Small Area Interval Estimation

Transcription

1 .. Small Area Interval Estmaton Partha Lahr Jont Program n Survey Methodology Unversty of Maryland, College Park (Based on jont work wth Masayo Yoshmor, Former JPSM Vstng PhD Student and Research Fellow of JSPS, Graduate School of Engneerng Scence, Osaka Unversty) June 12, 2013 () Census Bureau June 12, / 20

2 The Fay Herrot Bayesan Model Ref: Fay and Herrot (1979) For = 1,, m, where Level 1: (Samplng Dstrbuton): y θ N(θ, D ); Level 2: (Pror Dstrbuton): θ N(x β, A) m : number of small area; y : drect survey estmate of θ ; θ : true mean for area ; x : p 1 vector of known auxlary varables; D : known samplng varance of the drect estmate; The p 1 vector of regresson coeffcents β and model varance A are unknown. () Census Bureau June 12, / 20

3 Emprcal Bayes (EB) Estmator of θ Let  be a consstent estmator of A, for large m. An EB of θ s gven by where ˆB = D Â+D ˆβ = ˆβ(Â) ˆθ EB = (1 ˆB )y + ˆB x ˆβ. () Census Bureau June 12, / 20

4 Confdence Interval for θ An nterval, denoted by I, s called a 100(1 α)% nterval for θ f P(θ I β, A) = 1 α, β R p, A R +, where the probablty P s wth respect to the jont dstrbuton of {(y, θ ), = 1,, m} under the Fay-Herrot model; R + s the postve part of the real lne. () Census Bureau June 12, / 20

5 A General Form of Confdence Interval for θ Most of the ntervals proposed n the lterature can be wrtten as: where ˆθ s an estmator of θ ; (ˆθ + q 1 (α)ˆτ (ˆθ ), ˆθ + q 2 (α)ˆτ (ˆθ )) ˆτ (ˆθ ) s an estmate of the measure of uncertanty of ˆθ ; q 1 (α) and q 2 (α) are chosen sutably n an effort to attan coverage probablty close to the nomnal level 1 α. () Census Bureau June 12, / 20

6 Drect Confdence Interval The choce ˆθ = y leads to the drect nterval I D gven by I D : y ± z α/2 D, where z α/2 s the upper 100(1 α/2)% pont of N(0, 1). Remarks: The coverage probablty s 1 α; When D s large, the length s too large to make any reasonable concluson. () Census Bureau June 12, / 20

7 Synthetc Confdence Interval Ref: Hall and Mat (JRSS, 2006) where (x T ˆβ ( ) + q 1 (α) Â( ), x T ˆβ ( ) + q 2 (α) Â( ) ) ˆβ ( ) and  ( ) are consstent estmators of β and A, respectvely, based on all but the th area data. L [q 2(α)] L [q 1(α)] = 1 α where L s a parametrc bootstrap approxmaton of the dstrbuton L of θ x ˆβ ( )  ( ). Remarks: The coverage s 1 α + O(m 1.5 ). The method s synthetc (Rao 2005). Ths approach could be useful n stuatons especally when y s mssng for the th area. () Census Bureau June 12, / 20

8 Bayesan Credble Interval Assume β and A are known. where ˆθ B ˆθ B (A) = (1 B )y + B x β, B B (A) = σ (A) = AD A+D D D +A, I B (A) : ˆθ B (A) ± z α/2 σ (A), Remarks: θ y ; β, A N[ˆθ B (A), g 1 = σ 2 (A)]. The Bayesan credble nterval cuts down the length of the drect confdence nterval by 100 (1 1 B )% The maxmum beneft from the Bayesan methodology s acheved when B s large, () Census Bureau June 12, / 20

9 Herarchcal Bayesan Credble Interval A revew paper: Morrs and Tang (2012) Assume pror on β and A, e.g. π(β, A) 1, β R p, A R +. Obtan the posteror dstrbuton of θ y = (y 1,, y m ) and use ths posteror to construct credble nterval for θ. Computaton: MCMC Numercal Integraton; Laplace approxmaton. ADM approxmaton () Census Bureau June 12, / 20

