On Outlier Robust Small Area Mean Estimate Based on Prediction of Empirical Distribution Function

Transcription

1 On Outler Robust Small Area Mean Estmate Based on Predcton of Emprcal Dstrbuton Functon Payam Mokhtaran Natonal Insttute of Appled Statstcs Research Australa Unversty of Wollongong Small Area Estmaton Conference 3-5 September 2014, Poznań, Poland Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 1 / 31

2 Outlne 1 Introducton and Motvaton 2 Outler Robust Small Area Estmaton 3 Robust Random Effect Block Bootstrap (RREB) 4 Outler Robust Small Area Mean estmate Based on EDF 5 Numercal Evaluatons 6 Concludng Remarks and Further Researches Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 2 / 31

3 Introducton Outlers n the sample data are common n many surveys Typcal model-based survey estmaton methods are extremely senstve to sample outlers Estmates of the model parameters and predctons of populaton quanttes become unstable n the presence of outlers Use of outler robust approaches to ft the model s of nterest In the proposed robust approaches n the lterate, model parameter estmaton functons are usually modfed to make them less outler senstve (M-estmaton) Rchardson and Welsh (1995) Chambers and Tzavds (2006) Snha and Rao (2009) Chambers et al. (2014) Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 3 / 31

4 Introducton Outlers n the sample data are common n many surveys Typcal model-based survey estmaton methods are extremely senstve to sample outlers Estmates of the model parameters and predctons of populaton quanttes become unstable n the presence of outlers Use of outler robust approaches to ft the model s of nterest In the proposed robust approaches n the lterate, model parameter estmaton functons are usually modfed to make them less outler senstve (M-estmaton) Rchardson and Welsh (1995) Chambers and Tzavds (2006) Snha and Rao (2009) Chambers et al. (2014) Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 3 / 31

5 Outler Robust Approaches - Chambers (1986) Robust Projectve (non-representatve) All of the populaton outlers are n a sample taken from the populaton Estmate usng robust model parameter estmates to mtgate the nfluence of the outlers on the workng model In fact, the workng model s projected onto the entre non-sampled part of the populaton Robust Predctve (representatve) Some of the populaton outlers are n a sample taken from the populaton In ths scenaro, partcularly wth asymmetrc outlers, usng robust projectve estmator results n based estmates Addtonal outler robust bas correcton term needs to be take nto account Usng sample outler nformaton, the outler effects need to be predcted to obtan more relable estmates Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 4 / 31

6 The proposed Approach We propose an outler robust bounded bootstrap method to deal wth the nfluence of outlers on small area estmates under a mxed model Our approach s an outler robust extenson of the random effect block bootstrap approach proposed by Chambers and Chandra (2013) The proposed approach s called Robust Random Effect Block Bootstrap (RREB) In terms of mxed model parameter estmates, RREB leads to more relable mxed model parameter estmates than comparable robust approaches n the lterature Based on RREB, the predcton of the emprcal dstrbuton functon to acheve an estmator for small area means s presented Two types of Mean Squared Error (MSE) estmator for the proposed RREB-based predctors are proposed Numercal results ndcate that the RREB-based estmator s stable and leads to a relable populaton quantty predctor wth a smaller MSE under asymmetrc outlers Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 5 / 31

7 The proposed Approach We propose an outler robust bounded bootstrap method to deal wth the nfluence of outlers on small area estmates under a mxed model Our approach s an outler robust extenson of the random effect block bootstrap approach proposed by Chambers and Chandra (2013) The proposed approach s called Robust Random Effect Block Bootstrap (RREB) In terms of mxed model parameter estmates, RREB leads to more relable mxed model parameter estmates than comparable robust approaches n the lterature Based on RREB, the predcton of the emprcal dstrbuton functon to acheve an estmator for small area means s presented Two types of Mean Squared Error (MSE) estmator for the proposed RREB-based predctors are proposed Numercal results ndcate that the RREB-based estmator s stable and leads to a relable populaton quantty predctor wth a smaller MSE under asymmetrc outlers Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 5 / 31

