1000 Tartu, June 26-29, 2007

  9. Simulation study (2) Tartu, June 26-29, 2007

  10. Thank you! Tartu, June 26-29, 2007

  11. Exponential family (1) • Population distribution belongs to expontial family • With canonocal representation • And inclusion probabilities have the form Tartu, June 26-29, 2007

  12. Exponential family (2) • Then sample distribution belonds to the same family of distributions with canonical parameters Tartu, June 26-29, 2007

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