Robust Fay Herriot Estimators in Small Area Estimation Sebastian - - PowerPoint PPT Presentation

Robust Fay Herriot Estimators in Small Area Estimation

Sebastian Warnholz Statistical Consultancy – FU Berlin 5th May 2016

Outline

◮ Small Area Estimation ◮ Area Level Models ◮ Robust Area Level Models ◮ Example & Simulation Study

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Small Area Estimation

◮ SAE: Estimation of population parameters for small domains / areas ◮ Problem: Direct estimations may have insufficient precision

(variance)

◮ Estimations may be based on survey data which was not designed to

make predictions for small domains

◮ Very view or no sampled units are available within target domains

◮ Methods used in SAE borrow strength to improve domain

predictions by

◮ using additional data sources ◮ exploiting correlation structures (space and time) ◮ often models , S3RI Research Seminars 3

Models in SAE

◮ Area level models:

◮ Use information on the area level, e.g. aggregates like a direct

estimator

◮ Are used when unit level information is not available ◮ May be useful to reduce computational complexity

◮ Unit level models:

◮ Use the sampled observations directly ◮ May provide more precise parameter estimates due to increased

number of observations

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Area Level Models

◮ Fay and Herriot (1979):

◮ ¯

yi = θi + ei; ei ∼ N(0, σ2

ei); i = 1, . . . , D

◮ θi = x⊤

i β + vi; vi ∼ N(0, σ2 v)

◮ And combined, an estimator for the population mean can be derived:

ˆ θFH

i

= ˆ γi¯ yi + (1 − ˆ γi)x⊤

i ˆ

β with ˆ γi = ˆ σ2

v

ˆ σ2

v + σ2 ei

◮ When σ2

ei >> ˆ

σ2

v we rely more on the synthetic estimator

◮ When σ2

ei << ˆ

σ2

v the direct estimator is preferred

◮ σ2

ei is assumed to be known under the model – in practice we may

use the sampling variance

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Outliers in Area Level Models

¯ yi = x⊤

i β + vi + ei ◮ Area level outliers are outliers in the random effect: vi – i.e. all

units within a domain are outlying

◮ Here a robust method can be beneficial

◮ Unit level outliers are outliers in ei – single units

◮ We may use estimated sampling variances for σ2

ei; then the FH model

will automatically plug-in the synthetic estimator

◮ When the sampling variances are unreliable they may be replaced

using a more stable estimate based on generalised variance functions

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Robust Area Level Methods – Review

◮ When framed as a violation of the distributional assumption (of vi):

◮ Transform the response, i.e. the direct estimator – Sugasawa and

Kubokawa (2015)

◮ Replace the distribution (e.g.) ◮ generalised normal: Fabrizi and Trivisano (2010) ◮ t-distribution: Bell and Huang (2006) ◮ Cauchy distribution: Datta and Lahiri (1995)

◮ When we still believe in the normal distribution:

◮ Use influence functions in the context Hierarchical Bayes: Ghosh,

Maiti and Roy (2008)

◮ Use influence functions in the context of linear mixed models: Sinha

and Rao (2009)

◮ M-Quantile regression: Chambers and Tzavidis (2006)

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Robust Area Level Methods – Method

◮ Here the method by Sinha and Rao (2009) is adapted for area level

models framed as linear mixed model y ∼ N(Xβ, ZVvZ⊤ + Ve

• V

)

◮ Restrict the influence of the residuals in ML estimation equations.

E.g. for the regression parameters we use: X⊤V−1U

1 2 ψ(U− 1 2 (y − Xβ)) = 0

instead of X⊤V−1 (y − Xβ) = 0

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Robust Area Level Methods – Method

◮ Solving these robust estimation equations leads to outlier robust

parameter estimates, ˆ βψ and σ2,ψ

v

, and outlier robust predictions: ˆ vψ

i

ˆ θRFH

i

= x⊤

i ˆ

βψ + ˆ vψ

i ◮ In the setting of linear mixed models this representation is the

robust empirical best linear unbiased prediction (REBLUP)

◮ The MSE of these predictions can be computed using a parametric

bootstrap or an approximation based on the results of Chambers, Chandra and Tzavidis (2011)

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Robust Area Level Methods – Extensions

◮ Framed as linear mixed effects models we can incorporate spatial

and temporal correlation in the random effects:

◮ Simultanous autoregressive process – Pratesi and Salvati (2008) ◮ Random intercept + temporal autocorrelation – Rao and Yu (1994) ◮ Combining spatial and temporal correlation – Marhuenda et.al.

(2013)

◮ The same idea for robust predictions can be used for these methods

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Robust Area Level Methods – Optimisation

◮ Sinha and Rao (2009) derived Newton-Raphson algorithms based

• n a Taylor series expansion of the estimation equations (unit level

models)

◮ Schmid (2011) minimised the squared estimation equations for

variance components – more stable

◮ Schoch (2012) uses a IRWLS algorithm for β and a robust method

• f moments estimator for the variance parameters – more stable for

starting values

◮ Chatrchi (2012) uses a fixed point algorithm for variance

components – slow but stable for starting values

◮ For area level models:

◮ IRWLS algorithm for the regression parameters ◮ Fixed-point algorithm for the random effects ◮ For variance components: ◮ Fixed point algorithm for variances ◮ Newton-Raphson for correlation parameters , S3RI Research Seminars 11

Robust Area Level Methods – Software

◮ R-packages:

◮ rsae – implements the methods by Schoch (2012) for unit level

models

◮ saeRobust (about to be released) – implements the presented

methods for

◮ Standard RFH ◮ Spatial RFH ◮ Temporal RFH ◮ Spatio-Temporal RFH , S3RI Research Seminars 12