8 Outler Robust Small Area Estmaton A Revew on Lteratures Usually the area-specfc sample szes are small and so outlers n the sample data have a sgnfcant effect on nference for any partcular area Chambers and Tzavds (2006) proposed an M-quantle approach that s robust to the presence of ndvdual outlers Snha and Rao (2009) proposed an outler robust EBLUP (REBLUP) usng the robust model fttng approach of Rchardson and Welsh (1995), as well as a bootstrap MSE estmator (BOOT) Chambers et al (2014) proposed a bas-corrected verson of both the REBLUP and the M-quantle estmators. They also provded two analytcal MSE estmators (CCT, CCST) for these robust methods Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 6 / 31

9 Outler Robust Small Area Estmaton A Revew on Lteratures Usually the area-specfc sample szes are small and so outlers n the sample data have a sgnfcant effect on nference for any partcular area Chambers and Tzavds (2006) proposed an M-quantle approach that s robust to the presence of ndvdual outlers Snha and Rao (2009) proposed an outler robust EBLUP (REBLUP) usng the robust model fttng approach of Rchardson and Welsh (1995), as well as a bootstrap MSE estmator (BOOT) Chambers et al (2014) proposed a bas-corrected verson of both the REBLUP and the M-quantle estmators. They also provded two analytcal MSE estmators (CCT, CCST) for these robust methods Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 6 / 31

10 Outler Robust Small Area Estmaton A Revew on Lteratures Usually the area-specfc sample szes are small and so outlers n the sample data have a sgnfcant effect on nference for any partcular area Chambers and Tzavds (2006) proposed an M-quantle approach that s robust to the presence of ndvdual outlers Snha and Rao (2009) proposed an outler robust EBLUP (REBLUP) usng the robust model fttng approach of Rchardson and Welsh (1995), as well as a bootstrap MSE estmator (BOOT) Chambers et al (2014) proposed a bas-corrected verson of both the REBLUP and the M-quantle estmators. They also provded two analytcal MSE estmators (CCT, CCST) for these robust methods Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 6 / 31

11 Outler Robust Small Area Estmaton A Revew on Lteratures Usually the area-specfc sample szes are small and so outlers n the sample data have a sgnfcant effect on nference for any partcular area Chambers and Tzavds (2006) proposed an M-quantle approach that s robust to the presence of ndvdual outlers Snha and Rao (2009) proposed an outler robust EBLUP (REBLUP) usng the robust model fttng approach of Rchardson and Welsh (1995), as well as a bootstrap MSE estmator (BOOT) Chambers et al (2014) proposed a bas-corrected verson of both the REBLUP and the M-quantle estmators. They also provded two analytcal MSE estmators (CCT, CCST) for these robust methods Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 6 / 31

12 Outler Robust Small Area Estmaton Small area varable of nterest, covarate, and area-specfc random effect are defned under a lnear mxed model as below y j = x T j β + u + e j ; u e Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 7 / 31

13 Outler Robust Small Area Estmaton Small area varable of nterest, covarate, and area-specfc random effect are defned under a lnear mxed model as below y j = x T j β + u + e j ; u e Usng non-sample covarate nformaton (at the populaton level), the predcton of the emprcal dstrbuton functon of the area to acheve the mean estmate s ˆF = N 1 j s I (y j t) + I (ŷ j t) j r Tzavds et al. (2010) - The estmator of the area mean (EBLUP) s just the expected value functonal as below ˆȳ = td ˆF (t) = N 1 [ ] n ȳ s + (N n )ˆȳ r D Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 7 / 31

14 Outler Robust Small Area Estmaton Snha and Rao (2009) approach: Robust EBLUP (REBLUP) of the area mean s defned by ˆȳ r SR = x T r ˆβ SR + û SR The unknown mxed model parameters are estmated usng the robust method proposed by Rchardson and Welsh (1995) Chambers and Tzavds (2006) approach: M-quantle of the area mean s defned by ˆȳ r MQ = x T r ˆβ MQ The ˆβ MQ s estmated M-quantle regresson coeffcent whch s robust to nfluence of the outlers Chambers et al. (2014) approach: They proposed the bas corrected verson of the above robust approaches Here we propose a new outler robust method usng RREB based on Chambers et al. (2014) approach Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 8 / 31

