Estimation of covariance matrix under informative

sampling

Julia AruUniversity of Tartu and Statistics Estonia

Tartu, June 25-29, 2007

Tartu, June 26-29, 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

),...,,( 21 kyyyy)(ypf

)(

)()|()1|()(

p

spps E

fEIff

yyyy

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

)''exp()|( ybAyyy cEp

1

/ 2

1 1( | , ) exp ( ) ' ( )

2(2 ) | |p kf

y μ Σ y μ Σ y μ

Σ

1 1( 2 ) Σ A

Tartu, June 26-29, 2007 6

MVN case (2)

• Sample distribution is then again normal with parameters

1 1 1' ( ' ')( 2 ) λ μ Σ b Σ A

1 1( 2 ) Ω Σ A

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

1 1 1' ( ' ')( 2 ) λ μ Σ b Σ A1 1( 2 ) Ω Σ A

• 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

)'0 ,0(μ

R̂

1

1

r

rΣ

12/1

2/11A )'1 ,1(b

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Simulation study (2)

R̂r 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

1

( | ) ( ) exp ( ) ( ) ( )d

p i ii

f h g T B

y θ y θ y θ

* *

1

( | ) ( ) exp ( ) ( )d

p i ii

f h T B

y η y y η

1

( | ) exp ( )d

p i ii

E r pT

y y

Tartu, June 26-29, 2007 12

Exponential family (2)

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

*i i ip