Estimation of covariance matrix under informative sampling

1
Estimation of covariance matrix under informative
sampling

• Julia Aru
• University of Tartu and Statistics Estonia

Tartu, June 25-29, 2007
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Outline

• Informative sampling
• Population and sample distribution
• Multivariate normal distribution and exponential
  inclusion probabilities
• Conclusions for normal case
• Simulation study

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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

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Population and sample distribution

• Vector of study variables
• Population distribution
• Sample distribution

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MVN case (1)

• Population distribution multivariate normal with
  parameters µ and S
• Inclusion probabilities
• Matrix A is symmetrical and such that
  
• is positive-definite

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MVN case (2)

• Sample distribution is then again normal with
  parameters

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Conclusions for MVN case

• If variables are independent in the population (S
  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

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Simulation study (1)

• Population is bivariate standard normal with
  correlation coefficient r
• Inclusion probabilities
• Repetitions 1000, population size 10000, sample
  size 1000

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Simulation study (2)
r R
-1 -1 -1
-0.8 -0.26 -0.25
-0.6 0.02 0.01
-0.4 0.16 0.17
-0.2 0.26 0.27
0 0.33 0.33
0.2 0.40 0.39
0.4 0.46 0.46
0.6 0.54 0.53
0.8 0.67 0.68
1 1 1
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Thank you!
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Exponential family (1)

• Population distribution belongs to expontial
  family
• With canonocal representation
• And inclusion probabilities have the form

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Exponential family (2)

• Then sample distribution belonds to the same
  family of distributions with canonical parameters