15 Outler Robust Small Area Estmaton Snha and Rao (2009) approach: Robust EBLUP (REBLUP) of the area mean s defned by ˆȳ r SR = x T r ˆβ SR + û SR The unknown mxed model parameters are estmated usng the robust method proposed by Rchardson and Welsh (1995) Chambers and Tzavds (2006) approach: M-quantle of the area mean s defned by ˆȳ r MQ = x T r ˆβ MQ The ˆβ MQ s estmated M-quantle regresson coeffcent whch s robust to nfluence of the outlers Chambers et al. (2014) approach: They proposed the bas corrected verson of the above robust approaches Here we propose a new outler robust method usng RREB based on Chambers et al. (2014) approach Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 8 / 31

16 Outler Robust Small Area Estmaton Snha and Rao (2009) approach: Robust EBLUP (REBLUP) of the area mean s defned by ˆȳ r SR = x T r ˆβ SR + û SR The unknown mxed model parameters are estmated usng the robust method proposed by Rchardson and Welsh (1995) Chambers and Tzavds (2006) approach: M-quantle of the area mean s defned by ˆȳ r MQ = x T r ˆβ MQ The ˆβ MQ s estmated M-quantle regresson coeffcent whch s robust to nfluence of the outlers Chambers et al. (2014) approach: They proposed the bas corrected verson of the above robust approaches Here we propose a new outler robust method usng RREB based on Chambers et al. (2014) approach Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 8 / 31

17 Outler Robust Small Area Estmaton Snha and Rao (2009) approach: Robust EBLUP (REBLUP) of the area mean s defned by ˆȳ r SR = x T r ˆβ SR + û SR The unknown mxed model parameters are estmated usng the robust method proposed by Rchardson and Welsh (1995) Chambers and Tzavds (2006) approach: M-quantle of the area mean s defned by ˆȳ r MQ = x T r ˆβ MQ The ˆβ MQ s estmated M-quantle regresson coeffcent whch s robust to nfluence of the outlers Chambers et al. (2014) approach: They proposed the bas corrected verson of the above robust approaches Here we propose a new outler robust method usng RREB based on Chambers et al. (2014) approach Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 8 / 31

18 Robust Random Effect Block Bootstrap (RREB) Chambers and Chandra (2013) developed a procedure to ft a lnear mxed model usng a random effect block bootstrap (REB) We propose an outler robust extenson of the REB dea that can be used to ft a lnear mxed model n the presence of outlers Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 9 / 31

19 RREB Procedure Gven the herarchcal structure of the lnear mxed model we can calculate 1. Margnal resduals r j = y j x T j ˆβ; (r = y X ˆβ) Area average resduals r. = n 1 n r j ; r (2) = { r. } j=1 Zero-centred and standardsed area average resduals r (2) SC = r (2) C = r (2) av( r (2) )1 D [ D 1 (r (2) C ) T r (2) C ] 1/2 r (2) C ˆσ u Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 10 / 31

20 RREB Procedure Outler robust area-specfc (level 2) resduals r (2)R = ψ c2 (r (2) SC ); c 2 = 2ˆσ u Zero-centred and standardsed ndvdual level resduals r (1) C = (r r (2)R 1 nd 1) av((r r (2)R 1 nd 1))1 n r (1) SC = Outler robust ndvdual level (level 1) resduals [ n 1 (r (1) C ) T r (1) C ] 1/2 r (1) C ˆσ e r (1)R = (r (1)R ) = ψ c1 (r (1) SC ); c 1 = 2ˆσ e Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 11 / 31

21 RREB Procedure 2. Bootstrap errors defned by samplng wth replacement from each set of robust resduals (ndependently at level 2, block samplng at level 1) r (1)R r (2)R = (r (2)R ) = srswr(r (2)R, D) = (r (1)R j ) = srswr(r (1)R j=srswr({1,,d},1), n ) r (1)R = r (1)R 3. Robust bootstrap sample data (yj R, x j ) are generated va y R j = x T j ˆβ + r (2)R + r (1)R j Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 12 / 31

