Full bias-correction of spatial robust small area estimators - PowerPoint PPT Presentation

Introduction Estimation methods Simulation study Summary and Outlook Full bias-correction of spatial robust small area estimators Session: SAE Using Time Series or Spatial Models SAE 2013, Bangkok Timo Schmid Monday, September 2, 2013 Timo Schmid 1 (26) Full bias-correction of spatial robust small area estimators

Introduction Estimation methods Simulation study Summary and Outlook Contents Introduction Estimation methods Simulation study Summary and Outlook Timo Schmid 2 (26) Full bias-correction of spatial robust small area estimators

Introduction Estimation methods Simulation study Summary and Outlook Motivation ◮ Classical small area models are based on strong distributional assumptions, which are often not fulfilled in the case of business data. ◮ Especially in business surveys, outliers and skewed distributions are very common in the sample data. ◮ Skewed distributions and outliers are violating the strong assumptions of small area models. ◮ These phenomena have great impact on the estimators and lead to a substantial bias especially within small sample sizes. ◮ Beyond that, spatial dependencies occur very often in business data (e.g. similar industry segments). ◮ Thus, there is a need to investigate spatial outlier robust small area models. Timo Schmid 3 (26) Full bias-correction of spatial robust small area estimators

Introduction Estimation methods Simulation study Summary and Outlook Robust ”Plug-In” methods Assumption: All non-sampled values are not outliers ◮ The sample includes all outliers of the population. ◮ Chambers et. al. (2013) called this approach projective because they project the working model onto the whole non-sampled part of the population. ◮ Examples: ◮ Robust EBLUP (Sinha and Rao, 2009) ◮ M-Quantile methods (Chambers and Tzavidis, 2006) ◮ Spatial robust EBLUP (Schmid and M¨ unnich, 2013) ◮ These ”Plug-In” methods may suffer from a bias in situations with representative outliers or non-symmetric contamination. References: Chambers (1986) and Chambers et al. (2013) Timo Schmid 4 (26) Full bias-correction of spatial robust small area estimators

Introduction Estimation methods Simulation study Summary and Outlook Bias-corrected robust methods Assumption: Some non-sampled units are outliers ◮ The sample includes only some of the outliers in the population. ◮ Chambers et. al. (2013) called this approach predictive because they use the sample outlier information to predict contamination on the target variable. ◮ Define a robust bias correction to the robust ”Plug-In” estimators. Two concepts: Locally vs. fully bias corrections. ◮ Examples for the REBLUP: ◮ Locally: CCST (Chambers et al., 2013) ◮ Fully: CHAM (Dongmo-Jiongo et al., 2013) ◮ Fully: CB (Dongmo-Jiongo et al., 2013) References: Chambers (1986) and Chambers et al. (2013) Timo Schmid 5 (26) Full bias-correction of spatial robust small area estimators

Introduction Estimation methods Simulation study Summary and Outlook Bias-corrected robust methods Assumption: Some non-sampled units are outliers ◮ The sample includes only some of the outliers in the population. ◮ Chambers et. al. (2013) called this approach predictive because they use the sample outlier information to predict contamination on the target variable. ◮ Define a robust bias correction to the robust ”Plug-In” estimators. Two concepts: Locally vs. fully bias corrections. ◮ Concepts for the SREBLUP: ◮ Locally: SCCST ◮ Fully: SCHAM ◮ Fully: SCB References: Chambers (1986) and Chambers et al. (2013) Timo Schmid 5 (26) Full bias-correction of spatial robust small area estimators

Introduction Estimation methods Simulation study Summary and Outlook Contents Introduction Estimation methods Simulation study Summary and Outlook Timo Schmid 6 (26) Full bias-correction of spatial robust small area estimators

Introduction Estimation methods Simulation study Summary and Outlook Basic model and notations General linear mixed model: y = X β + Zv + e ◮ Individual level covariates X and area level covariates Z ◮ SAR model: Area random effect v ∼ N (0 , G ) with � − 1 G = σ 2 ( I − ρ W )( I − ρ W T ) � v ◮ W describes the neighbourhood structure of the areas i ◮ ρ ∈ [ − 1 , 1] defines the strength of the spatial relationship among the areas ◮ Error term e ∼ N (0 , R ) = N (0 , diag ( σ 2 e )) ◮ Variable of interest is y ∼ N ( X β , V ) with V θ = R + ZGZ T ◮ Target variable is y i Timo Schmid 7 (26) Full bias-correction of spatial robust small area estimators

Introduction Estimation methods Simulation study Summary and Outlook The benchmark: EBLUP Empirical best linear unbiased predictor (EBLUP) for y i is: � � � � � � � � ij ˆ ˆ y i = N − 1 = N − 1 ( x T β + z T y ij + y ij ˆ y ij + ij ˆ v i ) i i j ∈ s i j ∈ r i j ∈ s i j ∈ r i where ˆ ( X T V − 1 θ X ) − 1 ( X T V − 1 β = θ y ) GZ T V − 1 θ ( y − X ˆ ˆ = β ) v ˆ θ is e.g. the REML- or ML-Estimator of the variance component. Reference: Rao (2003). Timo Schmid 8 (26) Full bias-correction of spatial robust small area estimators

