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

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Presentation transcript:

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

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

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

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

Tartu, June 26-29, 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, 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, Simulation study (2) rR

Tartu, June 26-29, Thank you!

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

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