22 RREB Procedure 4. A mxed model s ftted to these bootstrap sample data to obtan bootstrap model parameter estmates ˆφ R = { ˆβ R, ˆσ u 2 R, ˆσ e 2 R } 5. Repeat step 2-4 to obtan B sets of bootstrap model parameter estmates RREB leads to less based model parameter estmates n the contamnated case A theorem presented that shows RREB model parameter estmates are consstent under the NULL model - s not presentng Note that RREB varance components estmates are stll sgnfcantly based - but ths bas s much smaller than that of RREML An adaptve algorthm proposed whch reduces ths bas of the RREB varance components estmates - s not presentng Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 13 / 31

23 RREB Procedure 4. A mxed model s ftted to these bootstrap sample data to obtan bootstrap model parameter estmates ˆφ R = { ˆβ R, ˆσ u 2 R, ˆσ e 2 R } 5. Repeat step 2-4 to obtan B sets of bootstrap model parameter estmates RREB leads to less based model parameter estmates n the contamnated case A theorem presented that shows RREB model parameter estmates are consstent under the NULL model - s not presentng Note that RREB varance components estmates are stll sgnfcantly based - but ths bas s much smaller than that of RREML An adaptve algorthm proposed whch reduces ths bas of the RREB varance components estmates - s not presentng Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 13 / 31

24 RREB Small Area Estmator Gven RREB mxed model parameter estmates, the RREB estmator of the area mean s as below [ ] ˆȳ RREB = N 1 n ȳ s + (N n )ˆȳ r RREB, ˆȳ RREB r = x T r ˆβ RREB + û RREB The area random effect needs to be predcted outsde of the bootstrap process, so { } { û RREB = ˆσ e 2(b)RREB + ˆσ u 2(b)RREB 2(b)RREB ) 1ˆσ u ȳ s x T s ˆβ RREB} B B 1 (n 1 b=1 Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 14 / 31

25 Bas-corrected RREB Small Area Estmator Chambers et al. (2014): Based on Welsh and Ronchett (1998), the bas-corrected verson of the robust predcton of the emprcal dstrbuton functon of the area s ˆF BC = N 1 I (y j t) + ((ŷ ψ k + ϑφ j (r j)) t) ; ψ < φ, j s j s k r ϑ φ j (r j) = ω ψ j φ {(y j x T j ˆβ ψ û ψ )/ω ψ j We use ŷk RREB and ϑ RREB j n the above functon to obtan the bas-corrected verson of the RREB predcton of the emprcal dstrbuton functon of the area There are two RREB bas correctons; external and nternal } Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 15 / 31

26 Bas-corrected RREB Small Area Estmator External RREB Bas-Correcton The predcted emprcal dstrbuton functon of the external bas-corrected RREB s ˆF RREB XBC = N 1 I (y j t) + j s j s k + ϑ (φ)rreb j (r j )) t), ϑ (φ)rreb j { (r j ) = ωj RREB φ (y j x T j k r ((ŷ RREB ˆβ RREB û RREB } )/ωj RREB, ω RREB j s a RREB estmator of the scale of the resduals Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 16 / 31

27 Bas-corrected RREB Small Area Estmator External RREB Bas-Correcton The predcted emprcal dstrbuton functon of the external bas-corrected RREB s ˆF RREB XBC = N 1 I (y j t) + j s j s k + ϑ (φ)rreb j (r j )) t), ω RREB j ˆȳ RREB XBC ϑ (φ)rreb j { (r j ) = ωj RREB φ (y j x T j k r ((ŷ RREB s a RREB estmator of the scale of the resduals = ˆȳ RREB ˆȳ RREB XBC = D + (1 n N 1 )n 1 ˆβ RREB û RREB td ˆF RREB XBC (t), ωj RREB j s { φ (y j x T j } )/ωj RREB, ˆβ RREB û RREB } )/ωj RREB Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 16 / 31

28 Bas-corrected RREB Small Area Estmator Internal RREB Bas-Correcton ˆȳ RREB IBC = N 1 { n ȳ s + (N n )ˆȳ RREB BC INT }, ˆȳ RREB BC INT = B 1 B [ x T r ˆβ (b)rreb + n 1 b=1 γ (b) ] + û RREB Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 17 / 31