CBS Data Example

◮ The target statistic is the mean tax turnover of 20 industry sectors

in the Netherlands

◮ Available is a synthetic population with 63981 observations

◮ Based on the Structural Business Survey (SBS) which is an annual

survey in the Netherlands conducted by CBS

◮ In this example one sample is drawn similar to the design in the

SBS:

◮ Stratified for the size class (employee) of firms ◮ SRSWOR within each stratum ◮ Large firms are selected with probability one

◮ Sample sizes range between 9 and 1052; 5074 overall ◮ This is repeated 500 times and compared to the population

parameters

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Modeling Strategy

¯ yi = β0 + β1¯ yi,t−1 + vi + ei

◮ ¯

yi is the direct estimator based on the HT estimator

◮ ¯

yi,t−1 is the true tax turnover from the previous period

◮ The sampling variances under the FH model, σ2 ei, are either based

• n the estimated standard error of the direct estimator; or

smoothed using a generalised variance function

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QQ Plots

−0.04 −0.02 0.00 0.02 Random Effects

RFH FH

−3 −2 −1.00 1 2 −2 −1 1 2 theoretical residuals / sqrt(samplingVar) −2 −1 1 2 theoretical

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Coefficient of Variation

20 40 60 5 10 15 20 domain (sorted by increasing CV of direct) CV in %

direct eblup reblup

direct eblup reblup

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RBIAS & RRMSE

Direct FH RFH FH.GVF RFH.GVF −20 −10 RBIAS in % 20 40 60 80 RRMSE in %

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Discussion

◮ Outlier robust predictions may be beneficial to address area level

• utliers

◮ Unit level outliers? ◮ MSE estimation is problematic in scenarios where the estimated

variance of the random effect is very small

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Thank you for your attention! Sebastian Warnholz (Sebastian.Warnholz@fu-berlin.de)

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Bibliography

◮ Bell / Huang (2006): Using the t-distribution to Deal with Outliers

in Small Area Estimation, Proceedings of Statistics Canada Symposium 2006: Methodological Issues in Measuring Population Health

◮ Chatrchi (2012): Robust Estimation of Variance Components in

Small Area Estimation, MA thesis, School of Mathematics and Statistics, Carleton University, Ottawa, Canada

◮ Datta / Lahiri (1995): Robust Hierarchical Bayes Estimation of

Small Area Characteristics in the Presence of Covariates and Outliers, Journal of Multivariate Analysis 54, pp. 310–328

◮ Ghosh / Maiti / Roy (2008): Influence functions and robust Bayes

and empirical Bayes small area estimation. Biometrika 95.3,

• pp. 573–585

◮ Fabrizi / Trivisano (2010): Robust Linear Mixed Models for Small

Area Estimation, Journal of Statistical Planning and Inference 140, 433–43

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Bibliography

◮ Fay / Herriot (1979): Estimation of income for small places: An

application of james-stein procedures to census data, Journal of the American Statistical Association 74 (366), 269–277

◮ Gershunskaya (2010): Robust Small Area Estimation Using a

Mixture Model, Section on Survey Methods, JSM, 2783-2796

◮ Marhuenda / Molina / Morales (2013): Small area estimation with

spatio-temporal Fay-Herriot models, Computational Statistics and Data Analysis 58, pp. 308–325

◮ Pratesi / Salvati (2008): Small area estimation: the EBLUP

estimator based on spatially correlated random area effects, Statistical Methods & Applications 17, pp. 113–141

◮ Rao / Yu (1994): Small-Area Estimation by Combining Time-Series

and Cross-Sectional Data, Canadian Journal of Statistics 22.4,

• pp. 511–528

◮ Schmid (2011): Spatial Robust Small Area Estimation applied on

Business Data, PhD thesis, University of Trier

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Bibliography

◮ Schoch (2012): Robust Unit-Level Small Area Estimation: A Fast

Algorithm for Large Datasets, Austrian Journal of Statistics 41 (4),

• pp. 243–265

◮ Sinha / Rao (2009): Robust small area estimation, The Canadian

Journal of Statistics 37 (3), 381–399

◮ Sugasawa / Kubokawa (2015): Parametric transformed Fay–Herriot

model for small area estimation, Journal of Multivariate Analysis 139, 295-311

◮ Wolter (2007): Introduction to Variance Estimation, Springer

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Mean-difference Plot

−1 1 2 1 2 (direct + reblup) / 2 direct − reblup

RFH

−1 1 2 1 2 3 (direct + eblup) / 2 direct − eblup

FH

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Quality Measures

◮ Relative Root Mean Square Error:

RRMSE m

i

=

• 1

R

R

• r=1

ˆ

θm

i,r − θi,r

θi,r

2

◮ Relative Bias:

RBIASm

i

= 1 R

R

• r=1

ˆ

θm

i,r − θi,r

θi,r

• ,

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Estimation Equations

◮ For variance parameter δl:

ψ(r)⊤U

1 2 V−1 ∂V

∂δl V−1U

1 2 ψ(r) − tr

• KV−1 ∂V

∂δl

• = 0

◮ For random effects:

Z⊤V−1

e U

1 2

e ψ

• U

− 1

2

e

(y − Xβ − Zv)

• − V−1

v U

1 2

u ψ

• U

− 1

2

u v

• = 0

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Algorithms

β(m+1) =

• X⊤V−1W1(β(m))X

−1 X⊤V−1W1(β(m))y

v(m+1) =

• Z⊤V−1

e W2(v(m))Z + V−1 v W3(v(m))

−1

× Z⊤V−1

e W2(v(m)) (y − Xβ)

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