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A. Demnati and J. N. K. Rao Statistics Canada / Carleton University

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### Demnati –Rao Approach( Survey Methodology, 2004 )

### Demnati –Rao Approach( Survey Methodology, 2004 )

### Model parameters(Symposium, 2005)

### Multiple Weight Adjustments

### Concluding Remarks

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

looking for a method of variance estimation that### 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

- General formulation

- Finite population parameters

- Model parameters

- Estimator for both parameters

- Variance estimators associated with and are different

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;

- 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

- Example – Ratio estimator of

- is a better choice over customary

- Royall and Cumberland (1981)

- Särndal et al. (1989)

- Valliant (1993)

- Binder (1996)

- Skinner (2004)

Also in Survey Methodology, 2004:### 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

- 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),

Example: Ratio estimator when y is assumed to be random

- 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

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

Under ratio model,

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

- 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 performance

- Ratio estimator:

- We calculated:

- Simulated

- and its components and

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

- 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 performance

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

Simulation 2: Conditional performance

Figure 3: Conditional relative bias of variance estimators

Simulation 2: Conditional performance

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

, and for nominal level of 95%

Generalized Linear Model ### 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

- 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

- 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

- 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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