29 Bas-corrected RREB Small Area Estmator Internal RREB Bas-Correcton ˆȳ RREB IBC = N 1 { n ȳ s + (N n )ˆȳ RREB BC INT }, ˆȳ RREB BC INT = B 1 B [ x T r ˆβ (b)rreb + n 1 b=1 γ (b) ] + û RREB γ (b) s the bas correcton term of area n bootstrap replcaton b. Ths term s gven by γ (b) = { ω (b)rreb j φ (y (b)lb j x T j j s ˆβ (b)rreb û (b)rreb } )/ω (b)rreb j Here φ assumes as an dentty functon, so the bas-correcton term leads to γ (b) = y (b)lb j x T j ˆβ (b)rreb û (b)rreb Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 17 / 31

30 RREB-IBC Small Area Estmator y (b)lb s less bounded (LB) values for RREB bootstrap samples based on a less restrctve nfluence functon ( ϕ > ψ) LB-RREB procedure: Gven the mxed model margnal resduals (r = y X ˆβ) Smlarly to RREB procedure, the zero-centred and standardsed area average resduals s r (2) SC = [ D 1 (r (2) C ) T r (2) C ] 1/2 r (2) C ˆσ u Then, LB outler robust area-specfc (level 2) resduals s r (2)LB = ϕ h2 (r (2) SC ); h 2ˆσ u, (h 2 c 2 ) The zero-centred and standardsed ndvdual level resduals s [ ] r (1) SC = n 1 (r (1) C ) T r (1) C 1/2 r (1) C ˆσ e Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 18 / 31

31 RREB-IBC Small Area Estmator LB outler robust ndvdual level (level 1) resduals s r (1)LB = (r (1)LB ) = ϕ h1 (r (1) SC ); h 1ˆσ e, (h 1 c 1 ) 2. Bootstrap errors defned by samplng wth replacement from each set of robust resduals (ndependently at level 2, block samplng at level 1) r (1)LB r (2)LB = (r (2)LB ) = srswr(r (2)LB, D) = (r (1)LB j ) = srswr(r (1)LB j=srswr({1,,d},1), n ), r (1)LB = r (1)LB 3. LB robust bootstrap sample data (yj LB, x j ) are generated va y LB j = x T j ˆβ + r (2)LB + r (1)LB j Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 19 / 31

32 RREB-IBC Small Area Estmator Therefore, the nternal bas-corrected RREB mean estmate for area s gven by ˆȳ RREB BC INT = B 1 B [ x T r ˆβ(b)RREB + n 1 b=1 γ (b) ] + û RREB where γ (b) = { (y (b)lb j x T j j s } ˆβ (b)rreb û (b)rreb ) Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 20 / 31

33 RREB-based MSE Estmaton We propose two approaches to estmatng the MSE of the RREB-based predctor of the populaton mean Usng the Prasad and Rao (1990) method of MSE estmaton Usng the observed varablty n the RREB bootstrap replcatons Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 21 / 31

34 Bootstrap-based Prasad-Rao type MSE Estmator We use bootstrap estmates of the varances and covarance of the RREB estmates of the varance components V RREB u = B 1 B b=1 (ˆσ 2(b)RREB u ˆσ 2RREB u ) 2 C RREB ue V RREB e = B 1 B b=1 = B 1 Leadng to ˆτ RREB = (ˆθ RREB, V RREB u B b=1 (ˆσ 2(b)RREB u, V RREB e (ˆσ 2(b)RREB e ˆσ 2RREB e ) 2 ˆσ u 2RREB )(ˆσ e 2(b)RREB ˆσ e 2RREB ), Cue RREB ), then MSE RREB PR (ˆȳ RREB ) = g 1 (ˆθ RREB ) + g 2 (ˆθ RREB ) + 2g 3 (ˆτ RREB ) Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 22 / 31

35 Bootstrap MSE Estmator Ths MSE estmator uses the observed bootstrap varablty of ˆȳ RREB, and s gven by MSE RREB (ˆȳ RREB ) = B 1 B (ˆȳ (b)rreb b=1 ˆȳ RREB ) 2 Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 23 / 31

