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Empirical Likelihood confidence intervals under unequal

probability sampling

Yves G. Berger

Omar De La Rivar Torres

Design-based inference without re-sampling, linearisation and variance estimation

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Outline

● Issues with standard Confidence Intervals.● A new Empirical Likelihood approach:

➔Point estimation➔Estimation of Confidence Intervals

● Simulations● European Income & Living Condition Survey

2009 (NET-SILC2, EUROSTAT)● Concluding remarks

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Issues with standardConfidence Intervals

● Skewed variables

→ Skewed sampling distributions

→ Poor coverages of Standard CI

→ Linearised variance estimates can be poor

● Example: Income/wealth variables

Domains Extreme values

Measures of poverty, Quantiles

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Example: Confidence interval of a 10% quantile

● Skewed population (exponential)● Skewed sampling distribution

95% CI based upon: n = 80N= 800

n = 80N = 150

Linearisation 98% 99.8%Rescaled Bootstrap 97% 99%Direct Bootstrap 93% 90%Woodruff 93% 94%Proposed Empirical Likelihood 96% 94%

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Example: Estimation of a mean with auxiliary variables

● Skewed data N = 150

n = 40 n = 80Standard 91% 93%Pseudo - EL1 94% 94%Pseudo - EL2 87% 89%Proposed EL 95% 94%

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Example: Persistent risk of Poverty (European Income & Living Condition

Survey 2009)● Male 25 yo – 44yo Standard Emp. Likelihood

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New Empirical Likelihood Approach

Does not involve● variance estimates● Linearisation● Re-sampling● normality of the point estimator● negligible sampling fractions

Remark: Pseudo-EL proposed EL

Pseudo-EL relies on variance

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The parameter of interest

● Population parameter solution of estimating equations

● Examples: Mean, Total, Ratio, Quantiles,

M-estimator, Poverty indicators, regression,

Winsorisation ...

does not need to be differentiable!

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Proposed Empirical Likelihood Approach

● Empirical likelihood function:

● = Unit mass of unit

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Proposed Empirical Likelihood Approach

● Maximise

Under the constraint

Design + auxiliary ● Example:

auxiliary variables

strat. variables

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Empirical log-likelihood ratio function (deviance)

"Reduced" "Full"

● Maximum under

● Maximum under

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Maximum Empirical Likelihood Estimator

● Maximum EL Estimator of minimises

Maximum EL Estimator is the solution of

The maximise under

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Maximum Empirical Likelihood Estimator

● Maximise

under the constraint and

● Solution:

● Consider that

always holds

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Examples of Maximum Empirical likelihood estimators

● Example 1: "model" with just an intercept

Hájek Estimator

Greg if contains auxiliary variables

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Examples of Maximum Empirical likelihood estimators

● Example 2: Ratio "model"

Hortvitz-Thompson estimator

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Maximum Empirical Likelihood Estimator

● Example 3: Auxiliary variables within

Optimal GREG● Example 4:

Kim (2009) EL

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Pps sampling (with replacement)

● Under regularity conditions

under pps sampling

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Empirical Likelihood Confidence Intervals (pps sampling)

● Confidence intervals (Wilks' type)

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Empirical Likelihood Confidence Intervals

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EL relies on normality of the estimating equation when !

Remark: with auxilliary variables, Greg

instead of HT● The point estimator does not have to be normal

or unbiased

stronger & harder to justify

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Without auxilliary variables

"Reduced" "Full"

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With auxilliary variables

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With auxilliary variables + Stratification

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πps sampling(without replacement)

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πps sampling(without replacement)

under Hájek (1964) asymptotic framework "High entropy"

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πps sampling(without replacement)

● reduce the effect on the CI of units with large (finite population corrections)

● not needed

● not adjusted by parameters that need to

be estimated

● Can be extended with auxilliary variables

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Simulations

● Population data (skewed) Rao & Wu (2006)

● and ~ exponential●

●

●

●

● Value to control correlation( , )

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Coverage Prob. Mean. N=800. No Auxil. Var.

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Coverage Prob. Mean. N=150. No Auxil. Var.

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Coverage Prob. Mean. N=150. With Auxil. Var.

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Variance Length CI. Mean. N=150. With Auxil.

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Coverage Prob. 1St Quartile. N=800. No Auxil.

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Variance Length CI. 1St Quartile. N=800. No Auxil.

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Coverage Prob. 1St Quartile. N=150. No Auxil.

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Variance Length CI. 1St Quartile. N=150. No Auxil.

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Persistent risk of Poverty (European Income & Living Condition Survey 2009)

● Male 25 yo – 44yo (multi-stage designs)

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New EL versus Bootstrap● Does not need re-sampling. "Simpler than

bootstrap"● Wider class of parameters compared to

bootstrap. ● More stable CI than direct bootstrap● Better coverage than bootstrap● EL include design information ( , stratication,

clusters).● EL intervals take into account of the bias of the

point estimator

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New EL versus Pseudo-Empirical likelihood

● New EL ≠ Pseudo-EL● The pseudo-EL function is not a standard EL

function● CI Relies on variances (design effect)● May need N for totals and counts● Limited range of parameter with pseudo-EL. E.g.

no pseudo-EL CI for quantiles (only woodruff)● More stable CI than pseudo-EL● Design information through a design effect

(estimated)● Range preserving and good coverages

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New EL versus Calibration

● Equivalent point estimator.● EL can be used without auxilliary information● EL can be used for testing, CI, p-values● EL can be used with "calibration weights"

(same point estimates)

● Calibration relies on CLT & variances● Calibration relies on a distance function

disconnected from mainstream statistics

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Extensions

● Multi-stage samplng (OK for small sampling fractions)

● Rao-Hartley-Cochran design● Modelling: design naturally included, random

effect no needed● Conditional Estimating Equations

Example

Can't be solved with estimating equations

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Extensions

● Re-weighting (Total nonresponse)

● Random Hot-deck imputation

● Calibration on known quantiles

or on distribution functions

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STD Bootstrap

RS DirectEmp. Lik.

Design based √ √ √ √

Does not rely on normality of Point estimator × √ × √

Does not need variance estimates × √ × √

Does not need re-sampling √ × × √

Does not need linearisation × √ √ √

Range preserved × √ × √

Take into account sampling distribution × √ × √

Take into account of the design √ √ ?√ √

Suitable with large sampling fractions √ × √ √Complex Parameters √ √ ? √

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Concluding Remarks● Does not involve variance estimation &

Linearisation● Design based (non-parametric)● Flexible and general approach (complex

parameters, modelling)● Does not rely on normality of the point

estimator● Better coverage for confidence intervals

(better inference)● EL intervals take into account of the bias

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References

BERGER & DE LA RIVA TORRES (2012).

http://eprints.soton.ac.uk/337688/

BERGER & DE LA RIVA TORRES (2012).

Proceedings of the Survey Research Method Section of the American Statistical Association, Joint Statistical Meeting, San Diego

OSIER, BERGER and GOEDEMÉ (2013)

Standard error estimation for the EU-SILC indicators of poverty and social exclusion. Eurostat “Methodologies and Working papers” series

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Regularity conditions●

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