10 Emprcal Bayes Confdence Interval Ref: Cox (1975) where I Cox (Â) : ˆθ EB (Â) ± z α/2 σ(â), x T β = µ s estmated by the sample mean ȳ = m 1 m =1 y and A by the ANOVA estmator:  ANOVA = max { (m 1) 1 m =1 (y ȳ) 2 D, 0 }. Remarks: Lke the Bayesan credble nterval, the length of the Cox nterval s smaller than that of the drect nterval. The dstrbuton of θ ˆθ EB s not a standard Normal. Thus, t s not g 1 (Â) approprate to use the Normal quantle z α/2 as the cut-off ponts. The Cox emprcal Bayes confdence nterval ntroduces a coverage error of the order O(m 1 ), not accurate enough n most small area applcatons. length of the nterval s zero when  ANOVA = 0 () Census Bureau June 12, / 20

11 Other EB Confdence Intervals Replace z α/2 by z α /2 to reduce coverage error (Cox 1975). Replace σ(â) by a measure of uncertanty that captures uncertanty due to estmaton of the hyperparameters β and A (e.g., g 1 + g 2 + 2g 3 ) (Ref: Morrs (1983) Prasad and Rao (1990)) Replace z α/2 by z α/2 c (Â) to reduce the coverage error to O(m 1.5 ) (Datta et al. 2002; Basu et al. 2003; Yoshmor 2013) Parametrc bootstrap (Lard and Lous 1987; Carln and Lous 1996; Chatterjee et al. 2008) () Census Bureau June 12, / 20

12 Parametrc Bootstrap Confdence Interval Ref: Chatterjee, Lahr and L (AS, 2008) Use the dstrbuton of θ ˆθ EB to approxmate the dstrbuton of θ ˆθ EB. σ ( ) σ (Â) Compute ˆβ and Â; Draw bootstrap sample from the followng bootstrap model: () y nd N(θ, D ) ()θ θ nd N(x ˆβ, Â) Compute ˆβ and  from y. Then we have ˆθ EB and σ 2( ) = A D A +D ; Compute (θ ˆθ EB )/σ ( ). = (1 ˆB )y + ˆB x ˆβ, Remarks: Need strctly postve estmate of A (L and Lahr, JMVA 2010) In smulatons, the parametrc bootstrap based on EB performs better than the parametrc bootstrap Hall-Mat method based on synthetc estmator n terms of length (Yoshmor, 2013). () Census Bureau June 12, / 20

13 . Parametrc Bootstrap Confdence Interval.. CI PB = [ˆθ EB + q 1 σ (Â), ˆθ EB. Theorem. Ụnder reg. cond. Pr(θ CI PB ) = 1 α + O(m 3/2 ), + q 2 σ (Â)]. () Census Bureau June 12, / 20

14 A Research Queston Can we fnd an emprcal Bayes confdence nterval of θ that has the followng propertes? Desred propertes: coverage error of order O(m 1.5 ); length smaller than that of the drect method; does not rely on smulaton-based heavy computatons. () Census Bureau June 12, / 20

15 A Generalzaton of Cox EB Confdence Interval where  h I Cox EB (Âh ) : ˆθ (Âh ) ± z α/2 σ (Âh ), s obtaned by maxmzng the followng adjusted resdual lkelhood: L ;ad (A) h (A) L RE (A), wth respect to A over (0, ); h (A) s a general area specfc adjustment factor; L RE (A) s the standard resdual lkelhood functon. () Census Bureau June 12, / 20

16 A Hgher-Order Expanson of Coverage We obtan the followng expanson under certan regularty condtons: where P(θ I Cox (Â h )) = 1 α + zϕ(z) a + b (h (A)) + O(m 1.5 ), m a = m tr(v 2 ) [ 4D A(A + D ) 2 + b = 2m D tr(v 2 ) A(A + D ) log(h(a)) A β = ˆβ(A) = (X V 1 X ) 1 X V 1 y (1 + z 2 )D 2 2A 2 (A + D ) 2 ] md A(A + D ) x Var( β)x () Census Bureau June 12, / 20