36 Model-based Smulatons Set-up Assumed mxed model (Chambers et al., 2014): y j = x j + u + e j, X d LN(1, 0.5), u e D = 40, N = 100, n = 5, Table: Outler scenaros Scenaro Area effect dstrbuton Indvdual effect dstrbuton [0,0]: no outler u N(0, 3) e N(0, 6) [0,e]: ndvdual outlers only u N(0, 3) e δn(0, 6) + (1 δ)n(20, 150) [u,0]: area outlers only u (1 : 36)N(0, 3) + (37 : 40)N(9, 20) e N(0, 6) [u,e]: both outlers u (1 : 36)N(0, 3) + (37 : 40)N(9, 20) e δn(0, 6) + (1 δ)n(20, 150) δ Bernoull; Pr(δ = 1) = 0.97 Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 24 / 31

37 Smulated realsatons of the populaton based on the outler scenaros [0,e] [u,0] [u,e] Populaton Data Set Colored Groups Populaton Data Set Colored Groups Populaton Data Set Colored Groups Varable of Interest Covarate Sample Data Set Colored Groups Varable of Interest Varable of Interest Covarate Sample Data Set Colored Groups Varable of Interest Varable of Interest Covarate Sample Data Set Colored Groups Varable of Interest Covarate Covarate Covarate Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 25 / 31

38 Medan relatve bas (%) - RREB vs REML over smulatons σ 2 u σ 2 e Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 26 / 31

39 Numercal Evaluatons Table: Model-based smulaton results for predctors of small area means Results (%) for the scenaros and areas Predctor [0,0] 1:40 [0,e] 1:40 [u,0] 1:36 [u,0] 37:40 [u,e] 1:36 [u,e] 37:40 Medan values of RB EBLUP REBLUP RREB REBLUP-BC RREB-XBC RREB-IBC Medan values of RRMSE EBLUP REBLUP RREB REBLUP-BC RREB-XBC RREB-IBC Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 27 / 31

40 Table: Model-based smulaton results for performance of MSE estmators - RB Results (%) for the scenaros and areas Predctor Estmator [0,0] 1:40 [0,e] 1:40 [u,0] 1:36 [u,0] 37:40 [u,e] 1:36 [u,e] 37:40 Medan values of RB EBLUP PR CCT CCST REBLUP CCT CCST BOOT RREB PR RREB REBLUP-BC CCST BOOT RREB-IBC RREB Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 28 / 31

41 Table: Model-based smulaton results for performance of MSE estmators - RRMSE Results (%) for the scenaros and areas Predctor Estmator [0,0] 1:40 [0,e] 1:40 [u,0] 1:36 [u,0] 37:40 [u,e] 1:36 [u,e] 37:40 Medan values of RRMSE EBLUP PR CCT CCST REBLUP CCT CCST BOOT RREB PR RREB REBLUP-BC CCST BOOT RREB-IBC RREB Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 29 / 31

42 Conclusons Based on proposed RREB, both outler robust projectve and robust predctve small area estmators presented The nternal bas-corrected RREB developed and t works well n terms of both estmaton and performance partcularly under asymmetrc outler contamnaton scenaro RREB-based small area estmator s less based and more stable than comparable proposed estmators n the lterature Although RREB s a bootstrap-based approach (teratve) t s easy to mplement and qute computatonally effcent Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 30 / 31

43 Conclusons Based on proposed RREB, both outler robust projectve and robust predctve small area estmators presented The nternal bas-corrected RREB developed and t works well n terms of both estmaton and performance partcularly under asymmetrc outler contamnaton scenaro RREB-based small area estmator s less based and more stable than comparable proposed estmators n the lterature Although RREB s a bootstrap-based approach (teratve) t s easy to mplement and qute computatonally effcent Investgaton and studyng on other populaton quanty estmates e.g., quantles, Gn coeffcent based on the outler robust predcton for the emprcal dstrbuton functon of small area s an nterest Extendng RREB to deal wth count and bnary data under a generalsed lnear mxed model GLMM s a topc for further research Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 30 / 31

44 Acknowledgement Prof. Ray Chambers Centre for Statstcal and Survey Methodology Natonal Insttute for Appled Statstcs Research Australa (NIASRA) Unversty of Wollongong Payam Mokhtaran (UOW) Outler Robust Small Area Estmaton SAE 2014 Poznań 31 / 31