Introduction Estimation methods Simulation study Summary and Outlook Robust EBLUP ψ and ˆ Basic idea : Substitute ˆ v with robust estimators ˆ v ψ , β and ˆ β y ψ leading to a robust estimator ˆ ij . Robustified ML-Equations: 1 X T V − 1 U 2 ψ ( r ) = 0 α ( β ) = 2 V − 1 ∂ V 2 ψ ( r ) − tr ( V − 1 ∂ V 1 1 ψ T ( r ) U V − 1 U Φ( θ l ) = K ) = 0 ∂θ l ∂θ l ◮ ψ is an influence function ◮ r = U − 1 2 ( y − X β ) and U = diag ( V ) REBLUP for y i is: ψ + z T � � � � � � y i = N − 1 ˆ � y ψ = N − 1 � ( x T ij ˆ v ψ y ij + ˆ y ij + ij ˆ i ) β i ij i j ∈ s i j ∈ r i j ∈ s i j ∈ r i Reference: Fellner (1986), Sinha and Rao (2009). Timo Schmid 9 (26) Full bias-correction of spatial robust small area estimators

Introduction Estimation methods Simulation study Summary and Outlook Spatial REBLUP ψ, sp and Basic idea : Substitute ˆ v with spatial robust estimators ˆ β and ˆ β v ψ, sp , leading to a spatial robust estimator ˆ y ψ, sp ˆ . ij Robustified spatial ML-equations: X T V − 1 U 1 / 2 ψ ( r ) = 0 α ( β ) = − tr ( V − 1 ∂ V K ) + U 1 / 2 ψ ( r ) V − 1 ∂ V V − 1 U 1 / 2 ψ ( r ) T = 0 Φ( θ l ) = ∂θ l ∂θ l − tr ( V − 1 ∂ V ∂ρ K ) + U 1 / 2 ψ ( r ) V − 1 ∂ V ∂ρ V − 1 U 1 / 2 ψ ( r ) T = 0 Ω( ρ ) = Spatial REBLUP for y i is: y i = 1 = 1 ψ, sp + z T � � � � � � � y ψ, sp � ij ˆ v ψ, sp ˆ ( x T y ij + ˆ y ij + β ij ˆ ) ij i N i N i j ∈ s i j ∈ r i j ∈ s i j ∈ r i Reference: Schmid and M¨ unnich (2013). Timo Schmid 10 (26) Full bias-correction of spatial robust small area estimators

Introduction Estimation methods Simulation study Summary and Outlook Bias-corrections for the REBLUP Locally : Chambers et. al. (2013) used an approach similar to the one of Welsh and Ronchetti (1998) for a robust prediction of the empirical distribution function of y leading to + (1 − n i ) 1 CCST REBLUP � � � ω ψ y ψ ij ) /ω ψ ˆ = ˆ y y ij ψ c ( y ij − ˆ i i ij N i n i j ∈ s i Fully : Dongmo-Jiongo et al. (2013) used ideas similar to Chambers (1986) and the fact that the EBLUP can be written as a weighted linear function of the sample leading to CHAM REBLUP � y ψ ˆ ˆ + N − 1 � � y = y ψ k 1 ( w j − 1)( y ij − ˆ ij ) i i i j ∈ s i m m + N − 1 � � y ψ + N − 1 � v ψ � � � � ψ k 1 w j ( y hj − ˆ hj ) ψ k 2 ̟ h ˆ i i h j ∈ s h h =1 h � = i h =1 Reference: Chambers et al. (2013) and Dongmo-Jiongo et al. (2013) Timo Schmid 11 (26) Full bias-correction of spatial robust small area estimators

Introduction Estimation methods Simulation study Summary and Outlook Bias-corrections for the SREBLUP Locally : The approach of Chambers et. al. (2013) can be extended to the case of spatial correlation leading to + (1 − n i ) 1 SCCST SREBLUP � � ˆ = ˆ � ω ψ, sp y ψ, sp ) /ω ψ, sp ψ c ( y ij − ˆ y y i i ij ij ij N i n i j ∈ s i Fully : The ideas of Dongmo-Jiongo et al. (2013) can be applied for the SREBLUP leading to SCHAM SREBLUP ˆ ˆ + N − 1 � ( w ψ, sp y ψ, sp � � = ψ k 1 − 1)( y ij − ˆ ) y y i i i j ij j ∈ s i m m + 1 + 1 � � w ψ, sp y ψ, sp � ̟ ψ, sp v ψ, sp � � � � ψ k 1 ( y hj − ˆ ) ψ k 2 ˆ j hj h h N i N i h =1 h � = i j ∈ s h h =1 Reference: Schmid et al. (2013) Timo Schmid 12 (26) Full bias-correction of spatial robust small area estimators

Introduction Estimation methods Simulation study Summary and Outlook Ideas behind the SCHAM estimator Basic idea: The spatial EBLUP of Petrucci et al. (2005) can be written as a weighted linear function of the sample leading to SEBLUP ˆ = N − 1 � w sp j y j . y i i j ∈ s Calculation leads to: SEBLUP SREBLUP ψ, sp − ˆ ˆ ˆ + N − 1 � w sp j ˆ v ψ, sp � �� y j − x T � = − 1 y y β i i j i j ∈ s i m m ψ, sp − ˆ + N − 1 � � w sp j ˆ v ψ, sp + N − 1 � v ψ, sp � y j − x T � ̟ h ˆ . β j h h h =1 j ∈ s h h � = i h =1 Reference: Schmid et al. (2013) Timo Schmid 13 (26) Full bias-correction of spatial robust small area estimators