17 A Second-order Effcent Emprcal Bayes Confdence Interval: Choce of h (A) For small area, we suggest an adjusted REML estmator of A where the adjustment factor satsfes the followng dfferental equaton: a + b (h (A)) = 0. Let  denote the soluton to the above. Then our proposed emprcal Bayes confdence nterval for θ s gven by I YL ( ) : ˆθ EB ( ) ± z α/2 σ ( ). Snce σ ( ) < D, the length of ths nterval, lke the orgnal Cox nterval I Cox ( ANOVA ), s always less than that of the drect nterval I D. () Census Bureau June 12, / 20

18 Choce of h (A) when OLS of β s used h (A) = A (1+z2)/4 (A + D ) (7 z2)/4 exp[ tr(v 1 )x (X X ) 1 X VX (X X ) 1 x /2] [ m =1 (A + D )] x (X X ) 1 x /2 C. where C s a generc constant free of A. For the balanced case D = D ( = 1,, m) h (A) = A (1+z2 )/4 (A + D) (7 z2 )/4+mx (X X ) 1 x /2 C. where C s a generc constant and free from A. In ths balanced case, we show the 4+p unqueness of the soluton  f m > 1 x (X X ) 1 x. () Census Bureau June 12, / 20

19 Smulaton Results: The Fay-Herrot Model wth x T β = 0 and Unequal Samplng Varances Table : Coverage Probablty and Average Length Pattern G Cox.RE CLL.LL Cox.YL Drect a (2.4) 94.5 (2.7) 95.3 (2.8) 95.1 (3.3) (2.3) 94.5 (2.5) 95.3 (2.6) 94.9 (3.0) (2.1) 94.6 (2.4) 95.2 (2.4) 95.2 (2.8) (2.0) 94.9 (2.2) 95.2 (2.2) 95.1 (2.5) (1.8) 94.3 (1.9) 95.0 (2.0) 94.7 (2.1) b (3.3) 94.5 (4.0) 95.8 (4.3) 94.9 (7.8) (2.3) 94.5 (2.5) 95.1 (2.6) 95.0 (3.0) (2.1) 94.6 (2.4) 95.3 (2.5) 94.9 (2.8) (2.0) 94.7 (2.2) 95.3 (2.2) 95.1 (2.5) (1.1) 94.7 (1.2) 95.0 (1.2) 95.0 (1.2) () Census Bureau June 12, / 20

20 Smulaton Results: Samplng Varances and Covarate from the 1999 SAIPE data Table : Coverage Probablty and Average length State Ds leverage Cox.RE CLL.LL Cox.YL Drect DC (4.2) 92.7 (14.5) 95.3 (20.3) 94.6 (20.8) DE (4.0) 98.6 (9.4) 98.4 (11.0) 95.1 (17.1) MS (4.0) 98.2 (9.4) 97.4 (11.1) 94.3 (16.6) LA (3.9) 97.8 (9.5) 97.5 (11.1) 95.8 (16.3) ME (3.9) 97.6 (9.5) 97.4 (11.1) 94.9 (15.8) MT (3.9) 97.5 (9.2) 97.4 (10.7) 94.2 (15.5) NM (3.9) 98.5 (9.0) 98.3 (10.4) 95.5 (15.0) MO (3.9) 98.2 (9.0) 99.0 (10.4) 96.3 (14.9) WV (3.8) 98.1 (8.9) 97.4 (10.4) 94.8 (14.8) RI (3.8) 98.3 (8.9) 97.7 (10.4) 94.3 (14.7) OR (3.8) 97.2 (8.9) 97.3 (10.3) 93.8 (14.5) ND (3.8) 96.6 (9.2) 96.4 (10.7) 95.7 (14.1) VT (3.8) 96.4 (9.2) 96.1 (10.7) 94.6 (14.1) SC (3.8) 98.0 (8.7) 98.0 (10.1) 95.2 (14.1) ID (3.8) 97.7 (8.8) 97.2 (10.2) 94.4 (14.0) AL (3.8) 98.3 (8.6) 98.1 (10.0) 95.6 (13.7) KS (3.8) 97.6 (8.7) 97.3 (10.1) 94.6 (13.6) GA (3.7) 97.3 (8.7) 97.1 (10.0) 95.1 (13.4) () Census Bureau June 12, / 20