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Linearization Variance Estimators for Survey Data: Some Recent Work. A. Demnati and J. N. K. Rao Statistics Canada / Carleton University. A Presentation at the Third International Conference on Establishment Surveys June 18-21, 2007. Montréal, Québec, Canada June 20, 2007.

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Linearization Variance Estimatorsfor Survey Data: Some Recent Work

A. Demnati and J. N. K. RaoStatistics Canada / Carleton University

A Presentation at the Third International Conference onEstablishment SurveysJune 18-21, 2007

Montréal, Québec, CanadaJune 20, 2007


Situation

Situation

  • is simple

  • is widely applicable

  • has good properties

  • provides unique choice

  • for estimators

  • of nonlinear finite population parameters

  • SM, 2004

  • defined explicitly or implicitly

  • SM, 2004

  • using calibration weights

  • SM, 2004

  • under missing data

  • JSM, 2002 and JMS, 2002

  • using repeated survey

  • FCSM, 2003

  • of model parameters

  • Symposium, 2005

  • of dual frames

  • JSM, 2007


Demnati rao approach
Demnati –Rao Approach

  • General formulation

  • Finite population parameters

  • Model parameters

  • Estimator for both parameters

  • Variance estimators associated with and are different


Demnati rao approach survey methodology 2004
Demnati –Rao Approach( Survey Methodology, 2004 )

  • Write the estimator of a finite population parameter as

with

if element k is not in sample s;

if element k is in sample s;


Demnati rao approach survey methodology 20041

Demnati –Rao Approach( Survey Methodology, 2004 )

  • A linearization sampling variance estimator is given by

with

: variance estimator of the H-T estimator

of the total

is a (N×1) vector of arbitrary number


Demnati rao approach survey methodology 20042

Demnati –Rao Approach( Survey Methodology, 2004 )

  • Example – Ratio estimator of

For SRS and


Demnati rao approach survey methodology 20043

Demnati –Rao Approach( Survey Methodology, 2004 )

  • Example – Ratio estimator of

  • is a better choice over customary

  • Royall and Cumberland (1981)

  • Särndal et al. (1989)

  • Valliant (1993)

  • Binder (1996)

  • Skinner (2004)


Demnati rao approach1

Demnati –Rao Approach

  • Calibration Estimators:

  • the GREG Estimator

  • the “Optimal” Regression Estimator

  • the Generalized Raking Estimator

  • Two-Phase Sampling

  • New Extensions:

  • Wilcoxon Rank-Sum Test

  • Cox Proportional Hazards Model


Model parameters symposium 2005

Model parameters(Symposium, 2005)

  • Finite-population assumed to be generated from a superpopulation model

  • Inference on model parameter

  • Total variance of :

: model expectation and variance

: design expectation and variance

i) if f ≈ 0 then

ii) if f ≈ 1 then

where f is the sampling fraction. For multistage sampling, the psu sampling fraction plays the role of f.

In case i),


  • for

  • Define

  • We have

where Ad is a 2×N matrix of random variables with kth column:

  • We get

where Ab is a 2×N matrix of arbitrary real numbers with kth column:

where is an estimator of the total variance of


and

when

  • A variance estimator of is given by

with

where

Note that is an estimator of model covariance

when and when


= model variance + sampling variance

where

and

  • Under SRS,

where


Note: remains valid under misspecification of

  • Hence,

Note: g-weight appears automatically in

and the finite population correction 1-n/N is absent in


Simulation 1 unconditional performance
Simulation 1: Unconditional performance

  • We generated R=2,000 finite populations , each of size N=393 from the ratio model

where

are independent observations generated from a N(0,1)

are the “number of beds” for the Hospitals population

studied in Valliant, Dorfman, and Royall (2000, p.424-427)

  • One simple random sample of specified size n is drawn from each generated population

  • Parameter of interest:


Simulation 1 unconditional performance1
Simulation 1: Unconditional performance

  • Ratio estimator:

  • We calculated:

  • Simulated

  • and its components and


Simulation 1 unconditional performance2
Simulation 1: Unconditional performance

Figure 1: Averages of variance estimates for selected sample sizes compared to simulated MSE of the ratio estimator.


Simulation 2 conditional performance
Simulation 2: Conditional performance

  • We generate R=20,000 finite populations , each of size N=393 from the ratio model

using the number of beds as

  • One simple random sample of size n=100 is drawn from each generated population

  • Parameter of interest:

  • We arranged the 20,000 samples in ascending order of -values and then grouped them into 20 groups each of size 1,000


Simulation 2 conditional performance1
Simulation 2: Conditional performance

Figure 2: Conditional relative bias of the expansion and ratio estimators of


Simulation 2 conditional performance2
Simulation 2: Conditional performance

Figure 3: Conditional relative bias of variance estimators


Simulation 2 conditional performance3
Simulation 2: Conditional performance

Figure 4: Conditional coverage rates of normal theory confidence intervals based on

, and for nominal level of 95%


G weighted estimating functions model parameter

g-weighted estimating functions: model parameter

  • is the solution of weighted estimating equation:

  • is solution

  • Special case: (GREG)

  • Linear Regression Model

  • Logistic Regression Model


Simulation 3 estimating equations
Simulation 3: Estimating equations

  • We generated R=10,000 finite populations , each of size N=393 from the model

  • Using the number of beds as

  • leads to an average of about 60% for z

  • One simple random sample of size n=30 is drawn from each generated population

  • Parameter of interest:

  • Population units are grouped into two classes with 271 units k having x<350 in class 1 and 122 units k with x>=350 in class 2

  • Post-stratification: X=(271,122)T



Multiple weight adjustments

Multiple Weight Adjustments

  • Weight Adjustments for

  • Units (or complete) nonresponse

  • Calibration

  • Due to lack of time, not presented in the talk,

but it is included in the proceeding paper


Concluding remarks

Concluding Remarks

  • We provided a method of variance estimation for estimators:

  • of nonlinear model parameters

  • using survey data

  • defined explicitly or implicitly

  • using multiple weight adjustments

  • under missing data

  • The method

  • is simple

  • is widely applicable

  • has good properties

  • provides unique choice

Thank you Very Much


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